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ed Cog ES 
 
 OF 
 
 APPROXIMATE COMPUTATIONS 
 
 BY 
 JOSEPH J. SKINNER, C.E. 
 
 INSTRUCTOR IN MATHEMATICS IN THE 
 SHEFFIELD SCIENTIFIC SCHOOL 
 OF YALE COLLEGE 
 
 
 
 NEW YORK 
 
 HENRY HOLT AND COMPANY 
 1876 3 
 

 
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 ART. 
 2—The two general problems in approximate computa- 
 tions, - - - - - 
 
 3—Limits of approximate numbers, ~ - 
 4—Definition of absolute error, - - - 
 Definition of relative error, ~ - 
 5—Prop. 1. Determination of superior limit of relative 
 error of result from given limit of absolute error, 
 PROB. 2. Determination of superior limit of absolute 
 error of result from given limit of relative error, 
 PROB. 3. Determination of allowable limit of relative 
 error from assigned limit of absolute error, - 
 
 PROB. 4. Determination of allowable limit of absolute 
 error from assigned limit of relative error, - 
 6é—Relative error independent of position of decimal point, 
 
 7—Order of units affected by a given relative error, 
 8—Rejection of redundant figures of a result, - 
 9—PROB. 5.—To determine how many exact figures of a 
 result must be computed so that its relative error 
 shall not exceed an assigned limit, - - 
 1I—PROB. 6. Necessary approximation of quantities for 
 __ addition, the approximation of the result being as- 
 Signed, pe = = = = 
 13—Approximation of the sum of approximate quantities, 
 14—PROB. 7. Necessary approximation of quantities for 
 subtraction, the dusters of the result being 
 assigned, - - - - 
 15—Approximation of the Aifference of aes ae te quan- 
 tities, 
 16—PROB. 8. To form a product by peadecd multiplica- 
 tion, exact within a unit of the zth order of decimals, 
 19—PROB. 9. Necessary number of decimals in factors for 
 abridged multiplication, the product being required 
 to the zth order of decimals, 
 21—Extension of the rule for Prob. 9, where the limit of 
 error is a unit of higher order than decimals, 
 
 - 
 
 PAGE 
 
 O ONDA DA wm & WWwWNN 
 
 —_ 
 
 IT 
 
 13 
 17 
 
 18 
 
 19 
 
 20 
 
 24 
 29 
 
iv CONTENTS. 
 
 ART. PAGE, 
 
 22—PROB. 10. Absolute approximation of the product of 
 two approximate factors, - - - 
 
 23—Absolute approximation of the product of any number 
 of approximate factors, - - ~ 
 
 24—PROB. 11. To perform an abridged multiplication so 
 that the error of the process shall be less than that 
 due to the errors of the factors, - - 
 
 Mode of indicating limits of error of OEE: quan- 
 tities in practice, - - - 
 25—PROB. 12. Relative approximation of the product of 
 any number of approximate factors, - - 
 26—Absolute approximation of the square of an approxi- 
 mate number, - - - - 
 27—Absolute approximation of the cube of an approximate 
 number, ma = - ~ - 
 
 28— Relative approximation of the square and cube and ath 
 power of an approximate number, - 
 
 29—PROB. 13. Order of units of the pene significant 
 figure of a quotient, - 
 
 30—PROB. 14. To form a aenent by abridged division; 
 exact within a unit of the zth order, - 
 
 32—PrRoOB. 15. Absolute approximation of the quenee. of 
 an approximate dividend and divisor, 
 
 33—PROB. 16. To perform an abridged division so that the 
 error of the process shall be less than that due to 
 the errors of the quantities, - - < 
 
 34—Determination of the error of an appro quoucn 
 by inspection, - 
 
 35—PROB. 17. Relative approximation of the quotient of 
 an approximate dividend and divisor, - 
 
 a 
 37—Approximate computation of the fraction eo 
 
 38—-PROB. 18. Extraction of sels root by abndeed 
 method, - - 
 
 39—Square root of unity plus or minus a Sencil fraction, 
 
 40—PROB. 19. Absolute approximation of squee root of 
 an approximate number, ~ - 
 
 41—Square root generally to be found with as many exact 
 figures as there are in the number, - ~ 
 
 42—PROB. 20. Relative approximation of square root of an 
 approximate number, ~ - - 
 
 43—Error of square root found by inspection, ~ 
 44—PROB. 21. Extraction of cube root by abridged method, 
 45—Cube root of unity plus or minus a small fraction, 
 
 31 
 32 
 
 32 
 33 
 36 
 39 
 40 
 4I 
 43 
 Aa 
 49 
 
 5° 
 52 
 G3 
 55 
 
 58 
 63 
 
 64 
 66 
 
 . 67 
 
 68 
 71 
 75 
 
CONTENTS. 
 
 ART. 
 
 46—PROB. 22. Absolute approximation cf cube root of an 
 approximate number, - - - 
 
 47—Cube root generally to be found with as many exact 
 figures as there are in the number, - - 
 
 48—PROB. 23. Relative approximation of cube root or #th 
 root of an approximate number, - - 
 
 49—Error of cube root found by inspection, - 
 
 50—-Logarithmic computations, ~ ~ - 
 
 51— Natural trigonometric functions, - - 
 
 52— PROB. 24. Computation of the value of any complex 
 monomial whose factors may be taken as accurately 
 as we please, - - - = 
 
 53— Computation of complex polynomials, - 
 
 54—Computation of complex expressions containing factors 
 known with only a limited degree of approximation, 
 
 v 
 
 PAGE. 
 
 76 
 
 86 
 92 
 
 94 
 

 

 
 neg 
 
 PRINCIPLES” 
 
 OF 
 
 APPROXIMATE COMPUTATIONS. 
 
 INTRODUCTORY. 
 
 1. It is frequently necessary to perform arithmet- 
 ica] operations upon quantities whose numerical val- 
 ues cannot be taken with absolute accuracy, either 
 because these values are incommensurable with unity 
 or because they are the results of measurements made 
 in Astronomy, Physics, Chemistry, Engineering, etc., 
 with instruments capable of giving only a limited de- 
 gree of precision. When the ordinary rules of arith- 
 metic are applied to a computation involving such 
 approximate quantities the operation is often very 
 long and tedious, and when it is finished the operator 
 may be in doubt as to the degree of accuracy of his 
 final result; that is, as to the amount of error to 
 which this result is liable from the errors of the quan- 
 tities employed. The object of the following pages 
 is to present some simple rules for conducting com- 
 putations involving approximate quantities, in such a 
 manner as to require the fewest figures and to show 
 at once the degree of accuracy of the result. The 
 methods are partly suggested, as also a few of the 
 
“2 APPROXIMATE COMPUTATIONS. 
 
 examples, by the small French treatises of Martin, 
 Babinet and Housel, etc., but many changes have 
 been made in the statement and demonstration of the 
 rules, and much not found in those treatises has been 
 added. 
 
 2. In practice we commonly either know before 
 beginning a computation what degree of accuracy 
 will be sufficient in the result, or else we wish to com- 
 pute a result with all the accuracy possible. Our 
 computations accordingly come under one or the 
 other of the following general problems: 
 
 First. A set of quantities being given, whose values may 
 be taken as accurately as we please, to computea result with 
 any required accuracy. 
 
 Second. <A set of quantities being given, each to a certain 
 
 degree of approximation, to compute a result as accurately as 
 the data allow. 
 
 It might happen that the data of a problem were 
 only known with a limited approximation, while yet 
 the result should be demanded with a certain required 
 accuracy. If, however, it should be found by the 
 methods which will be applied to the first of these 
 general problems that the errors of the data must 
 render it impossible to get the result as accurately as 
 required, the example would then simply have to be 
 made a case of the second general problem. 
 
 3. In computations involving approximate quan- 
 tities it is important to adopt the following principle, 
 viz.: to overstate, rather than understate, the error 
 at any time committed. We shall thus be sure of 
 not claiming for our results greater accuracy than 
 they really possess. 
 
 By a superior limit of an approximate number we 
 mean merely some number known to be larger than 
 the true value of the number. By an zxferior limit 
 of an approximate number we mean merely some 
 
INTRODUCTOR ¥. 3 
 
 number known to be smaller than the true value of 
 the number. Thus if the true value of an approxi- 
 mate number were known to lie between 800 and 700, 
 the former would be a superior and the latter an in- 
 ferior limit of the approximate number. 
 
 4. By an absolute error we mean the amount 
 ~which must be added to or subtracted from an ap- 
 proximate number to obtain the exact number. We 
 are generally satisfied with stating that the error does 
 not exceed a certain amount; that is, we assign a su- 
 perior limit to the possible error. More often than 
 otherwise this limit is taken at a unit or half a unit of 
 the order of the last figure retained in the approxi- 
 mate number. [or example, the length of the cir- 
 cumference of a circle whose diameter is I, being 7 = 
 Relat 0205,.4,,, 1) wevtake 3.4475, asthe. approxi- 
 mate value we make an error less than 0.0001; if 
 we take 3.1416 for the approximate value the error is 
 less than 4+ X 0.0001. In all cases in which we know 
 the figure which would follow those retained we may 
 reduce the error to at most Za/f a unit of the lowest 
 order retained. For this purpose we increase by I 
 the last figure retained whenever the following figure 
 would equal or exceed 5, as illustrated in the value 
 OL lis 3.1410. 
 
 It may then often happen that the right hand fig- 
 ure of an approximate number will differ from the 
 corresponding figure of the exact number. In what 
 follows, however, when we speak of a number as hav- 
 ing 2 exact figures we shall mean that the error is less 
 than half a unit of the order of the zth figure from 
 the left; thus the number 3.1416 will be regarded as 
 the value of z with five exact figures. 
 
 By a relative error we mean the fraction expressing 
 the ratio which the absolute error of an approximate 
 
4 APPROXIMATE COMPUTATIONS. 
 
 number bears to the exact number; for example, if 
 instead of the number 403 we take an approximate 
 value, say 400, the relative error of this approximate 
 value will be q#z. If then 4 represent the exact 
 value of a number, & the absolute error of an ap- 
 proximate value of the number, and & the relative 
 error of the same approximate value, we have the 
 following exact relation between the quantities VV, & 
 and R: 
 
 R=S; (1) 
 or, foe iV (2) 
 
 The relative error of a number is often of more im- 
 portance than the absolute error; for an absolute er- 
 ror that would in some cases be of no consequence 
 might in others be many times greater than could be 
 tolerated. For example, we should not often expect 
 a large building to be laid out so as to correspond to 
 the specifications within less than a millimetre, but an 
 error of a millimetre in the diameter of a portion of 
 a thermometric tube would render the instrument 
 worthless. 
 
 5. In practical examples, since it is often impossi- 
 ble to obtain the exact value of a quantity sought, 
 the following problems occur with regard to absolute 
 and relative errors: 
 
 PROBLEM I. A superior limit of the absolute error of a given 
 
 result being known, it is required to assign a superior limit to 
 the relative error of the same result. 
 
 RULE 1. Divide the known limit of the absolute 
 error by an inferior limit of the result. 
 
 Thus if A’ be the required superior limit of the 
 relative error of a result, the known limit of the ab- 
 solute error being /&, and V — x being an inferior 
 limit of the result, we shall have 
 
INTRODUCTORY. 5 
 
 Oe: 
 tf 
 i rca (3) 
 
 The reason for this rule is obvious. If it were pos- 
 sible to obtain the exact value of the result, the rela- 
 tive error of an approximate result would be given 
 by equation (1). But equation (3) will give a larger 
 value than equation (1). Hence, (Art. 3), Rule 1 will 
 give a superior limit of the relative error. To find 
 RF’ neither VV nor x need be exactly known, but we 
 may take for V — # any convenient number known 
 to be less than the true value of V. For example, 
 ewelave tie limber 73.42... as the result of a 
 computation, and know that the absolute error is less 
 than 0.10 we may evidently be sure that the relative 
 error is less than 9; = 74,7; that is, this is a supe- 
 rior limit of the relative error. 
 
 
 
 
 
 PROBLEM 2. A superior limit of the relative error of a given 
 result being known, it is required to assign a superior limit to 
 the absolute error of the same result. 
 
 RULE 2. Multiply the known limit of the relative 
 error by a superior limit of the result. 
 
 Thus if 4” be the required superior limit of the ab- 
 solute error of a result, the known limit of the rela- 
 tive error being #, we have 
 
 E' = R(N + x). (4) 
 
 It is hardly necessary to explain the reason for the 
 rule. Equation (2) would give the exact value of 
 the absolute error of an approximate result if it were 
 possible to obtain VV exactly. But equation (4) will 
 give a larger value than equation (2). Hence Rule 2 
 will give a superior limit of the absolute error. For 
 example, if 47.25... be the result of a computation, 
 and it be known that the relative error is less than 
 Tooo, we may evidently be sure that the absolute er- 
 
6 APPROXIMATE COMPUTATIONS. 
 
 ror is less than 7399; that is, this is a superior limit 
 of the absolute error. 
 
 PROBLEM 3. If a computation is to be made, so that the ab- 
 solute error of the result shall not exceed a limit Z, assigned in 
 advance, it is required to determine how large a relative error 
 
 we are at liberty to make in the work, or, in other words, to as- 
 sign an aflowad/e limit to the relative error of the result. 
 
 RULE 3. Divide the assigned limit of the absolute 
 error by a sitperior limit of the result. 
 
 Thus if A” be the required allowable limit of the 
 relative error to be made, we shall have 
 
 th ene, £ 
 A OTPeN Se: (5) 
 
 The reason for the rule is obvious. If it were pos- 
 sible to know the exact value of the result, we should 
 evidently, from equation (1), be at liberty to make a 
 
 relative error in the work equal to nia We shall 
 therefore, a fortiort, not exceed the assigned limit of 
 
 error if R” does not exceed the smaller value Wiscca dtl 
 dy Faz 
 given by Rule 3. For example, suppose we were to 
 make any computation so that the absolute error of 
 the result should not exceed 0.002, and suppose the 
 result to be 7.8..., we should evidently be at liberty 
 to make a relative error in the computation equal to 
 0.092; for from equation (2) the limit of the absolute 
 ier would then be £ = 2°22 Xx 7.3.x ae 
 that is, the actual absolute error would be less than 
 the limit 0.002, assigned in advance. 
 
 PROBLEM 4. If a computation is to be made so that the rela- 
 tive error of the result shall not exceed a limit 2, assigned in 
 
 advance, it is required to determine an allowable limit’ to the 
 absolute error of the result. 
 
INTRODUCTORY. a 
 
 RULE 4. Wultiply the assigned limit of the rela 
 tive error by an inferior limtt of the result. 
 Thus if 4” be the required allowable limit of the 
 absolute error to be made, we shall have 
 Ai h(N 2): (6) 
 The reason for the rule is simple. If the absolute 
 
 error of the result is made not to exceed 4” = 
 R(N — x) 
 ’ 
 
 
 
 
 
 R(N — x), the relative error cannot exceed 
 
 which is evidently less than “, the limit assigned in 
 advance. For example, suppose we were to make a 
 computation so that the relative error of the result 
 should not exceed zo'o9, and suppose the result to be 
 3.2...3; we should evidently be at liberty to make 
 an absolute error in the computation equal to zien = 
 0.003; for from equation (1) the relative error would 
 then be $:-$22 < 3)/57; that is, the actual relative er- 
 ror would be less than zo'5a, the limit assigned in ad- 
 vance. 
 
 In the application of Rules 3 and 4 to practical ex- 
 amples we should evidently have to begin by deter- 
 mining, from inspection or rough calculation, some 
 value that might safely be taken for V Becta Nae 
 as the case might be. 
 
 6. If an approximate quantity be given, the abso- 
 lute error of which is stated to be less than a certain 
 number of units of the lowest order retained, it is evi- 
 dent that the relative error of the quantity will be in- 
 dependent of the position of the decimal point. For 
 Suoeoeawe Nave the numbers 7.024 ..., 702.4... 
 and 0.07024..., stated to be correct within three of 
 the lowest units retained in each, the limits of their 
 relative errors will, by Rule 1, be respectively 
 
8 APPROXIMATE COMPUTATIONS. 
 
 0.00 O. 0.0000 
 3 sath and OSes 
 
 b 
 
 i 700’ 0.07 
 and each of these limits is equal to yoo. If there- 
 fore a number is given, whose absolute error does not 
 exceed ove unit of the zth order, we may evidently 
 find a limit of the relative error by rejecting the dec- 
 imal point from the number, and also all the figures 
 following the zth, replacing with zeros all the other 
 figures except one or two at the left, and dividing 
 unity by the result regarded as a whole number. For 
 example, suppose it be known that the absolute error 
 of the number 3.1415126 does not exceed a unit of 
 the order of the 5th figure from the left, we may say 
 at once that the relative error is less than soda. 
 Also, if a result be given with a certain number of 
 exact figures, since its absolute error will then by defi- 
 nition not exceed sa/f a unit of the lowest order re- 
 tained, the limit of the relative error may evidently 
 be found by disregarding the decimal point, doubling 
 the number, replacing all the figures but one or two 
 at the left by zeros, and dividing 1 by the result. 
 Thus, if the number 76.24 be a result whose absolute 
 error is less than half a unit of its lowest order, we 
 may say at once that the relative error cannot 
 exceed zs300- 
 
 
 
 ’ 
 
 %. It may be shown that if the numerator of the 
 relative error of a given result be unity, and its de- 
 nominator contain z entire figures, then when the first 
 significant figure of the result is less than that of the 
 denominator of the relative error, the absolute error 
 of the result is less than a unit of the zth order, count- 
 ing to the right from the highest significant figure of 
 the result, inclusive; and when the first figure of the 
 result is equal to or greater than that of the denomi- 
 
INTRODUCTORY. 9 
 
 nator of the relative error, the absolute error will still 
 be less than a unit of the (7 — 1)th order of the re- 
 sult, counting as before. Suppose, for example, that 
 the relative error of a result be known to be less than 
 S000: Since the position of the decimal point in the 
 result is immaterial in this connection, we may sup- 
 pose, if we please, that the result has but one figure 
 in its entire part. Then, if this figure be less than 6, 
 the absolute error will evidently by Rule 2 be less 
 than go'so = 0.001; that is, the denominator of the 
 relative error having 4 places the absolute error is less 
 than a unit of the order of the 4th figure, counting 
 from the highest significant figure of the result inclu- 
 sive. But if the units figure of the number were 
 equal to or greater than 6, we should still have the 
 absolute error less than 7335 < 0.01; that is, the de- 
 nominator of the relative error having 4 places, and 
 the first significant figure of the result being greater 
 than that of this denominator, the absolute error is 
 less than a unit of the 3d place, counting from the 
 highest significant figure of the result inclusive. 
 
 Since, from equation (1), the absolute error is pro- 
 portional to the relative error, it also follows that when 
 the first significant figure of a given result is less than 
 that of half the denominator of its relative error, the 
 numerator of this error being 1, and its denominator 
 having z entire figures, the absolute error is less than 
 half a unit of the zth order, counting as before, and 
 when the first significant figure of the result is equal 
 to or greater than that of half the denominator of the 
 relative error, the absolute error is less than half a 
 unit of the (z — 1)th order, counting in the same way. 
 Thus, if the relative error of a result were equal to 
 Zoo0, and the result were equal say to 38.7449..., 
 the absolute error would be less than half a unit of the 
 
 
 
10 APPROXIMATE COMPUTATIONS. 
 
 Ath place from the left; but if the result were equal say 
 to 945.347 ..., the absolute error would still be less 
 than half a unit of the 3d place from the left. 
 
 8. It does not follow from the principles of the pre- 
 ceding article that if the absolute error of a result 
 having more than 7 figures is known to be less than a 
 unit or half a unit of the zth place we may reject all 
 the figures that follow the zth and still have the abso- 
 lute error of the result within the same limits. For, 
 take the example given above, in which the relative 
 error was supposed to be geoo, and the result 
 38.7449 ..., we should know that the absolute error 
 was less than half a hundredth; but if we take simply 
 38.74 for the result, we evidently make a new error, 
 nearly equal also to half a hundredth; and if the two 
 errors happened to be in the same direction the error 
 of the result 38.74 might evidently be double the error 
 of the result 38.7449... If then we wish to retain 
 only a part of the figures of any approximate result 
 already obtained, and to be able to state that it is still 
 correct within a unit or half a unit of the lowest order 
 retained, we have this caution to observe, viz: to con- 
 sider whether the new error made by rejecting figures, 
 plus the error already made in the computation, will 
 not exceed a unit or half a unit of the lowest order re- 
 tained. 
 
 It is evident that if we wish to retain only a limited 
 number of figures in a result to be computed, we can 
 make the final error less than a wzz¢t of the lowest order 
 to be retained, by assigning, as the limit of error to be 
 made in the computation, “a/f a unit of this order; 
 for the error to be made by rejecting the figures that 
 would follow need never be greater than another half 
 unit of the same order (Art. 4). But we cannot in 
 a complicated computation always be sure of assigning 
 
INTRODUCTOR Y. Il 
 
 in advance a small enough limit to the error so as to 
 be able to reject all the figures after the zth and still 
 to say that the error is less than Za/f a unit of the zth 
 order; for we are sometimes liable to make an addi- 
 tional error nearly equal to a half unit of this order 
 when we come to reject the figures that would follow. 
 
 9. PROBLEM 5. The numerator of an assigned limit * to the 
 relative error of a required result being 1, and the denominator 
 
 being entire, it is required to determine how many exact figures 
 of the result must be computed. 
 
 RULE 5. /f the first significant figures of the re- 
 guired result be equal to or greater than those of half 
 the denominator of the assigned relative error, compute 
 as many figures in the result, counting from its high- 
 est significant one inclusive, as there are figures tn the 
 said half denominator ; otherwise compute one addt- 
 tional figure. 
 
 We may demonstrate the correctness of this rule as 
 follows: Let the assigned limit of relative error of a 
 result be, say, zo oa. Since the position of the decimal 
 point in the result-is not important in this connection, 
 we may fix the limit of absolute error of the result at 
 half a semple unit, and determine how large the result 
 would have to be to bring the relative error within 
 the assigned limit. It is evident from the definition of 
 relative error that any number greater than 3500, 
 exact within half a simple unit, would have a relative 
 
 ] 
 
 erogees than =. 54 = 700d, Dub thatany number 
 less than 3500, the absolute error being half a simple 
 unit, would have a relative error greater than yp'o7 } 
 consequently if the first figures of the result were equal 
 
 * In speaking of a limit of error, either relative or absolute, we shall hereafter 
 mean the superior limit numerically, unless otherwise stated. Also, when a result 
 is required with z exact figures we mean that all the figures following the zth are 
 
 to be rejected, the zth figure being adjusted, if necessary,so that the absolute error 
 shall be less than half a unit of this order. Art. 4. 
 
12 APPROXIMATE COMPUTATIONS. 
 
 to or greater than 35, only four places would be 
 needed, but if the first figures of the result were less 
 than 35 another place would have to be taken. For 
 example, if the square root of 2 were required, so 
 that the relative error should be less than <>, we 
 should have to find four exact figures of the root, viz. : 
 1.414; but if the square root of 0.0019 were required 
 with the same limit of relative error, we should need 
 only three exact significant figures, viz.: 0.0436. 
 
 It is obvious that if the limit of absolute error of a 
 required result were placed at a unit, instead of half a 
 unit, of the lowest order to be retained, the number 
 of figures necessary in the result could be determined 
 by Rule 5, if the word a/f, wherever it occurs in the 
 rule, were struck out. For example, if it be asked 
 
 how many figures it is necessary to retain in V 19, 
 so that if it be simply known that the absolute error is 
 less than a wzz¢t of the lowest order retained, the rela- 
 tive error shall be less than <5, we see that three 
 figures, viz. : 4.35, are not enough, but that four, viz. : 
 4.358, are sufficient. 
 
 10. It is worth noticing, that if exact quantities 
 are added to or subtracted from approximate quanti- 
 ties the absolute error of the result will be the same 
 as if only the approximate quantities were taken; and 
 also that if an approximate quantity be multiplied or 
 divided by an exact number the relative error will re- 
 main unchanged. Thus if z be taken equal to 3.1415, 
 the sum of this plus or minus any exact number will 
 have the same absolute error; and the product or 
 quotient of w= 3.1415 by any exact number will 
 have the same relative error; as is evident from Art. 6. 
 
ADDITION. 
 
 11. Prosenm 6. A set of quantities being proposed, whose 
 values may be taken as accurately as we please, it is required to 
 determine how far each must be computed, so that the absolute 
 error of their sum shall not exceed half a unit of the zth order. 
 
 RULE 6. Jf there are not more than ten quantities 
 to be added, compute each so that tts absolute error 
 shall not exceed half a unit of the order next below 
 the nth. If there are more than ten quantities to be 
 added, indicate by the plus or minus sign the direction 
 of the error of each quantity when computed as just 
 stated, or else compute in cach still another figure. 
 
 The rule needs but little explanation. If there are 
 not more than ten errors in either direction, each not 
 greater than, say, 0.005, or half a hundredth, the sum 
 of all these errors will not exceed 0.05, or half a tenth. 
 
 If we wish to indicate the direction of the error of 
 an approximate quantity, we place a plus sign after 
 it when its error would have to be added to give the 
 true result, and a minus sign in the opposite case. 
 
 EXAMPLE I. Find the sum of the square roots of 
 prmerreerDe rss 22 0 8 O., 75 S410, Ly. b2, 413,04; 108, 
 17, and 18, so that the absolute error of the result 
 shall be less than 0.0005. 
 
 Taking each with four exact decimals, we have 
 
14 APPROXIMATE COMPUTATIONS. 
 
 1.4142 -- 
 1.7321 — 
 22301 — 
 2.4495 
 2.64538 
 2.8284 
 3.1623 
 3.3166 
 3.4641 
 3.6056 
 3-7417 
 3.8730 
 Aolost 
 4.2426 
 42.8351 
 
 There being in the above example not more than 
 eight errors in one direction, each less than 0.00005, 
 the error of the sum cannot exceed 0.0004, which is 
 ‘less than the assigned limit 0.0005. Observe, also, 
 that in this particular example, by finding this smaller 
 limit, viz.: 0.0004, since the last figure of the sum 
 happens to be 0.0001, we are able to reject this and 
 still to say that 42.835 is correct within 0.0005, or that 
 this is the answer with five exact figures. But if we 
 take the sum of the same numbers, omitting the last 
 one, we find it to be 38.5925. We aré therensre 
 evidently unable, without knowing the direction of 
 the error, to reject the last figure of this result, and 
 still have the answer contain five exact figures. 
 
 In general when we wish to retain all the exact 
 figures of an approximate result, and no more, we 
 must find between what limits the true result lies, and 
 then retain only those figures which would be kept 
 for numbers at either limit. For example, in the case 
 just cited, where the sum of thirteen numbers is 
 
 Wed Peerage 4 
 
 ++ | 
 
ADDITION. 1s 
 
 38.5925, eight of them are taken too large and five 
 too small; hence the true result must lie between 
 38.5921 and 38.5928. We can therefore take 38.59 
 as the result with four exact figures, but by our defi- 
 nition neither 38.592 nor 38.593 as the result with 
 five exact figures, for the computation does not shaw 
 which of these results is the more accurate. 
 
 EXAMPLE 2. es the series 
 
 
 
 3 3° 72 seeunraee 
 retaining only eight decimals in re result, and make 
 the error of the result less than a unit of the eighth 
 decimal place. 
 
 Observing the remark near the end of Art. 8, and 
 taking nine terms, each exact to the ninth decimal 
 place, we have, 
 
 0.666666667 
 24691358 
 1646091 
 130642 
 11290 
 
 1026 
 
 96 
 
 2 
 
 I 
 0.693147180 
 Answer, 0.69314718. 
 
 [ee 
 
 J++4+4 | 
 
 EXAMPLE 3. Compute the expression V44+ 
 
 Vo0.08 + Vo0.06 + 1.7435013...., with three exact 
 decimals in the result. 
 
 Taking four exact decimals in each term the sum 
 will be found to be 8.9044. But the possible error of 
 this result is greater than 0.0001. Hence the last 
 decimal cannot be rejected without determining still 
 
16 APPROXIMATE COMPUTATIONS. 
 
 another one. (Art. 8, at end). Let the next decimal 
 be determined. 
 
 12. It is clear that if any other limit of absolute 
 error be assigned for a sum than that stated in Prob- 
 lem 6, the allowable limit of absolute error for each 
 of the quantities to be added may be found by divid- 
 ing the assigned limit of absolute error of the sum by 
 the number of quantities to be added. 
 
 EXAMPLE 4. Compute the expression 352.7856... 
 + 7¥2+ V7 + 25.00082... + 0.000074..., with 
 
 a relative error in the result not to exceed zadas. 
 
 We see that the sum will be greater than 375. 
 Hence, by Rule 4, we are allowed to make an absolute 
 error in the sum equal to s%$30, and as there are five 
 numbers to be added we are evidently at liberty to 
 make an absolute error in each of them equal to 
 xX sthta = socos, Which 1s ‘greater thansoue 
 Hence, if the thousandths place in each number be 
 found within a unit of that order, the result will satisfy 
 the conditions. Taking the first four numbers with 
 that approximation, and neglecting the fifth, we have 
 
 352.785 
 1.414 
 2.645 | 
 
 25.000 
 
 381.844 
 
 with an absolute error less than 0.005, and a relative 
 error, therefore, less than sou%aw < sod0~- 
 
 EXAMPLE 5. Calculate the sum of the reciprocals. 
 of-the numbers 3/°7;-0; 11; 13,774,. 17-5100 
 22, 23, 26, 27, 28, and 29, so that the relative error 
 of the result shall be less than sggov. 
 
ADDITION. 17 
 
 13. It is evident that if it be required to find the 
 sum or several quantities which can each be obtained 
 with only a limited degree of approximation, if we do 
 not know the direction of the errors of any of them, 
 we must take as the limit of the absolute error of the 
 sum, the sum of the possible absolute errors of the 
 quantities. If the absolute error of one of the quanti- 
 ties greatly exceeds those of the others we should 
 commonly not take the trouble to find the exact sum 
 of the errors; but we may take some approximate 
 value of it that we can see would exceed it. For 
 example, in adding the numbers 76.3, 18.71, and 
 528.345, supposed approximate within half a unit of 
 the lowest order in each, we should be satisfied with 
 saying that the sum could be found exact within 0.06. 
 
 EXAMPLE 6. Add the following numbers, supposed 
 approximate within 2 units of the lowest order in 
 each, and assign a limit to the absolute and relative 
 error of the sum: 35.278, 26.435, 18.7346, 21.0064, 
 3.2176, 0.2142, 0.00125, 
 
 Z 
 
SUBERACIION: 
 
 14. PRoBLEM 7. Two quantities being proposed whose 
 values may be taken as accurately as we please, it is required to 
 determine how far each must be computed so that the absolute 
 error of their difference shall not exceed an assigned limit. 
 
 RULE 7. Compute each quantity so that tts absolute 
 error shall not exceed half the limit of error assigned 
 Jor the difference. 
 
 The rule needs no demonstration. And it is also 
 evident that if the errors of the two quantities were in 
 the same direction, these errors would not need to be 
 made smaller than the limit assigned for the error of 
 the difference. Thus, if a difference of two numbers 
 be required within a unit of a given order, it will be 
 sufficient to compute each to that order of units when- 
 ever it can be seen that their errors will be in the 
 same direction. 
 
 EXAMPLE 7. Compute the expression V7 — V5 
 so that the absolute error of the result shall not ex- 
 ceed 0.001. 
 
 The answer may be either 0.409 or 0.410. 
 
 EXAMPLE 8. Compute the expression V95 — V5, 
 with a relative error less than z5dos. 
 
 By calculating the first two figures of each root we 
 find the difference will be greater than 7. Hence we 
 are at liberty to make an absolute error in the result 
 equal to 0.0007. But not knowing beforehand the 
 
SUBTRACTION. 19 
 
 amount of the errors that would result by taking both 
 numbers too small or both too large, with three deci- 
 mals each, nor the direction of the errors if each were 
 to be taken to the nearest 3d decimal, we have to 
 determine in each the 4th decimal, giving 
 9.7468 — 
 TPN GAN ee 
 7-5107 
 Since the errors of the two quantities happen now 
 to be in the same direction, the absolute error of the 
 result cannot exceed 0.00005. Then the true differ- 
 ence lies between 7.5106 and 7.5108. Either of these 
 limits would give for the answer with four exact 
 figures, 7.511, the absolute error being then less than 
 half a thousandth, and the relative error, therefore, 
 (Art. 6) less than tsdou < roo0d aS required. 
 EXAMPLE 9. Compute the expression “V3 — V2, 
 with a relative error less than zoos. 
 Answer, 0.02804. 
 415. It hardly needs to be pointed out that if two 
 numbers can be obtained, each with only a limited 
 degree of approximation, the limit of the absolute 
 error of the difference of the numbers will have to be | 
 taken equal to the sum of the limits of the errors of 
 the numbers, unless it is known that their errors are 
 in the same direction; in this case the larger of the 
 two limits of error may be taken as the limit of error 
 of the difference. For example, if the numbers 3.725 
 and 1.834 are each approximate within half a thou- 
 sandth, but the direction of the errors not known, 
 there will be an uncertainty of a thousandth in the 
 difference, 1.891. If, however, we have the numbers 
 3.725 and 1.8342 each known to be foo small by not 
 more than half a unit of its lowest order, the error of 
 the difference 1.8908 cannot exceed half a thousandth. 
 
 
 
MULCTIREVCAT ON: 
 
 16. Prosiem 8. Two factors being proposed, whose values 
 may be taken as accurately as we please, it is required to form 
 their product so that its absolute error shall not exceed a unit of 
 the zth order of decimals. 
 
 RULE 8. Jake either factor for the multiplier, and 
 write wt with its figures in reverse order under the 
 multiplicand, and in such a position that the original 
 simple untts figure of the multiplier shall come under 
 the (n+ 1)th decimal of the multiplicand. Begin 
 each partial product with the product of the multiply- 
 wing figure into the figure of the multiplicand directly 
 over tt, rejecting the figures of the multiplicand to the 
 right of this, but correcting this product, tf necessary. 
 by adding to tt the number of units of the same orde 
 nearest to what would be added tf the rest of the multi- 
 plicand were used, and place the partial products with 
 thetr right hand figures in a vertical line. The sum 
 of the partial products will have n+ 1 decimals, the 
 last of which must generally be regarded as doubtful. 
 
 EXAMPLE 10. Compute the product 
 
 763-05403698956 X 25.4463057845 
 with an absolute error less than 0.0001. 
 
MULTIPLICATION. 21 
 
 Operation: 
 
 BES a eee 
 
 548750 3 644.52 
 
 erscor on] fies 
 
 381 5 27018 
 
 20m 22 101 
 
 BOs 22516 
 
 4578 32 
 
 228 92 
 
 3 82 
 
 oe 
 
 6 
 
 1941 6.906 34. 
 
 This method of abridged multiplication is ascribed 
 to Oughtred. The explanation of it is very simple. 
 In the above example the original units figure 5 of the 
 multiplier stands under the fifth decimal figure of the 
 multiplicand; hence, considering first the second par- 
 tial product its right hand figure will be of the fifth 
 order of decimals. But the original tens figure of 
 the multiplier stands under the sixth decimal of the 
 multiplicand, and the first deczmal of the multiplier 
 under the fourth decimal of the multiplicand, so that 
 the partial products made with these figures of the 
 multiplier will also begin at the right with the fifth 
 decimal place; and so for all the partial products, 
 since the position value of the figures of the reversed 
 multiplier diminishes towards the left, in the same ra- 
 tio as that of the figures of the multiplicand increases. 
 The position of the decimal point in the result is easy 
 to fix. It may in any case be determined by the last 
 sentence of Rule 8. If the multiplier in any example 
 has no entire figures, supply the place of simple units 
 with a zero, 
 
22 APPRIXIMATE COMPUTA TIONS. 
 
 Let us consider the possible error of a result found 
 by Rule 8. If the rule with regard to correcting the 
 partial products to allow for the part of the multipli- 
 ~ cand each time rejected is carefully followed, it is evi- 
 dent that the error of each partial product cannot 
 exceed half a unit of its lowest order; and since the 
 right hand figure of each partial product is of the 
 same order as that of the final result, the error of this 
 result, whenever the number of partial errors does 
 not exceed twenty, cannot be greater than ten units 
 of the lowest order obtained; that is, it will not ex- 
 ceed ove unit of the next higher order, which is the 
 limit assigned in the statement of the problem. 
 
 ‘,. In the example worked above, the last partial prod- 
 uct Comes from using the figure 8 of the multiplier ; 
 
 and if the next ficure of the “multiplier at the left had 
 been large enough to give more than half a unit of 
 the lowest order in the result, we should have put in 
 another partial product equal to one unit of that or- 
 der. But since the number of partial products does 
 not exceed ten, the total error of the result cannot 
 exceed five units of its lowest order, and we may 
 therefore reject the last figure 4 of the result, and the 
 answer will still be correct within 0.0001, the assigned 
 limit of error. 
 
 1%. It is easy in practice to indicate the direction 
 of the errors of the partial products by the plus and 
 minus signs, and thus, when the work is done, to 
 often determine a much smaller limit of error than 
 that assigned in advance. 
 
 EXAMPLE Ioa. Make the product of the same fac- 
 tors as in Example Io, with the same limit of error, 
 but taking the former multiplier for the multiplicand. 
 
MULTIPLICATION, 23 
 
 25.4463 057845 
 6598963 0450.367 
 
 178 I241 405 — 
 
 15 2677 835 
 
 7633 892 
 
 127232 
 
 103179 
 
 76 
 
 15 
 
 Be 
 
 19416.90 636 
 
 +++ | 
 
 
 
 
 By thus marking the direction of the 
 observe that five of the partial products are 
 and three are too small; hence the error of the 
 cannot be more than 23 units of the lowest order ob- 
 tained; which is only a quarter of the limit of error 
 assigned in advance. But without determining a still 
 smaller limit of error, we could not decide from either 
 this operation or that of example 9, or both of them, 
 whether, if we wished to reject the last decimal of the 
 result, the one before it should be left a 3 or changed 
 to 4. 
 
 18. EXAMPLE II. Multiply 0.995....by 9.95.... 
 so that the absolute error of the result shall not ex- 
 ceed 0.01. 
 
 By the rule for arrangement we have 
 
 00h he 
 pe tb RO 
 8955 
 89 6 
 50 
 9.90 I 
 
 But let us examine the possible error of this result, 
 
 
 
24 . APPROXIMATE COMPUTATIONS, 
 
 supposing the 5 in each factor to be liable to an e: ror 
 of half a unit of its own order. The first partial 
 product would evidently be liable to an error of 44 
 units of the order of the last place in the product; 
 the second partial product would be liable to an error 
 of nearly a unit of the same order, since, besides the 
 error of the multiplicand, we make an additional error 
 in rejecting a figure of this partial product; and, 
 making a similar observation, the last partial product 
 might be wrong by about 5 units of the same order. 
 It is therefore possible that we have exceeded the as- 
 signed limit of error. In fact, if the true values of 
 the numbers were 0.9945 and 9.945, their exact prod- 
 uct would be 9.8903025, a result differing from the 
 approximate one found above, by 0.0106975, or a 
 trifle more than the assigned limit of error. 
 
 It is thus seen that in the application of Rule 8 it 
 will not in all cases be enough to know within a half 
 unit of its own order the figure of the multiplicand 
 standing over the right hand figure of the reversed 
 multiplier, and that of the multiplier standing under 
 the highest figure of the multiplicand. But a single 
 additional figure in each factor will be amply sufficient. 
 For if the multiplicand has one exact figure to the 
 right of the multiplier, and the multiplier one to the 
 left of the multiplicand, we may always reduce the 
 errors of the first and last partial products each to a 
 single unit of the lowest order obtained in the prod- 
 uct; and since the assigned limit of error is 10 of 
 these lowest units there is still room for 16 additional 
 partial products with errors all in the same direction, 
 each less than half of one of these units. 
 
 19. PRosLeM 9. The product of two factors being required 
 within a unit of the zth order of decimals, it is required to state 
 a rule for determining the number of decimai places to be com- 
 puted in each factor. 
 
- 
 
 MULTIPLICATION. 25 
 
 RULE 9. Either factor being regarded as the multt- 
 plicand, compute one more decimal place in the other 
 factor than the whole number of significant figures 
 in the multiplicand above the nth order of decimals in- 
 clustve.* 
 
 This rule follows from the arrangement of the fac- 
 tors by Rule 8. According to that Rule the units 
 figure of the multiplier will stand under the (z + 1)th 
 decimal of the multiplicand. Rule 9g will then give 
 us one figure of the multiplier to the left of the high- 
 est significant figure of the multiplicand, as required 
 by the last section. Thus if it were required to multi- 
 ply 0.007425625 by 99.284376 with an absolute error 
 less than a unit of the 4th decimal place, the arrange- 
 ment by Rule 8 would be 
 
 0.0074 25625 
 67 3482.99 
 
 from which it is evident that the three left hand figures 
 of the multiplier will not be used in forming the prod- 
 uct. Striking them off, the number of deczmads re- 
 maining in the multiplier, viz.: three, will exceed by 
 one the number of significant figures in the multipli- 
 cand above its 4th decimal place inclusive, viz.: two. 
 The number of necessary figures in either factor will 
 not be altered by making the former multiplier the 
 multiplicand. For, making this change we should 
 have 
 
 99.2843 76 
 52652 4700.0 
 
 In this arrangement the two left hand figures of the 
 
 * Whenever we speak of significant figures in this way we of course include any 
 zeros that come in below the highest significant figure. Thus the number o.0010702 
 would be regarded as having five significant figures. 
 
26 APPROXIMATE COMPUTATIONS. 
 
 new multiplier may be struck off, and the number of 
 decimals left in it, viz.: seven, will again exceed by 
 one the number of significant figures in the new 
 multiplicand above its 4th decimal place inclusive, 
 Wide sio 
 
 EXAMPLE 12. Determine by Rule 9 how many 
 decimal places would have to be calculated in V¥9057 
 
 and ¥V0.0093 so that their product by Rule 8 should 
 not be wrong by more than a unit of the 5th decimal 
 place. 
 
 Regarding V¥9057 as the multiplicand, it will have 
 seven figures above the 5th decimal place inclusive. 
 Hence the number of decimal places to calculate in 
 V0.0093 will be eight. Regarding vo0.0093 as the 
 multiplicand it will have four significant figures above 
 the 5th decimal place inclusive. Hence the number of 
 decimal places to calculate in V9057 will be five. Let 
 
 the figures be obtained and the product formed by 
 Rule 8. 
 
 
 
 20. Rule 9 is framed to cover safely all cases. 
 But in a great majority of examples, viz.: where the 
 sum of the highest significant figures of the two fac- 
 tors, plus one-half the number of partial products, does 
 not exceed 18, we may do with one less exact figure ; 
 ’ and if the sum of the highest significant figures of the 
 two factors, plus one-half the number of partial prod- 
 ucts, is less than 8, we shall even then generally get 
 the result with an error less than fa/fa unit of the 
 wth order. It is easy in any special example to re- 
 cognize the possible error committed. 
 
 EXAMPLE 13. Compute the expression V0.0003 
 x v1000 with a relative error less than yodu0- 
 
MULTIPLICATION. 27 
 
 The square root of 0.0003 is more than 0.017, and 
 that of 1000 is more than 30. Hence the product will 
 be more than o.5. By Rule 4 we are at liberty then 
 to make an absolute error in the result equal to zO-eee 
 = 0.00005, that is, half a unit of the ath decima 
 place. The sum of the highest significant figures of 
 the factors, I and 3, is so small that we shall probably 
 be safe if we work as if the allowable error were a 
 whole unit of the 4th decimal place, and take also one 
 less decimal in each factor than Rule 9 would require. 
 (The reason why the error of a result is less when the 
 sum of the highest significant figures of the factors is 
 small, is evidently that the sum of the errors of the 
 first and last partial products will then also be small). 
 Since vV0.0003 will have three significant figures 
 above the 4th.decimal place inclusive, we must then 
 know three exact decimals in V1000; and we may 
 determine the requisite number of decimals in 
 V0.0003 in a similar way, or by simply considering 
 that we want the same number of significant figures 
 in it as in ¥1000, which, from what has just been 
 found, must be five. We then want the 6th decimal 
 place in V0.0003. 
 
 The multiplication will be as follows: 
 
 0.54 774 
 
28 APPROXIMATE COMPUTATIONS. 
 
 In taking each factor in this example to the nearest 
 unit of the lowest order retained, each is too large. 
 If we suppose the multiplicand too large by half a 
 unit of its lowest order, the first partial product would 
 also be too large by half a unit of its lowest order, 
 since the multiplying figure is 1. And if we suppose 
 the multiplier too large by half a unit of its lowest cr- 
 der, the last partial product would be too large by 
 about 14 units of the same order as before, since the 
 highest figure of the multiplicand is 3. The errors 
 of the second and third partial products cannot in- 
 crease the error of the result to more than 3 units of 
 its lowest order; hence the true result cannot be less 
 than 0.54771. On the other hand, if the errors of 
 the factors are as near zero as possible the sum of the 
 three partial products which could then be too small 
 will be less than a unit of the lowest order of the re- 
 sult. Hence the true result cannot be greater than 
 0.54775. We may then, if we please, reject the last 
 decimal obtained in the result, and the answer, 0.5477 
 will have four exact figures, and a relative error less 
 than szgo0 < rodeos, as required. 
 
 EXAMPLE 14. Compute the expression 100 7 V2, 
 with a relative error in the result not exceeding zoq00- 
 
 The product ES esas 400. ‘Then the allowable 
 absolute error is 7é$89 = 0.04. If we work then as 
 if the limit of error were ove unit of the second deci- 
 mal place, instead of four, we may safely determine 
 the requisite number of decimals by taking one less 
 than Rule 9 would give. Regarding 100 z7 as a sin- 
 gle factor, it will have five figures including the 2d 
 decimal place, and we therefore need five decimals in 
 
 v2. Computing these figures, and taking the value 
 of 100 w to correspond we have 
 
MULTIPLICATION. 29 
 
 314.159 + 
 + 124 14.1 
 314159 + 
 125664 — 
 3142 — 
 an RY [eae 
 Coca 
 Seats 
 444.28 8 
 If the 6th decimal figure of the multiplier be sup- 
 posed equal to 5, there would be another partial 
 product not exceeding 2 of the lowest units of the 
 product. As for the partial products written, only 
 two are too small. Hence the result cannot be too 
 small by more than 3 of its lowest units. And it 
 cannot be too large by more than 2 of these units. 
 Therefore the true value of the result lies between 
 444.291 and 444.286. Hence the result to the near- 
 est hundredth is 444.29, with a relative error less than 
 
 poo eae ao ee See 
 440000 =< 100000° 
 
 21. It is easy to see that if the highest significant 
 figure of either factor is in the zth decimal place, then 
 by Rule g there would be 2 decimals to calculate in 
 the other factor. If the highest significant figure of 
 one factor is # places below the wth there will then be 
 2 — p decimals to be calculated in the other factor; 
 and the extension of Rule 9g to cases in which the as- 
 signed limit of error is a unit of higher order than 
 decimals is therefore easy. 
 
 EXAMPLE 15. How many figures must be taken in 
 the factors 9843.768 and 947.84321, so that the error 
 - of the product shall be less than a million ? 
 
 The highest figure of the: first factor stands in the 
 third place below millions, hence the required num- 
 
30 APPROXIMATE COMPUTATIONS. 
 
 ber of decimals in the other factor is 2 — 3 = —1; 
 that is, we may neglect the units figure. ‘The highest 
 figure of the second factor is in the fourth place be- 
 low millions, hence the number of decimals required 
 in the first factor is 2 — 4 = —2; that is, we may 
 neglect the tens. We have then 
 
 98 
 .059 
 88 
 a 
 93 
 Since the original units figure of the multiplier 
 would thus come under the place of hundred thou- 
 sands in the multiplicand, the right hand figure of the 
 product obtained is hundred thousands; hence the 
 product is 9300000, with an error less than a million. 
 
 EXAMPLE 16. How many figures must by this ex- 
 tension of Rule 9 be computed in each of the factors, 
 
 V98734216 and vV0.0093, so that the error of the 
 product by Rule 8 shall be less than a unit of tens 
 place. 
 
 The simple units in the value of the first factor may 
 be disregarded, and we want four decimal places in 
 the value of the second factor. Let the factors be 
 
 found with this approximation, and their product 
 formed by Rule 8. 
 
 EXAMPLE 17. How many figures must be com- 
 
 puted in V758425 and vV0.00009, so that the error 
 of their product by Rule 8 shall be less than 100? 
 
 In Article 52 will be found a general rule for deter- 
 mining the necessary approximation of each factor 
 where an expression contains more than two factors, 
 either as multipliers or divisors or both. 
 
MULTIPLICATION, 31 
 
 22. PROBLEM Io. Two factors being given, each to a cer- 
 tain degree of approximation, it is required to assign a superior 
 limit to the absolute error of their product. 
 
 RULE 10. Multiply a superior limit of each factor 
 by the absolute error of the other factor. The sum of 
 the two products thus obtained may be taken for the 
 limit of the absolute error of the product of the fac- 
 Lors. 
 
 Demonstration. Suppose there be given the two 
 approximate factors, a’ and 6’, whose absolute errors 
 are a and f, we are to determine a limit of the error 
 of the product, a’0’. In the most unfavorable case 
 the errors will be in the same direction; if then a and 
 6 are the true values of the factors, suppose a’ =a +a, 
 of =b+ 6. Multiplying these two equations, mem- 
 ber by member, the product will be 
 
 all!’ =ab+ apt bat af, 
 and the absolute error of the product will be 
 
 a’! — ab = apt bart af. (7) 
 Now since a and £ are usually very small compared 
 with a and @, the product af will be very small com- 
 pared with af and da, so that we have for the abso- 
 lute error of the product very nearly 
 
 a’l’ — ab= af + ba. (8) 
 
 In finding the limit of error by Rule 10 we shall evi- 
 dently get a larger limit than would be given by the 
 right hand member of equation (8), since instead of a 
 and 6 we take by the rule quantities known to be 
 larger than them. Hence the rule is a safe one, not- 
 withstanding the neglect of the very small quantity 
 
 afi. 
 
 EXAMPLE 18. Determine the limit of absolute er- 
 ror in the product of the factor 784.2817, supposed 
 to have an absolute error not exceeding 0.0004, by 
 
32 APPROXIMATE COMPUTATIONS. 
 
 the factor 3.483, supposed to have an absolute error 
 not exceeding 0.006, 
 We may evidently assume as a safe limit, 
 
 800 X 0.006 + 3.5 X 0.0004 = 4.8 + 0.0014 
 
 The product may then be made with an error not ex- 
 ceeding 5 simple units. 
 
 23. If we have three approximate factors, a’ = 
 ata, W=6+, and ¢=c+y, their product 
 will be 
 
 a’b'c' = abe + aby + ach + bea, 
 if we neglect terms containing each more than one 
 error as a factor. The absolute error of the product 
 will then be, very nearly, 
 
 a’b'c' — abc = aby + acB + bea, (9) 
 and if in practice we substitute in the right hand 
 member of this equation superior limits of the ap- 
 proximate quantities, we may evidently take for the 
 limit of absolute error of the product of three approx- 
 imate factors the sum of the products obtained by 
 multiplying the error of each factor by the product 
 of superior limits of the other two factors. The meth- 
 od may evidently be extended to any number of fac- 
 tors. Hence, we may take for the limit of the abso- 
 lute error of the product of any number of approxt- 
 mate factors the sum of the products obtained by mul- 
 tiplying the absolute error of each factor by the con- 
 tinued product of superior limits of all the other fac- 
 tors. If any of the products ady, etc., would be of 
 a much lower order than the highest, such may evi- 
 dently be disregarded. 
 
 24. The formula, a/b‘ — ab =a + ba, assumes 
 that the product of the approximate factors, a’d’, is to 
 be exactly formed. If the product is made by 
 
MULTIPLICATION. 33 
 
 abridged multiplication the errors of the process must 
 be allowed for. But it is always easy to reduce the 
 error resulting from the process of abridged multipli- 
 cation to very much less than that due to the errors 
 of approximate factors. 
 
 PROBLEM 11. To determine the necessary arrangement of 
 approximate factors for abridged multiplication, so that the er- 
 ror due to the abridged process shall be less than that due to 
 the error of either factor. 
 
 RULE 11. When the absolute error of the factor 
 having the greater relative error equals or exceeds § 
 of its lowest units, take either factor for the multt- 
 plier, reverse the order of its figures, and place tt un- 
 der the multiplicand as far to the right as possible 
 without having any of the significant figures of the 
 multiplier to the right of the multiplicand or any sig- 
 nificant figures of the multiplicand to the left of: the 
 multiplier. When the error of the factor having the 
 greater relative error ts between ¥ and § of its lowest 
 untts, put the reversed multiplier one place further to 
 the right. 
 
 The reason for the rule will be clear from an exam- 
 ple or two. We shall hereafter indicate the limit of 
 absolute error of approximate factors by writing it in 
 a special style of figure, following the factor, with the 
 plus or minus sign, this limit of error being under- 
 stood to be so many units of the lowest order retained 
 in the factor. In reversing the multiplier its limit of 
 error will come at the left. Where simply the plus . 
 or minus sign follows a number the error of the num- 
 ber will be understood to be not greater than half a 
 unit of the lowest order retained inthe number. ‘The 
 approximate factors of Example 18 will then be writ- 
 ten, 784.2817 + 4 and 3.483 + 6. If we form their 
 
 3 
 
34 APPROXIMATE COMPUTATIONS. 
 
 product by arranging them according to Rule II, we 
 shall have 
 
 297-21 Oier. SO 
 
 By this operation it is evident that the errors of the 
 first and second partial products due to the error of 
 the multiplicand amount to about 47 units of the low- 
 est order in the result, while the errors due to the 
 neglected parts of the last four partial products do not 
 exceed 2 of these units. And it is evident that if the 
 right hand figure of the reversed multiplier had been 
 the smallest possible, viz.: 1, and the error of the mul- 
 tiplicand as large as 5 of its lowest units, the error of 
 the result due to that of the multiplicand would have 
 been at least 5 of the lowest units of the result; 
 which would allow for Jo partial products besides the 
 first, with errors all in the same direction, before the 
 sum of the errors of the abridged process would equal 
 that due to the error of the multiplicand. The result 
 would not be essentially different if the former multi- 
 plicand were made the multiplier. For by Rule 11 
 
 we should have 
 
 VS4.201 7 gy 
 6 +3843 
 
 23523 + 
 3137 + 
 6257 vt- 
 a 
 4h 
 
 
 
 WW 
 Co 
 
 24 
 
 
 
MULTIPLICATION. 35 
 
 and here the possible error of the last partial product 
 is nearly all due to that of the same factor which 
 caused the greatest part of the error in the former 
 operation. 
 
 But let us look at one more example. Take the 
 factors 1124.267543 + 2, and 8425.7987 + 2. If we 
 were to make their product by arranging them like 
 this, 
 
 SAS JOO7 icy 
 Brea eAs7O2 12 Tl 
 
 it is evident that the amount of the error of their prod- 
 uct due to the errors of the factors would be only 
 about 2 of the lowest units obtained, whereas the 
 errors of the product arising from the abridged process 
 would be those due to the parts rejected from eight 
 partial products; the limit of the errors arising from 
 the abridged process could not then be placed in ad- 
 vance at less than 4 of the lowest units of the prod- 
 uct. To reduce the errors due to the abridged process 
 of multiplication to less than those due to the errors 
 of the factors, we need therefore, in accordance with 
 Rule 11, to move the multiplier one place to the 
 right. And this will be amply sufficient. For if we 
 arrange the factors thus, 
 
 BAOn 700 7 
 Bash 70 242 11 
 
 it is evident that the error of the product due to those 
 of the factors will be about 24 of the lowest units of 
 the product, while that due to the neglected parts of 
 partial products will be less than 5 of the same units. 
 In this example the highest significant figure of the 
 multiplier is the smallest possible, which reduces the 
 part of the error of the product due to that of the 
 
36 APPROXIMATE COMPUTATIONS. 
 
 multiplicand to a relatively small amount. And it is 
 clear that if the limit of absolute error of the factor 
 having the greater relative error were but half a unit 
 of its lowest order we might add a zero to the factor, 
 and call the error 5 units of the lowest order then re- 
 tained, which would bring the case under the first 
 part of Rule 11, already ioseand: If the relative 
 errors of two factors are the same, or nearly the same, 
 either factor may for the purposes of the rule be as- 
 sumed as having the greater relative error. The rule 
 is sufficient in all cases in which not more than ten 
 partial products have errors in one direction. 
 
 EXAMPLE 19. Multiply 3.14278 + by 0.00742 +, 
 and let the error from the abridged process be small 
 in comparison with the limit of error of the product 
 due to that of either factor. 
 
 Annexing a zero to each factor as just suggested, 
 and then arranging by Rule 11, we have 
 
 3.14278 0 =e 5 
 5 + 0 24700.0 
 21999 + 
 1257 + 
 630-— 
 On-biz6 
 G.0233 10) try 
 The error due to the abridged process is less than a 
 tenth of the possible error of the pages from that 
 of the multiplier. 
 
 EXAMPLE 20. Multiply 4.725 + 3 byo.1478 + 7 
 
 and assign a limit of error to the result. 
 
 25. PROBLEM 12. Two or more factors being given, each 
 to a certain degree of approximation, it is required to assign a 
 superior limit to the relative error of their product. 
 
MULTIPLICATION. : 37 
 
 RULE 12. Jake for the limit of the relative error 
 of the product, the sum of the superior limits of the 
 relative errors of the factors. 
 
 Demonstration. We have for two factors the very 
 nearly exact relation, equation (8), 
 
 a’! —ab= apt ba. 
 Dividing both members by ad we obtain 
 TH ore MIPS 
 ab eed: 
 
 The first member of this equation is by definition the 
 relative error of the product a’d’, while the second 
 member is the sum of the relative errors of the factors 
 a’ and 0’; and since by the rule we take instead of 
 these latter errors superior limits of them, it is evident 
 that Rule 12 will give a safe limit of the error of a 
 product of two factors. It follows then that we may 
 take for the limit of the relative error of the product 
 of any number of approximate factors, the sum of 
 superior limits of the relative errors of the factors. 
 For if the product of two factors be regarded as a 
 single factor, the limit of its relative error plus that 
 of a third factor may be taken for the limit of the 
 relative error of the product of the first product by 
 the third factor, and so on. 
 
 Rule 12 assumes, like Rule 10, that the product of 
 the factors is to be exactly formed. If made by the 
 abridged process the new errors introduced must be 
 allowed for. But it has been shown that the additional 
 absolute error introduced by the abridged process 
 may easily be made much less than that due to the 
 errors of the factors; hence the additional relative 
 error due to the abridged process may also be cor- 
 respondingly reduced. By placing the multiplier one 
 
 aes 
 a 
 
38 APPROXIMATE COMPUTATIONS. 
 
 or two places farther to the right than required by 
 Rule 11 the error from the abridged process will be 
 quite insignificant compared with that due to the 
 errors of the factors. By indicating the limit of error 
 of each partial product we may also if we please 
 determine the limit of error of the actual result in- 
 dependently of any rule. 
 
 EXAMPLE 21. Determine by Rule 12 the limit of 
 relative error in the product 
 (65.432 + 2) (6.21242 )2).(1.5632 eae) 
 and then compute the product. 
 The sum of the limits of the relative errors may 
 evidently be taken at 
 2 2 2 ES Sone sn a 
 
 SS 
 
 60000 60000 15000 60000 5000 
 
 Arranging the first two factors by Rule 11 we have 
 OS Ase iets 
 
 pee wiles ees) herd (8 
 202150210 4 120 
 12006 Ae ee 
 6543 + 
 Se LS hoes 
 PA CGA Pa wad 
 
 406.489 8 + 256 
 Call this result 406.490 + 26, and make the next 
 product as follows: 
 406.490 + 26 
 
 
 
 BMS ot US Stes 
 406.490 + 26 
 203, 2Ai5 see ag 
 24.329 0) 2b aie 
 1219 — 
 Sul shea 
 635.424 + 124 
 
MULTIPLICA TION. 39 
 
 The limit of absolute error of the result, as indicated 
 by the limits of error of the partial products, is 124 
 of the lowest units of the result; hence the relative 
 error of this result cannot exceed ¢3h44s5, which is a 
 
 trifle less than the limit determined above, viz.: s5\ss. 
 
 
 
 EXAMPLE 22. Determine by Rules 10 and 12 the 
 absolute and relative errors to be expected in the 
 product of the following approximate factors, and 
 then form the product and determine its error by in- 
 spection, as illustrated in example 21: 
 
 (756.32 + 3) (25.41 = 7) (0.3248 + 2). 
 
 26. It follows from Rule 10 or equation (8) that 
 the limit of the absolute error of the square of an ap- 
 proximate number may be taken equal to twice a su- 
 perior limit of the number, multiplied by the absolute 
 error of the number. For if in equation (8) we make 
 bo’ =a' and 6 =a we have 
 
 a’? —a = 2aa. (10) 
 And it follows from this that if the limit of absolute 
 error of an approximate number be a unit of the th 
 order counting down from the highest significant fig- 
 ure inclusive, the absolute error of the square of the 
 number will not exceed a unit of the (7 — 1)th order, 
 counting down in the square also from its highest sig- 
 nificant figure inclusive. For, supposing, as we may 
 for the present purpose, the number a’ to be entire, 
 and to contain z figures and be correct within a sim- 
 ple unit, we shall have a= 1; anda’? will have either 
 2m or 2~—1 figures. But if a’ has only 2% —1 
 figures, the highest significant figure of a’ must be 
 less than 4, in which case 2aqa, or the absolute error 
 of the square, will have only z figures. Hence, if 
 2aa be placed under a’? so that the lowest units of 
 
40 APPROXIMATE COMPUTATIONS. 
 
 the two come in the same vertical line, there will be 
 z—1 figures of a’? at the left of 2aa. Thus, sup- 
 pose a! = 31572 andaw=1. We have 
 a’? = 996791184 
 24a = 63144 
 
 Here, z being five, the error of the square is less than 
 a unit of the fourth order, counting from the left in 
 the value of a’”, If the first figure of a’ had been 
 5 or greater, 22a would have had one more figure, 
 but so also would a@’”, hence the same statement 
 would hold; and the truth of our proposition is evi- 
 dent. And since, if @ remain constant, 2aa@ will be 
 proportional to a, we may also state the principle that 
 if an approximate number be exact within “alfa unit 
 of the zth order from its highest significant figure, 
 the square of the number will be exact within half 
 a unit of the (z — 1)th order from its highest sig- 
 nificant figure. It is evident that the principles of 
 this paragraph assume that the square of the approx- 
 imate number is to be precisely formed. If it be 
 made by abridged multiplication, care must be taken 
 that the errors due to the abridged process are made 
 insignificant in comparison with 2@q@; and this is in 
 practice always easy. The principle stated at the 
 head of this article, or equation (10), will generally 
 give a somewhat smaller limit of error than the other 
 principles of this article, though it is convenient to 
 remember that we may expect in the square of an 
 approximate number as many exact figures less one, 
 as there are in the number itself. 
 
 EXAMPLE 23. Compute (0.0080715 + 7)? by 
 abridged multiplication, and determine the limit of 
 error of the result. 
 
 2%@. From the approximate value of the absolute 
 error of the product of three factors, equation (9), 
 
MULTIPLICATION. 41 
 
 Art. 23, by making the factors and their errors equal 
 we obtain 
 
 a’ 
 
 3— g? = 3a°a (11) 
 
 and if in practical examples we take in the right 
 hand member of this equation, instead of a, a superior 
 limit of a’, we may evidently assume as the limit of 
 absolute error of the cube of an approximate number, 
 the absolute error of the number, multiplied by three 
 times the square of the superior limit of the number. 
 It is hardly necessary to say that if the cube is form- 
 ed by abridged multiplication the new error intro- 
 duced must be added, or else made insignificant. 
 
 EXAMPLE 24. Determine the limit of absolute er- 
 ror of (3.456 + 2)° and (883.4 + 4)*, and perform 
 the multiplications. 
 
 For the absolute error of the first of these cubes 
 we may take 
 
 3a7a < 3(3.5) (3.5) (0.002) < 0.075. 
 
 28. By dividing both members of equation (10) 
 by a’, and those of equation (11) by a? we have the 
 approximate equations | 
 
 ae PE (12) 
 a 
 
 a 
 ——{— = 38 (13) 
 
 Jrom which tt ts evident that we may take for the 
 limits of the relative errors of the square and cube of 
 an approximate number respectively twice and three 
 times the limit of the relative error of the number, a 
 result evidently in harmony also with Rule 12. 
 
 EXAMPLE 25. Apply the principle just stated, to 
 determine the limits of the relative errors of the re- 
 
42 APPROXIMATE COMPUTATIONS, 
 
 sults of Examples 23 and 24, and from the relative 
 errors thus found deduce also the absolute errors. 
 For. the relative error of the first cubein Example 
 
 
 
 24 we have 3° ee , and since the cube will not 
 a 
 
 3400 
 exceed 42 we may take for the limit of absolute er- 
 
 6 X 42, 
 
 3400 
 the Iimit determined in Example 24. 
 
 ror, which is the least trifle less than 0.075, 
 
 It evidently follows from Rule 12, that we may take 
 for the limit of the relative error of the nth power of 
 an approximate number, n times a limit of the rela- 
 tive error of the number. If 2 were extremely large, 
 however, we should not assume the limit of the rela- 
 tive error of the number at its smallest possible value. 
 

 
 mining the order of units of the highest signif e ota 
 quotient. oe 
 
 RULE 13. Observe how many places to We right or 
 left the decimal point of the dividend would have to 
 be moved in order that the first figure of the quotient 
 should be simple units. If the point would have to 
 be moved n places towards the right the first signifi- 
 cant figure of the true quotient wrll be in the nth dec- 
 imal place. If the point would have to be moved n 
 places towards the left the first figure of the true 
 guotient will be n places above simple units, that ts, 
 the true quotient will have n+ 1 figures at the left 
 of the decimal point. 
 
 The rule may be illustrated by an example or two. 
 In what follows we shall arrange quantities in division 
 as we arrange them in algebra, viz.: with the divisor 
 at the right ‘of the dividend, and the quotient below 
 the divisor. Suppose then we have for division 
 
 0.1941690| _763.05436 
 os 
 0.0002 
 
 It is plain that if the decimal point of the dividend 
 were moved four places to the right the first figure of 
 the quotient would be 2 simple units. If we correct 
 that quotient by moving the decimal point back four 
 
44 APPROXIMATE COMPUTATIONS. 
 
 places to the left, the 2 will evidently be brought into 
 the fourth decimal place. If the dividend and divi- 
 sor were as follows: 
 7428.436 0.04269 
 ie 
 I 74008.8 
 
 it is clear that if the decimal point of the dividend 
 were moved five places to the left the first figure of 
 the quotient would be simple units. If we correct 
 that quotient by moving its decimal point five places 
 to the right the true quotient will evidently have six 
 figures at the left of the decimal point, the results in 
 each of these illustrations agreeing with the rule. 
 
 20. PROBLEM 14. A dividend and divisor being proposed, 
 whose values may be taken as accurately as we please, it is re- 
 quired to form their quotient so that its absolute error shall not 
 exceed a unit of the zth order from the decimal point. 
 
 RULE 14. Begin by determining the first signifi- 
 cant figure of the quotient, and the order of its untts. 
 Then count the number of figures that the quotient 
 must contain, from this figure inclusive down to the 
 nth inclusive. Beginning at the left, assume in the 
 divisor a number of significant figures greater by one 
 than the number just counted, and take at the left of 
 the dividend as many significant figures as there are 
 in the product of this assumed divisor by the first sig- 
 nificant figure of the quotient. Subtract this product 
 Srom the assumed dividend, and instead of annexing 
 any figure to the remainder, reject a figure at the right 
 of the divisor to determine the second figure of the 
 guotient. Continue the process of division by throw- | 
 ing off successively the figures of the divisor, until the 
 quotient contains one figure below the nth order. In 
 multiplying the portion of the divisor each time re- 
 tained, by the successive quotient figures, have regard 
 
DIVISION. 45 
 
 to the rejected part of the divisor so far as to make 
 each partial product tf possible exact within half of 
 one of its lowest units. Whether the extra figure 
 of the quotient can be discarded without passing the 
 assigned limit of error must be determined by the 
 special conditions of each example. 
 
 EXAMPLE 26. Compute the expression 
 194 16.9063468085... 
 763.05 403678956... 
 with an absolute error in the quotient less than a unit 
 of the 6th decimal place. 
 
 We see that the decimal point in the dividend 
 would have to be moved one place to the left for the 
 first figure of the quotient to be simple units, hence 
 there will be two entire figures in the quotient. 
 
 Therefore we need nine figures of the divisor and ten 
 of the dividend. We take then 
 
 19416.90635|76 3.054037 
 15261 08074|25 4 463058 
 4155 82561 
 3815 27018 
 349 55543 
 305 22161 
 35 33382 
 30 52216 
 4 81166 
 4 57832 
 23334 
 22892 
 442 
 382 
 60 
 
46 APPROXIMATE COMPUTATIONS. 
 
 We mark with a dot each figure when it is rejected 
 from the divisor. Let us consider the possible errors 
 introduced in this example. The error of the divi- 
 dend employed is less than half a unit of the lowest 
 order retained in it; the error of the assumed divisor 
 being also less than half a unit of its lower order, the 
 error of the first partial product cannot exceed a unit 
 of the lowest order in that, since the first quotient 
 figure, by which we multiply, is 2. If the direction 
 of the error of the divisor were not known, the limit 
 of error of the second partial product would also bé 
 a unit of its lowest order, since it would be doubtful 
 whether we ought to carry a 3 or a 4 to its lowest 
 figure, from the product of the rejected figure 7 of 
 the divisor by the second figure of the quotient. As 
 for the remaining partial products, none of their er- 
 rors can be more than half a unit of their lowest 
 order. But the remainder, 60, by which the 7th dec- 
 imal of the quotient is determined, cannot be in error 
 by more than the sum of the errors of the dividend 
 and the partial products preceding this remainder, 
 that is, > +1+1+3= 5. If the true remainder at 
 the foot ought then to be 55 instead of 60, the 7th 
 decimal of the quotient would be 7, but should be 
 followed by other figures; but if the true remainder 
 ought to be 65, the 7th decimal of the quotient would 
 still be 8, though followed by other figures. There- 
 fore the 7th decimal of the quotient actually obtain- 
 ed cannot be wrong by more than a unit of its own 
 order. Hence, a fortzori, the error of the quotient is 
 less than a unit of the 6th decimal place, as required. 
 In fact, from the limit of error now determined, we 
 may reject the 7th decimal of the quotient, either in- 
 creasing the previous one or not, without passing the 
 assigned limit of error. 
 
DIVISION. 4 7 
 
 31. It is plain that if the left hand significant fig- 
 ure of a divisor be the smallest possible, viz: 1, and 
 it be followed by one or more zeros, then when the 
 division has been continued until all the figures of the 
 divisor but the 1 have been rejected, and we are 
 ready to find the next figure of the quotient, an er- 
 ror in the remainder to be used in determining this 
 quotient figure, equal to 7 units of the order of the 
 remainder, will cause an error in the quotient also 
 equal to 7 units of the order of the quotient figure 
 to be found. But by the application of Rule 14 this 
 quotient figure will be of the order next below the 
 mth. In the most unfavorable case possible, the first 
 figure of the quotient will be the largest possible, viz: 
 9; for then the limit of error of the first partial prod- 
 uct may be 45 units of its order, that of the second 
 partial product 1 unit of the same order. If we add 
 to these the possible error of the dividend, } a unit 
 of the same order, we have still room for 8 more par- 
 tial products with errors of 3 a unit each, all in the 
 same direction, before the error of a remainder shall 
 equal 10 units of its lowest order. Hence, if Rule 
 14 is followed, the error of the quotient cannot exceed 
 IO units of the order next below the zth, or I unit 
 of the zth order, except there should be more than 
 10 partial products having errors in one and the same 
 direction, a case evidently not often likely to occur in 
 practice. And it is clear that if the figure of the 
 divisor following the last one to be used by the rule 
 is known correctly, we may reduce the error of the 
 first partial product to less than a unit of its lowest 
 order; and then, if the number of partial products 
 is less than 10, the error of a quotient found by 
 the rule will in such cases be less than Aa/f a unit 
 of the zth order; so that if we wish to reject the 
 
48 APPROXIMATE COMPUTATIONS. 
 
 extra figure of the quotient we can if necessary ad 
 just the previous figure so that the error of the quo- 
 tient shall still be less than a unit of the zth order. 
 
 It is not necessary to state a special rule for fixing 
 the number of decimals to be computed in a pro- 
 posed dividend and divisor so that a quotient shall be 
 correct within a unit of an assigned order; for the 
 application of Rule 14 is sufficient to determine the 
 required number. 
 
 EXAMPLE 27. Compute the expression ——, with 
 a relative error less than zodu07-. 2 
 
 The quotient will be greater than 2, hence, by Rule 
 4, we may make an absolute error = 0.0002. We 
 shall be safe then if we apply Rule 14 as if to find a 
 quotient with a limit of absolute error equal to a unit 
 of the 4th decimal place. We need then, by Rule 
 14, 6 figures each in the divisor and dividend. The 
 division will be as follows: 
 
 
 
 2 82842|> 22145 
 31317 
 28284 
 3033 
 2828 
 205 
 I41 
 64 
 mes 
 7 
 
 There are only 5 partial products preceding the last 
 remainder, and the error of this cannot exceed 4 
 
 
 
 
 
DIEISLON. 49 
 units of the same order, hence the error of the quo- 
 tient must be less than 3 units of the order of its fig- 
 ure 5. If we reject the 5, the answer will be 2.2214, 
 with an absolute error less than 0.0001, and a rela- 
 
 tive error less than s9$095, and, @ fortzorz, less than 
 1 
 T0000: 
 
 2 ; 
 , with 
 
 
 
 EXAMPLE 28. Compute the expression 
 a relative error in the quotient less than yodo07 - 
 
 EXAMPLE 29. Compute the expression 
 0.54674321 =a 
 V 0.0003 
 with a relative error in the quotient less than ys'o0 ° 
 
 
 
 32. PROBLEM 15. A dividend and divisor being given, each 
 to a certain degree of approximation, it is required to assign a 
 superior limit to the absolute error of their quotient. 
 
 RULE 15. Multiply a superior limit of the quotient 
 by the relative error of the divisor, divide the absolute 
 error of the dividend by an inferior limit of the dt- 
 visor, and take the sum of the two results for the re- 
 guired limit. 
 
 Demonstration. It is evident that the most unfa- 
 vorable case, or that in which the error of the quo- 
 tient will be the greatest, is that in which the errors of 
 the dividend and divisor are in opposite directions; 
 for example, the dividend too large and the divisor 
 too small. Suppose then that @ and 6 represent an 
 exact dividend and divisor, a’ and 0 the correspond- 
 ing approximate dividend and divisor, and a and 8 
 their absolute errors. In the most unfavorable case 
 we have 
 
 a ata 
 
 F 38 
 
50 APPROXIMATE COMPUTATIONS. 
 
 / 
 The absolute error of the quotient s would then be 
 
 ata a. a+rba a ap © 
 
 
 
 b—-f b- io — p) ae 
 batap_a@i pia 
 fiat ae Gee (15) 
 
 But Rule 15 will evidently give a superior limit of 
 the last member of this equation. Hence it is a safe 
 rule. 
 
 EXAMPLE 30. Determine the limit of absolute er- 
 4237.5 = 5 
 
 85.846 + 4 
 
 By Rule 15 we may evidently take 
 
 4500 0.004 4 O51 5 - LAs 
 80 80 80 160 
 EXAMPLE 31. Assign limits to the absolute errors 
 
 of the following indicated quotients: 
 20eeA 2 et 0 
 
 0.47328 + 7 
 5 334-725. + 3, 
 8874 
 s 0432 0 ae 
 784.3 = 4 
 In the second of these expressions the error of the de- 
 
 nominator is supposed to be 0, and in the last the 
 error of the numerator is supposed to be o. 
 
 33. It is assumed in Rule 15 that the division of 
 the approximate numbers will be exactly made, or, at 
 least, that if the abridged process of division is em- 
 
 
 
 ror of the quotient 
 
 
 
 
 
 
 
 <10508 
 
 
 
DIVISION. 61 
 
 ployed, care will be taken that the errors arising from 
 the contractions of the process shall be very small in 
 comparison with those due to the errors of the divi- 
 dend and divisor. 
 
 PROBLEM 16. A dividend and divisor being given,each to a 
 certain degree of approximation, it is required to state a rule 
 for dividing by the abridged process, so that the error of the 
 
 quotient due to this process shall be less than that due to the 
 errors of the quantities. 
 
 RULE 16. Begin the division as usual. Then, if 
 the sum of the errors of the dividend and of the first 
 partial product formed does not exceed as many units 
 of the lowest order employed in the dividend as half 
 the number of partial products to follow, add places 
 to the atvidend and divisor until tt does. 
 
 EXAMPLE 32. Calculate by the abridged process 
 the quotient 
 4.35278 + 
 3.14159 +’ 
 and let the error from the abridged process be less 
 than that due to the limits of error of the quantities. 
 
 ole «Jee 6 
 
 4.352780 + 5|3.141590 + 5 
 3141590 + 5]1.385534 + 5 
 I 211190 
 042477 2 2 
 268713 
 2oT Soe 
 17386 
 15708 — 
 1678 
 1571 — 
 107 
 AC ae 
 Dje a7 
 
 
 
2 APPROXIMATE COMPUTATIONS. 
 
 By thus adding a zero to the dividend and divisor we 
 make the limit of error of the dividend and first par- 
 tial product, due to the limits of error of the quanti- 
 ties, 10 units of the lowest order then in the dividend, 
 while the only partial products which are in error 
 from the abridged process are the last four, and the 
 sum of the errors of these does not exceed 2 units of 
 the same order as before, hence the error of the 
 quotient due to the use of the abridged process is 
 much less than that due to the limits of error of the 
 quantities. 
 
 34. We have illustrated in the above example a 
 method by which the limit of the absolute error of an 
 approximate quotient may be determined by an in- 
 spection of the work. By indicating the error of the 
 dividend and of each partial product, it is evident 
 that the sum of all these errors cannot exceed 14 of 
 the lowest order of units in the dividend actually 
 employed. Therefore, since the first figure of the 
 divisor is 3, the error of the quotient cannot exceed 5 
 units of the order of the figure 4 of the quotient, 
 determined by using the last remainder, 13. 
 
 EXAMPLE 33. Perform the division of Example 30, 
 having regard to Rule 16, and determine the limit of 
 error by inspection. 
 
 423795) 4:48 §OAG Cee 
 34338 + 2/4936 £2 
 803 7 
 772, O thers 
 
 a Tat 
 
 258 
 
 55 ic. o 
 
DIVISION. 53 
 
 It will be noticed that after finding what the last 
 figure of the quotient is to be we do not write the 
 corresponding partial product; but we merely con- 
 sider what the quotient figure in the place just found 
 would be if the remainder by which it is determined 
 were diminished or increased by the possible error of 
 that remainder. Thus in the above example, if the 
 remainder 53 were diminished by 8, the 4th quotient 
 figure would be 5 instead of 6; and if the remainder 
 53 were increased by 8, the 4th quotient figure 
 would be 7; but without a closer determination of 
 the limits of error of the partial products we should 
 have to assume that the 7 might be followed by other 
 figures. Therefore if we take 6 for the fourth figure 
 of the quotient, we place the limit of error at 2 units 
 of its order, so as not to understate the error, though 
 in reality we can see that the error could not be more 
 than a trifle over I unit of that order; which is the 
 limit determined by Example 30, supposing the divi- 
 sion were to be made exactly. 
 
 EXAMPLE 34. Calculate the expression 
 7 
 2.74384 £7 
 having regard to Rule 16. 
 
 EXAMPLE 35. Perform the divisions of Example 
 31, having regard to Rule 16, and determine the 
 limits of error by inspection; then compare these 
 limits with those found by Rule 15. 
 
 35. PROBLEM 17. A dividend and divisor being given, each 
 to a certain degree of approximation, it is required to assign a 
 superior limit to the relative error of their quotient. 
 
 RULE 17. Take for the limit of the relative error 
 of the quotient, the sum of the superior limits of the 
 relative errors of the dividend and divisor. 
 
54 APPROXIMATE COMPUTATIONS. 
 
 Demonstration. In the exact expression, equation 
 
 / 
 (15), for the absolute error of at viZen 7 E+ Swe 
 shall make but a very slight error if we substitute 0 
 for 6’. Making the substitution, and dividing by the 
 
 expression for the true quotient, we have the relative 
 a’ ‘ 
 
 error of Pp very nearly, 
 NA Rape 14 
 BOTS _ Bye 
 a Takg Wan (16) 
 b 
 
 Rule 17 will give a superior limit of the right hand 
 member of this equation. Hence it is a safe rule. 
 
 36. It is assumed in Rule 17, as in Rule 15, that 
 the division of the approximate numbers is to be ex- 
 actly performed, or that the additional error from the 
 abridged process is to be very small in comparison 
 with that due to the error of the dividend or divisor. 
 But we have shown in Rule 16 how to make the er- 
 ror from the abridged process less than that due to 
 the errors of the quantities; and it is evident that we 
 may in any case make the error of the abridged proc- 
 ess quite insignificant by adding to the dividend and 
 divisor one or two more places than required by 
 
 Rule 16. 
 
 EXAMPLE 36. Determine by Rule 17 the limit of 
 relative error of 
 WRA2E ocd 
 10.3864 £9 : 
 
 
 
 
 
 and from the relative error and a limit of the quo- 
 
DIVISION. 55 
 
 tient find, by Rule 2, a limit to the absolute error. 
 Then make the division by Rule 16, and determine 
 the error by inspection. 
 
 EXAMPLE 37. Apply the same process to the 
 quantities in examples 30 and 31. 
 
 3%. Besides the general abridged process of divi- 
 -sion already explained, there is a special formula 
 which may sometimes be used with advantage when 
 the divisor is but little greater or less than unity. Sup- 
 
 pose it to be required to compute the quotient 
 
 
 
 ’ 
 
 I 
 @ being any number, and 7x a small fraction. Find- 
 ing two terms of the quotient by algebraic division, 
 and adding the indicated quotient of the remainder, 
 we may write 
 a ax ia 
 opm Qa iy Q( laws) 1 ent 
 If we neglect the last term of this expression, and 
 take a(1—~*) for the quotient we shall make an abso- 
 2 
 
 
 
 
 
 
 
 ax é: 
 lute error equal to : The corresponding rela- 
 
 tive error will be 
 
 ax’ 
 
 10a eee 
 Mulan are Dad 
 a 
 I+2 
 
 Now when ~# is very small, x? will be very much 
 smaller, and in such cases we may take a(1—-+) as 
 
 
 
 
 
 : a : 
 the approximate value of eee By doing so we sub- 
 wer 
 
 stitute a multiplication for a division; and we may if 
 we please easily determine the limit of absolute error 
 
56 APPROXIMATE COMPUTATIONS. 
 
 committed, knowing that the approximate result is 
 too small, and its relative error equal to 2”. 
 
 EXAMPLE 38. Compute by the above method the 
 expression y-y¢77, and determine the limit of error 
 committed. 
 
 For y-a677 We substitute 2(0.9923) = 1.9846. The 
 relative error of this being (0.0077)? < 0.00006, the 
 absolute error cannot be greater than 2(0.00006) = 
 0.00012, (Rule 2); that is, the result 1.9846 is toc 
 small by a little more than a unit of the 4th decima! 
 place. 
 
 
 
 By treating the fraction in the same way as we 
 
 a 
 treated ——_, we may also deduce 
 itz y 
 
 
 
 
 
 
 
 ax 
 me ew naa 
 If we take aA a(i+-z), we make an absolute er- 
 
 ax ; 
 ror equal to : and a relative error equal to 
 —r 
 
 ax’ 
 
 | Broa 
 
 
 
 
 
 — 2 
 a eee 
 
 
 
 = 
 this error being in the same direction as before, and 
 having the same expression. 
 
 EXAMPLE 39. Divide 2 by 0.9923. 
 
 We substitute 2(1.0077) = 2.0154; the limit of the 
 relative error of this result being, as in the last exam- 
 ple, 0.00006, and the absolute error in fact < 0 00012, 
 though if the relative error were quite up to 0.0006 
 
DIVISION. 57 
 
 the absolute error would slightly exceed 0.00012, 
 since the quotient is greater than 2. 
 It is evident then that where x is a small fraction, 
 
 . a : \ 
 in place of -—— we may substitute a(1—-~), and for 
 
 I++ 
 
 yay We may substitute a(1+-), and the results will 
 
 in each case be a little too small, the relative error 
 being +’, and the absolute error being the quotient 
 multiplied by x”. 
 
 EXAMPLE 40. At a temperature, ¢ = 5° centi- 
 grade, and a pressure of I atmosphere, the volume, V, 
 of a mass of hydrogen being 1 litre, what will be its 
 volume, ’,, at zero, under the same pressure ? 
 
 We have from Physics the formula 
 
 V 
 
 Ke 1+74(0.0036613) 
 yo = I e I 
 °  1+5(0.0036613) | 1+0.0183065. 
 
 Say V, = 1X(1—0.0183065) = 0.9817 very nearly. 
 The relative and absolute errors of this result being a 
 little less than (0.02)? = 0.0004, we may be sure that 
 the answer within a unit of the 4th decimal place is 
 0.9820. 
 
 or 
 
 EXAMPLE 41. Suppose the volume of the gas ata 
 temperature of —5° to be 4 litres, what will be its 
 volume at zero? In the formulas of example 40, 
 make ¢= —5, and V=4 
 
SQUARE ROOT. 
 
 38. PROBLEM 18. Any exact number being given, it is re- 
 quired to find its square root by an abridged method, but so 
 that the absolute error of the root shail not exceed a unit of the 
 (2z+1)th order, counting to the right from the highest signifi- 
 cant figure of the root inclusive. 
 
 RULE 18. Employ the ordinary process of extract- 
 ing the square root until n+ 1 significant figures of 
 the root have been found. Form the next trial divisor 
 as usual, and in forming the new dividend bring 
 down only one new figure from the original number. 
 finish the work by dividing this dividend by the trial 
 divisor gust formed, contracting the latter one place at 
 the right after cach quotient figure ts found, and 
 placing the quotient figures in the root, until n addt- 
 tional figures have been found, observing not to tn- 
 crease the last one unless it would be followed by a 
 
 gure at least as large as 7. 
 
 EXAMPLE 42. Compute V10498.59325783, so that 
 the result shall contain nine figures and be exact 
 within a unit of the lowest order retained. 
 
 We put 2x+1 =9, whence ~+1=5. We shall 
 then find five figures by the ordinary process, and 
 four by division. 
 
SQUARE ROOT. 69 
 
 10498.59325783 ( 102.462643 
 I 
 202 ) 0498 
 404 
 2044) 9459 
 81 76 
 20486 ) 12 8332 
 eet 2050 
 20492) 54165 
 40984 
 Lod 
 12295 
 886 
 820 
 66 
 61 
 5 
 
 After finding five figures of the root, the next trial 
 divisor, 20492, is formed as usual; but since the next 
 figure of the root is not to be annexed to this trial 
 divisor before multiplying, we must evidently bring 
 down but one new figure to the dividend. In bring- 
 ing down this new figure, we do not increase it, what- 
 ever it is followed by in the original number, for the 
 contracted process tends to make the result a trifle in 
 excess, as will be seen below. ‘The last four figures 
 of the root are given by division. 
 
 To show that the error of a square root found by 
 Rule 18 is less than the limit assigned in the state- 
 ment of the problem, we may proceed as follows: 
 Since the process of extracting the square root of a 
 number which is partly decimal does not essentially 
 
 
 
 
 
60 APPROXIMATE COMPUTATIONS. 
 
 differ from that in which the number is entire, we may 
 confine the investigation of the principle to the ex- 
 traction of the roots of whole numbers. These roots, 
 however, may be entire, or they may be partly iec- 
 imal. 
 
 Suppose the square root of an entire number to be 
 separated into two parts, the part at the right being 
 made to contain z figures besides the decimal figures, 
 and the part at the left at least 2-+ 1 figures. De- 
 note the relative value of the left hand part by a, and 
 that of the remaining part by &. (By the relative 
 value of the left hand part we mean the value of 
 the z+ 1 figures with z zeros added). If WV be the 
 number whose root we are considering we shall then 
 have, 
 
 N= (at bv =a + 2ab + &. (17) 
 
 If we employ the ordinary process of extracting 
 the square root of the number J, until we have found 
 the part a, and after forming as usual the next divi- 
 dend bring down to this the remaining periods of the 
 number, this dividend will evidently be equal to 
 N—a’. Finding the value of V — a from equation 
 (17) and denoting it by R, we have 
 
 R= 2ab+ &, 
 Ie b 
 
 whence Oe ree es (18) 
 22. OR 
 
 Let us determine a superior limit of the value of 
 2 
 
 this last term, ae Since & has but xz figures above 
 a 
 
 the decimal point, 0’ cannot have more than 2n figures 
 above that point, whereas a, with its relative value, 
 must have at least 2n + 1 places above the decimal 
 point; and 2a@ must hence be more than twice as 
 
 ee 
 great as 0. Therefore the value of ae less than 
 
 atl 
 
SQUARE ROOT. 61 
 
 half a simple unit. It is thus evident from equation 
 (18) that if the dividend R were exactly divided by 
 2a, and the quotient taken for the part 4 of the square 
 root, the result would be too large, but by less than 
 half a untt. 
 
 In employing the abridged process of division for 
 
 finding the value of the quotient = the only addi- 
 
 tional error introduced is that due to the rejection of 
 some figures from a few partial products. <A limit to 
 the sum of these errors may be determined as follows: 
 Since the part a, without regard to its relative value, 
 contains z + 1 figures, the next trial divisor, which is 
 2a, will also contain at least z+ 1 figures. Now for 
 each of these z + 1 figures in the divisor a new figure 
 of the root can be found, the last one being therefore 
 one place to the right of the 2%-+ 1 figures to be 
 found by Rule 18. The error of the final remainder 
 by which this extra figure would be determined, due 
 simply to the successive cutting off of 2 figures from 
 the trial divisor, will not exceed z times the half of 
 one of the lowest order of units in this remainder. 
 And if the divisor 2a does not contain more than 
 m+ 1 figures, the left hand one must be at least 
 equal to 2; so that the error of the extra figure of the 
 root due to the cause we are now considering will not 
 exceed z times a quarter of a unit of its own order. 
 Therefore, if the total number of figures to be found 
 by Rule 18 do not exceed 13, z being then 6, if the 
 redundant figure be annexed, the total error of the 
 root will not exceed 7 units of the order of this figure, 
 allowing 5 for the error due to disregarding the part 
 
 3 in equation (18), 14 for the errors of the contract- 
 
 ed division, and 4 for a possible error in adjusting 
 
62 APPROXIMATE COMPUTATIONS. 
 
 this extra figure with reference to what would follow. 
 Hence, when this redundant figure would be 7 or 
 over, if we reject it without increasing the previous 
 figure the result will be too small, and if we reject it 
 by adding 1 to the previous figure the result will be 
 too large; but in neither case will the error be greater 
 than a unit of the lowest order then retained, which 
 will be the (27% + 1)th. In the great majority of ex- 
 amples the error of a square root found by Rule 18 
 will be less than fa/fa unit of this order; and the 
 direction of the error will also frequently be apparent, 
 
 bearing in mind that the exact quotient — would 
 make the result too large. at 
 
 EXAMPLE 43. Compute vV0.0006620567113, so 
 that the error of the result shall not exceed a unit of 
 the 8th decimal place. 
 
 We see that the first significant figure of the root 
 will be in the 2d decimal place, hence the error is re- 
 quired to be less than a unit of the order of the 7th 
 significant figure, giving z+ I = 4. 
 
 0.0006620567113 (0.025730462 
 a. 
 45 ) 262 
 oy ts 
 507 ) 3705 
 3549 
 5143 ) 15667 
 tes 
 5140) 2381 
 2058 
 323 
 309 
 14 
 
 
 
 
 
SQUARE ROOT. 63 
 
 After finding four significant figures of the root, 
 the next dividend, after annexing one new figure 
 from, the number, will not contain the trial divisor, 
 hence a zero is placed in the root, and the next figure 
 of the root is found by cutting off one figure of the 
 divisor. We have found one figure more in the root 
 than was required, and if we reject this redundant 
 figure we shall know that the result will be correct 
 within less than half a unit of the 8th decimal place. 
 
 EXAMPLE 44. Compute v2, so that the relative 
 error shall be less than a ten-millionth. 
 The allowable absolute error may be taken 
 1.4X 1 
 IQ0OOO0000 
 
 and since the first figure of the root is in units place 
 we may put 27 + 1 = 8, whencex+ 1=4 5. Where 
 z+ 1 happens to come a mixed number like this we 
 evidently have to take for 2+ 1 the next largest 
 whole number, in order to bring it under the rule. 
 
 Find therefore in V2 five figures by the ordinary 
 process, and the rest by contraction. 
 
 = 0.00000014, 
 
 EXAMPLE 45. Compute vs and V150, with twelve 
 decimals in each. 
 
 39. If it be required to extract the square root of 
 a number equal to unity plus or minus a small fraction, 
 we may at once write the root equal to unity plus or 
 minus one-half the difference between that and the 
 number; and if by this process # zeros follow the 
 decimal point in the root, we may get the root with 
 a limit of absolute error of half a unit of the order of 
 the 2zth decimal place. For if MV = (1 + 0)’, then in 
 equation (18), neglecting its last term we should have 
 
64 APPROXIMATE COMPUTATIONS. 
 
 jan tc la 
 
 have stated is evident. For if d is a decimal fraction 
 having z zeros following the decimal point, 2? would 
 2 
 
 That the limit of error is what we 
 
 
 
 have 27 such zeros, and therefore ROE the neglected 
 
 part of equation (18), would be less than half a 
 unit of the 27th decimal place. 
 
 Thus we may write V1.01239 = 1.006195, within 
 0.00005 ; and since we know from equation (18) that 
 _ the error committed makes the result too large, we 
 may take either 1.0061 or 1.0062 as the result within 
 0.0001. 
 
 Also we may write vVo.9821 = VI—0.0179 = 
 I—0.00895 = 0.99105, within 0.00005; and since 
 this result is also too large we may reject the last dec- 
 imal, and the result will be 0.9910, with an error less 
 than half a unit of its lowest order. To show that a 
 result found like this 0.99105 is too large, observe 
 that if V = (1—4)’, then equation (18) might be writ- 
 
 2 
 
 I—NV_, 6b : 
 IPE so that neglecting the last term 
 
 
 
 ten 06 = 
 
 of this equation makes 6 too small, and therefore 
 (1—4) too large. 
 
 EXAMPLE 46. Write the values of V1.01719 and 
 V0.98911 within 0.00005. 
 
 4Q. PROBLEM 19. Any approximate number being given, 
 with an absolute error not exceeding an assigned limit, it is re- 
 quired to assign a limit to the absolute error of the square root 
 of the number. 
 
 RULE 19. Zake for the limit of the absolute error 
 of the square root, the absolute error of the number, 
 divided by an inferior limit of twice the said square 
 root. 
 
SQUARE ROOT, 65 
 
 Demonstration. Suppose that a’? represents any 
 approximate number, and that @ =a+a is the 
 square root of it when precisely taken, a being the 
 error of the root a’ due to that of the number a’. 
 We have a’ = a + 2aa + a’, or, by transposing and 
 neglecting the very small quantity a’, we have, as in 
 equation (10), the following nearly exact equation, 
 
 a'*— a* = 24a, 
 ai? ait a? 
 
 whence a= Wet (19) 
 
 The numerator of the last member of equation 
 (19) is the absolute error of the number a@’’, and the 
 equation gives a little larger value of a than the true 
 value, hence Rule 19 will give a safe limit for a, pro- 
 vided the root of the approximate number is exactly 
 taken, or that the new error introduced by the con- 
 tracted method of finding the root be made insignifi- 
 cant, and this is in practice always easy. For, after 
 finding by Rule 19 as small a limit as is convenient, 
 to the error of the root if precisely taken, it will be 
 easy to see what value z + 1 must have in applying 
 Rule 18, so that the error due to the abridged process 
 shall be less, or as many times less, as we please. 
 
 EXAMPLE 47. Given zw = 3.14159265, so that the 
 absolute error, without stating its direction, is less 
 than half a unit of the 8th decimal place, determine 
 by Rule 19 a limit to the error of its square root if 
 precisely taken, and then find the root by the abridged 
 process, but let the new error introduced by that 
 process be less than a tenth of the limit found by 
 Rule 19. 
 
 By Rule 19 we may take for the limit of error 
 ©.000000005 
 
 2X17 —— < a.0000000015. The conditions of the 
 
66 APPROXIMATE COMPUTATIONS. 
 
 problem will then evidently be satisfied if we apply 
 Rule 18 so that the error of the process shall be less 
 than a unit of the roth decimal place, and since the 
 root has one entire figure, we have 2v+1= 11, and 
 #+1=6. Finding 6 figures by the ordinary proc- 
 
 ess, and 5 by division, we get V7 =1.77245 38499 + 16. 
 
 41. It results from equation (19) that if the limit 
 of absolute error of an approximate number be a unit 
 of the zth order, counting down from the highest sig- 
 nificant figure of the number, the square root may be 
 found with an error not exceeding 2 units of the zth 
 order, counting down in the root also from its high- 
 est significant figure; and unless the first significant 
 figure of the root is 3 or 4, the limit of error may be 
 made a szzg/e unit of the same order as before. For 
 in equation (19) suppose that a has z figures above 
 the decimal point, and that a’? — a’ equals a simple 
 
 : I ‘ : 
 unit; then a= be The largest value of q@ in this 
 a 
 
 equation will correspond to the smallest value of a; 
 but if a’ has z figures above the decimal point, a will 
 have either 3(z+1) or 32 figures above that point, 
 and if it has only 97 then the highest one must be at 
 least equal to 3; so that a will not exceed unity di- 
 vided by a number containing 42 figures, the first of 
 these being at least equal to 6. To take a definite 
 case, suppose an entire number, a’’, to contain 8 fig- 
 
 ; met 
 ures, and its error to be a simple unit; then a = — 
 2a 
 
 will be less than gpa <.0.0002. But the root will 
 have 4 figures besides the decimals; hence its error 
 will be less than 2 units of the 8th order from its 
 highest figure. And if the first figure of the root in 
 
SQUARE ROOT. 67 
 
 I 
 this example were as large as 5, then a= oF would 
 a 
 
 not exceed yoa00 = 0.0001. In making # an even 
 number we have taken the most unfavorable case. /Z 
 follows then that when an approximate number con- 
 tains n significant figures, and ts exact within a unit 
 of its lowest order, we may find the square root with 
 an error not excecding a unit of the order of tts nth 
 significant figure, except when the first significant 
 jigure of the root ts 3 or 4, and in those cases the limit 
 of error need not exceed 2 units of the same order. In 
 the majority of cases we may find as many exact sig- 
 nificant figures in the square root of an approximate 
 number as there are in the number. (Compare ex- 
 amples 47 and 50). 
 
 4%. PROBLEM 20. Any approximate number being given, 
 with a relative error not exceeding an assigned limit, it is re- 
 quired to assign a limit to the relative error of the square root 
 of the number. 
 
 RULE 20. Jake for the limit of the relative error 
 of the square root, one-half of the limit of the relative 
 ervor of the number. 
 
 Demonstration. Dividing both members of equa- 
 tion (19) by a, we have the nearly exact equation, 
 i em hg 2 
 
 ic oll Sua aa 0) 
 
 ae—_7 
 The factor - 2 being the relative error of the ap 
 
 
 
 proximate number a’’, and the slight error committed 
 in finding equation (19) being on the side of safety, 
 Rule 20 is evidently a safe one. It is also evidently 
 in harmony with the principles of Art. 28, 
 
 EXAMPLE 48. The absolute error of the number 
 
68 APPROXIMATE COMPUTATIONS. 
 
 85.4374 being supposed less than 4 of its lowest units, 
 assign a limit to the relative error of its square root, 
 and from this limit assign the limit of absolute error 
 of the root. 
 
 It is hardly necessary to say that Rule 20 assumes 
 that if the root is to be found by the abridged proc- 
 ess the new error introduced will be rendered insig- 
 nificant, or else allowed for. 
 
 43. Independently of Rules 18 and 1g it is easy 
 to find by inspection of the work in any example a 
 limit to the possible error committed in the square 
 root. We give an example or two. 
 
 EXAMPLE 49. Compute Vv 15. 
 15 (3.87298335 +3 
 9 
 68) 600 
 ol 
 797 ) 5600 
 5369 
 7742 ) 23100 
 15484 
 
 
 
 69704 + 
 6456 
 6107). 
 259 
 232 + 
 ey 
 PW 
 PD 2 
 After finding 4 figures of the root, and forming the 
 next trial divisor, 7744, if, in beginning the contrac- 
 
SQUARE ROOT, 69 
 
 tion by making the partial product of this divisor by 
 the root figure 9, we carry to that product 8 units of 
 its lowest order, which would be carried if the 9 were 
 annexed to the trial divisor, the partial product, 69704, 
 will be wrong by less than half a unit of its lowest or- 
 der; and if in getting the next partial product we 
 multiply by the next root figure 8 as if the figure 4 
 to be rejected in the divisor were a 6, that being the 
 nearest what it would be if the part of the root al- 
 ready found were doubled, the partial product 6197 
 is also correct within half a unit of its lowest order. 
 The other figures of the divisor being successively re- 
 jected, the final remainder 4, by which the figure 5 
 of the root is determined, cannot be wrong by more 
 than 2 units of its order; hence by the principle of 
 Art. 34 this last figure of the root cannot be wrong 
 by more than 3 of its units. 
 
 EXAMPLE 50. Compute V75.438175 + 2. 
 7% .438175 +2 (8.6855152 +2 
 6 
 
 
 
 ace: 
 166 ) 11 43 
 9 96 
 1728 ) 1 4781 
 I 3824 
 173605) 95775 
 86825 
 17370 ) 89500 
 86853 — 
 "2647 
 UE Se 
 QgIO 
 869 — 
 
 A4lI tear 
 
70 APPROXIMATE COMPUTATIONS. 
 
 The partial products in the contractions are treated as’ 
 in example 49. Only two of these have their errors 
 in the same direction; therefore the error of the final 
 remainder, 41, cannot be more than 21 of its units, al- 
 lowing 20 for the original error of the number. 
 Hence the last figure which we: have found in the 
 root cannot be wrong by more than 2 of its own or- 
 der of units. The limit of absolute error of the root 
 in this example if found by Rule 19 would be 
 9.000992 < 9,.0000002. 
 
 EXAMPLE 51. Compute V314.75435 + 6 so that 
 the error of the result shall be mainly due to that of 
 the number, and find the limit of the error by inspec- 
 tion of the process, as in examples 49 and 50. 
 
CUBE ROOT. 
 
 44, PRrop._eM 21. Any exact number being given, it is re- 
 quired to find its cube root by an abridged method, but so that 
 the absolute error of the root shall not exceed a unit of the 
 (2~+ 1)th order, counting to the right from the highest signifi- 
 cant figure of the root inclusive. 
 
 RULE 21. Employ the ordinary process of extract- 
 ing the cube root until n+1 figures of the root have 
 been found. Form the next trial divisor as usual, 
 except that no places are added to tt at the right, and 
 bring down to the new dividend no new figures from 
 the number. Then, of the first figure of this trial 
 divisor be less than 5, rezect all but the n+2 left hand 
 significant figures of the divisor, otherwise all but 
 n+1, aud reject from the dividend as many figures 
 less one as ave thus rejected from the divisor. Then 
 divide by the contracted method of division, placing 
 the quotient figures in the root, and reject all the 
 figures of the root that would follow the (2n+1)th, 
 without increasing the last one retained, whatever be 
 the value of the one which would follow. 
 
 EXAMPLE 52. Compute V 25, so that the absolute 
 error shall be less than a unit of the roth decimal 
 place. 
 
72 APPROXIMATE COMPUTATIONS. 
 
 25(2.9240177382 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1200 17000 
 69 Mae oe 16389 
 18 81 252300 611000 
 872 1744 254044 | 508088 
 Ciel A. 425579200 | 102912000 
 8764. 3505625614256 102457024. 
 8 16|256493280000 454976000000 
 877201 Neate 256494157201 
 112564950, 34403| 19848184,2799 
 17954052 
 1893532 
 1795465 
 98067 
 76949 
 21iiS 
 20520 
 598 
 
 The root having one entire figure, we have 27+1 
 =I1, whence x+1=6. After finding six figures of 
 the root by the ordinary process, and forming the 
 next trial divisor, viz., 256495034403,* observe that 
 if the divisor were to be completed as usual, it would 
 have but 2 places added to it, while the dividend 
 would have 3, so that in the contracted division we 
 must discard at first one more figure in the trial 
 divisor than in the dividend. Rejecting, therefore, 
 all but 2+2 figures of the divisor, and rejecting one 
 less in the dividend than in the divisor, we finish the 
 work by contracted division. 
 
 * Students who are not used to finding the cube root in the way we have begun the 
 work, will understand that this trial divisor is simply three times the square of the 
 first six figures of the root. 
 
CUBE ROOT. a3 
 
 To examine the limit of error of a cube root found 
 by Rule 21, we may proceed as in the case of square 
 root, by considering only the case in which the num- 
 ber is an entire number, but its root either entire or 
 partly decimal. If M represent the number whose 
 cube root is to be taken, 4 the right hand part of the 
 root, containing z figures besides the decimal part, 
 and a the relative value of the left hand part of the 
 root, with at least #+1 figures, besides the x places 
 corresponding to the part 4, we shall have 
 
 N=(a+by= a+ 3a°)0+3a0°+ 6’, 
 whence, V—a’= 30° + 3a0?+ 0%. 
 
 The first member of this equation is evidently the 
 remainder which would be left by the ordinary pro- 
 cess of extracting the cube root, after the part a had 
 been found, if all the remaining periods of the 
 number were brought down to the new dividend. 
 Putting R for M—a’, and transposing and dividing, 
 
 we obtain 
 i Nel ie 
 ee a) (21) 
 aes 
 To determine the greatest possible value of the last 
 
 2 3 
 
 term of this equation, viz., e ae ) observe that 0° 
 e324 
 
 
 
 has at most 22 figures at the left of the decimal 
 
 point, while @, with its relative value, has at least* 
 2 
 
 b 
 2n-+1 such figures; hence, — can never be greater 
 a 
 
 than 1, and will generally be considerably less than 
 
 that. Also 6° cannot have more than 37 figures at 
 
 the left of the decimal point, while 3@* cannot have 
 
 less than 2(27+1)—I = 4v+1 such figures; hence 
 
 3a must have +1 more figures than 6° has. The 
 3 
 
 ereatest value of - will evidently then be when z is 
 a 
 
 4 
 
74 APPROXIMATE COMPUTATIONS. 
 
 the smallest possible, viz., 1. If z=1, the value of 
 6 cannot exceed 10, nor that of a be less than 100; 
 
 
 
 
 
 i: 
 hence, -—; < Beinn i. We therefore have, in the 
 3a 30000 +=30 
 Ne eT 
 most unfavorable case possible, (- +—) < (: +2), 
 a 3a 30)": 
 
 It is easy to see, then, that in all ordinary cases the 
 
 value of this whole term will be considerably less than 
 
 1. If we even make @ as large as 99, and make a the 
 
 smallest possible for that value, viz., 10000, it may 
 3 
 
 2 
 
 b b 
 be proved by computation that ae a will be less 
 
 than I. 
 
 Hence, from equation (21), if the remainder R, 
 left after the ordinary process of finding z+1 figures 
 of the cube root, were exactly divided by 3a’, which 
 is the trial divisor for finding the next root figure, 
 the quotient would be in excess of the true value of 
 the part 4 of the root, but, in all ordinary cases, the 
 error would be considerably less than a unit of the 
 (2z+1)th order, counting down from the highest sig- 
 nificant figure of the root inclusive. And if the num- 
 ber of figures retained in the divisor be not less than 
 Rule 21 requires, the new error from contracting the 
 division will be insignificant. Therefore, with the 
 additional security that we correct a part of the error 
 by rejecting all the fizures of the root which would 
 follow the (2%+1)th, whatever be their magnitude, 
 we may accept Rule 21 as giving the result safely 
 within the assigned limit of error. 
 
 EXAMPLE 53. Compute 0.009, with an absolute 
 error less than 0.000000001. Also 5, with 10 deci- 
 
 mals, and W130, with 9 decimals. 
 For the first of these, make 2%+1=9, whence 
 ibis; 
 
 
 
 
 
CUBE ROOT. 75 
 
 0.009 (0.208008382 
 0.008 
 0.120000 | 1000000 
 0.608 4864/0.124864 998912 
 
 64/0.129792,0000] 1088000 ,000 
 1038336 
 49664 
 38938 
 10726 
 10383 
 343 
 
 4. If it be required to extract the cube root of a 
 number equal to unity plus or minus a small fraction, 
 we may at once write the root equal to unity plus or 
 minus one-third the difference between that and the 
 number; and if by this process x zeros follow the 
 decimal point in the root, we may write z more 
 figures of the root, the limit of absolute error being 
 a unit of the order of the last of these, or at most a 
 unit and one-thirtieth. For if V=(1+0)’, then, in 
 equation (21), neglecting its last term, we should have 
 
 cenit at , 
 & =at ——. That the limit of error is what we have 
 
 
 
 
 
 stated is evident from the application of Rule 21. 
 For, if we were finding as usual the cube root, say of 
 1.002, we should have the figure 1 of the root, and 
 the three zeros which would follow, by the ordinary 
 process, and by Rule 21 we could find three more 
 figures of the root by division with the next trial 
 divisor, which would evidently be 3, followed by 
 zeros. If V=(1+0)’, we should reject all the figures 
 of the result after the 27th decimal, without increas- 
 ing this, whatever it would be followed by. Thus 
 
 we may write 4/1.02147 = 1.0071, within 0.0001. 
 
 
 
76 APPROXIMATE COMPUTATIONS. 
 
 We may also write 4/0.97643 = /1—0.02357 = 
 I—0.0079 = 0.9921, Within 0.0001. “ObDservemana: 
 if, as in this case, V = (1—8)’, equation (21) would 
 
 
 
 —NV O° 
 become 0= : 3 °F (“—"), so that neglecting the last 
 
 term here tends to make 6 too small, and therefore 
 the final result (I—d) too large; hence, when 
 N=(1— 6)’, if we find in @ in all 2% decimals, no fol- 
 lowing ones must be rejected, without increasing by 
 I the last one retained in J, as just illustrated in 
 
 0.97643. 
 
 EXAMPLE 54. Write the values of 471.0157 and 
 40.98 within 0.0001. 
 
 The above method may be extended to the ex- 
 traction of the zth root of a number equal to unity 
 
 plus or minus a small fraction ; but the limit of error 
 increases slightly as 7 increases. 
 
 4G, PROBLEM 22. Any approximate number being given, 
 with an absolute error not exceeding an assigned limit, it is 
 required to assign a limit to the absolute error of the cube 
 root of the number. 
 
 RULE 22. Take for the limit of the absolute error 
 of the cube root, the absolute error of the number, 
 divided by an inferior limit of three times the square 
 of the said cube root. 
 
 Demonstration. Suppose that a” represents. any 
 approximate number, and that a’=a+a is the cube 
 root of it when precisely taken, a being the error of 
 the root a’ due to that of the number a”. We have 
 
 a’*—a’+3a°’a+ 3aa'+a’. 
 In finding the value of a from this equation, we 
 shall make a very slight error on the side of safety, by 
 
 neglecting the last two terms. We thus have, as in 
 equation (11), very nearly @°—a’=3a’a; whence 
 
CUBE ROOT. 77 
 
 Py hee 
 
 a= 
 3a" 
 
 (22) 
 
 The numerator of the last member of this equation 
 is the absolute error of the number a”; and Rule 
 22 substitutes for the denominator a little smaller 
 value; hence, if the cube root of the approximate 
 number be exactly found, or if the error from the con- 
 tracted process of finding the root be made inconsid- 
 erable, Rule 22 will give a safe limit of the total error 
 of the root. In practice we may find, by Rule 22, 
 as small a limit as convenient, and then easily see 
 what value z+1 must have in applying Rule 21, so 
 that the possible new error from the contracted pro- 
 cess of finding the root shall be small in comparison 
 with that due to the original error of the number. 
 
 EXAMPLE 55. If we take V 26=5.09, we make an 
 absolute error less than 0.01. Determine by Rule 22 
 
 a limit to the absolute error of 4/5.09+7, and then 
 find the root by the abridged method, but so that the 
 new error introduced shall not exceed a tenth of that 
 due to the error of 5.0947. 
 
 By a little trial we may see that the cube root will 
 exceed 1.7. The square of this is 2.89, and this mul- 
 tiplied by 3 gives 8.67. We may take then for the 
 required limit ®%°1—0.00125. -In order to make the 
 error from the abridged process of finding the root 
 less than a tenth of this, we may work then as if the 
 limit were 0.0001 ; and since the root will have one 
 entire figure, we have 2”%4+1=5. Find, therefore, 
 
 three figures of 4/5.09 by the ordinary method, and 
 two by division, and the result will be the value of 
 /20, within less than 14 units of thousandths place. 
 
 
 
 47. It results from equation (22), that, if the limit 
 of absolute error of an approximate number be a unit 
 
78 APPROXIMATE COMPUTATIONS. 
 
 of the zth order, counting down from the highest 
 significant figure of the number, the cube root may 
 be found with an error not exceeding 2 units of the 
 zth order, counting down in the root also from its 
 highest significant figure ; and unless the first signifi- 
 cant figure of the root is 4 or 5, the limit of error 
 may be made a szzg/e unit of the same order as before. 
 For in equation (22) suppose that a* has z figures 
 above the decimal point, and that a°—a’* equals a 
 
 
 
 : : I . 
 simple unit, then a= = The largest value of a@ in 
 
 this equation will correspond to the smallest value of 
 a. But if a’ has z figures above the decimal point, @ 
 will have either $(z+2) or 4(z+1) or 4 figures 
 above that point, the latter being the most unfavor- 
 able case. And if a has just 4 figures above the 
 decimal point, the left hand one must be as large as 
 
 4, followed by 6, since 100 > 4.6. To take a defi- 
 nite case, Suppose an entire number, a’, to contain 
 6 figures, and its error to be a simple unit. Then, 
 since the cube root a would be greater than 46, we 
 have 3a” > 6000; hence, a < gq4gy < 0.0002. But 
 the root a will have 2 figures besides the decimals ; 
 hence its error will be less than 2 units of the 6th 
 order from its highest figure. And if the first figure 
 of the root here were as large as 6, we should have 
 3a° > 10000, and a would therefore be less than 
 0.0001. Jt follows, then, that when an approximate 
 number contains n significant figures, and ts exact 
 within a unit of its lowest order, we may find the 
 cube root with an error not exceeding a unit of the 
 order of its nth significant figure, except when the first 
 significant figure of the root ts 4 or 5, and in those 
 cases the limit of error need not exceed 2 units of the 
 same order. In the majority of cases we may find 
 as many exact significant figures in the cube root of 
 
CUBE ROOT. 79 
 
 an approximate number as there are in the number. 
 (Compare Example 58, &c.) 
 EXAMPLE 56. Determine, by Rule 22, the limit of 
 
 absolute error of 4/0.000136425+7, and then find the 
 root by the abridged process, making the error of the 
 process, however, not more than a tenth of the limit 
 
 so determined. Do the same with 4/365.046+. 
 
 
 
 48, PrRopLEM 23. Any approximate number being 
 given, with a relative error not exceeding an assigned limit, it 
 is required to assign a limit to the relative error of the cube 
 root of the number. 
 
 RULE 23. Take for the limit of the relative error 
 of the cube root, one-third the limit of the relative 
 _ error of the number. 
 
 Demonstration. Dividing both members of equa- 
 tion (22) by a, we have 
 eh en lien o — 2. 
 
 meee a 
 3 
 
 3 
 a*—a,. 
 The factor meyer being here the relative error of 
 
 
 
 the approximate number a”, the limit of the relative 
 error of the cube root a’ may evidently be taken equal 
 to one-third that of the number, provided the root of 
 the approximate number be precisely found, or the 
 new error in finding it be made insignificant. 
 
 Rule 23 evidently agrees with the principles of 
 Art. 28. And it is clear that we may extend the 
 method to aroot of any index, avd take for the limit 
 of the relative error of the nth root of an approxt- 
 mate number, the nth part of the limit of the relative 
 error of the number. 
 
 EXAMPLE 57. Assign a limit to the relative error 
 
 of 4/842.731-.6, and from this limit find, by Rule 2,a 
 limit to the absolute error of the result. Also find 
 
80 APPROXIMATE COMPUTATIONS. 
 
 this latter limit by Rule 22, and compare the two an- 
 swers. 
 
 49. Byamethod similar to that of Art. 43, we 
 may determine the possible error ofa cube root by in- 
 specting the possible error of the last remainder. 
 
 EXAMPLE 58. Compute 4/1272.4386+7 as closely 
 as the limit of error allows. 
 
 Assuming that we can find eight figures in the 
 root, we will find five before beginning the contracted 
 division. 
 
 
 
 1272.4386+7 (10.836248 + 7 
 I 
 
 
 
 
 
 
 
 
 
 30000 272438 
 308 2464) 32404 | 2bort2 
 16 64|3499200 12726600 
 3243 9729|3508929 | 10526787 
 6 9|351866700 | 2199813000 
 32496 194976|352061676 | 2112370056 
 36.35225 0688 | 87442,944 
 Bibra e! <7: 
 16991 
 14090 + 
 2901 
 2818 + 
 83° 707 
 
 The final remainder, 83, is liable to error of 700 units 
 of its lowest order, from the original error of the num- 
 ber. Allowing 3 units of the same order for the re- 
 jected parts of the partial products, and adding the 
 83 to its possible error, 703, the quotient of the sum 
 by 352, the last divisor employed, will evidently be 
 but a trifle over 2 units of the same order as the last 
 figure 8 of the root, found by the same divisor. 
 The error due to neglecting the part of the divisor 
 
CUBE ROOT. SI 
 
 2 3 
 
 corresponding to ao = is, by Rule 21, less thana 
 
 tenth of a unit of the same order; hence the total 
 error of the root can be but little over two units of 
 that order. We call the limit 3, so as not to under- 
 state it. The limit, if found by Rule 22, would be 
 just about 2 of these units. 
 
 EXAMPLE 59. Compute 4/752.43275438 as closely 
 as the limit of error allows, and determine the limit 
 by inspection. 
 
 * 
 
 43 
 
LOGARITHMS AND TRIGONOMETRIC 
 FUNCTIONS. 
 
 $0. Ifwe examine a table of common logarithms, 
 we shall notice that the greatest difference between 
 the logarithms of any two consecutive numbers, of 
 figures each, is less than 5 units of the order of the 
 ath decimal of the logarithm, and that the least dif- 
 ference of any two such logarithms is about half of 
 one of these units. For example, the logarithms of 
 1000 and 1001 are respectively 3.000000 and 3.000434, 
 while the logarithms of 9998 and 9999 are 3.999913 
 and 3.999957. On the other hand, the greatest dif- 
 ference between any two numbers corresponding to 
 two logarithms that differ by a unit of the zth order 
 of decimals, is less than 3 units of the order of the 
 uth significant figure of the numbers, and the least 
 difference of any two such numbers is about a quar- 
 ter of a unit of the same order. For example, the 
 numbers corresponding to the logarithms 3.999800 
 and 3.999900 are 9995.4 and 9997.7, while the num- 
 bers corresponding to the logarithms 3.000000 and 
 3 OCCOTOO are 1000 and 1000.23. 
 
 The limits of difference stated above do not take 
 into account the slight inaccuracy which will some- 
 times occur in finding by interpolation logarithms or 
 numbers intermediate between those given directly 
 in the tables. But from the examples it is evident 
 
TRIGONOMETRIC FUNCTIONS. 83 
 
 that in finding the logarithm of an approximate num- 
 ber, an error in the number equal to a unit of the 
 order of its zth significant figure will be liable to 
 make the zth decimal of the logarithm uncertain ; 
 and an error in a logarithm equal to a unit of the 
 ath order of decimals will be liable to make the wth 
 significant figure of the number found from the loga- 
 rithm uncertain. Hence, if computations are made 
 with the ordinary six-figure logarithms, the result 
 cannot in general be depended on as accurate beyond 
 the 5th significant figure, though if the data to begin 
 with are exact the 6th significant figure of the result 
 may not be far out of the way. 
 
 But since it isclear from the examples given that 
 the amount of uncertainty in the logarithm of an ap- 
 proximate number consisting of z figures will vary 
 very much according to the value of the highest sig- 
 nificant figure of the number, it is not easy to givea 
 general rule which will always determine the smallest 
 obtainable limit of error in the result of a computa- 
 tion of approximate quantities by means of loga- 
 rithms. This limit is, however, easily determined in 
 any special example by indicating in the work the 
 possible error of each step of the computation, which 
 may be readily ascertained by observing what change 
 would occur in each logarithm taken from the table, 
 for a change in the approximate number equal to its 
 possible error, or what change would occur in a num- 
 ber to be found from an approximate logarithm, for 
 a change in this logarithm equal to its possible error. 
 The same method is evidently applicable with the 
 logarithms of trigonometric functions. 
 
 EXAMPLE 60. Given one side of a triangle a= 
 3500+ 7 feet, and the adjacent angles, B = 65°30’+ 
 30” and C = 85°30'+30”, compute the side 6 by 
 means of logarithms, and assign the limit of error. 
 
84 APPROXIMATE COMPUTATIONS. 
 
 We have the formula d = qsin B 
 sin 
 the angle 4 will be 29°+1’, and finding the logarithms, 
 
 we have: 
 
 log a .. (3500+7).. . 3-544068+ 725 
 log sin B (65°30’4307) . 9.959023+ 29 
 a.c.log sin A (29° + OL’). . 0.314429+ 229 
 log 6 (6569.3+5-9) . 3.817520+353 d=66 
 BIO. ae 
 Se) 
 
 We place at the right of each logarithm the change 
 which the tables would give, if the quantity whose 
 logarithm we are taking were changed by the 
 amount of its possible error. In looking out the 
 number for 4, corresponding to the sum of the 
 logarithms, we divide the possible error of this sum, 
 viz., 383, by the tabular. difference 66, giving the 
 possible error of 0 a little less than six feet. 
 
 Observing that 
 
 
 
 EXAMPLE 61. Compute in the same way the side ¢ 
 of the same triangle. 
 
 4. We may employ a like method in working 
 examples by means of natural trigonometric 
 functions. 
 
 EXAMPLE 62. Find the side 4, of example 60, by 
 means of natural functions, and assign its limit of 
 error. 
 
 We have, 
 
 nat sin (65°30’+ 30”)=o-909964 7 
 Multiply by a reversed, 7£ 0053 
 
 2729088+ 21 
 
 454098+ 4 
 
 O+ QI 
 
 3184°86-L 776 
 
TRIGONOMETRIC FUNCTIONS. 85 
 
 Divide by nat sin(29°+ 1’)=0-484814 26 
 3184:864776 | 048481426 
 2908864756 | 6569:346:0 
 27600 
 24241-- 16 
 3359 
 29094 2 
 450 290 
 436 
 14 
 When we reach the remainder 450, by which the 
 units figure of the quotient is obtained, we see that 
 the possible error of this remainder has amounted to 
 290 units of its lowest order. Dividing this possible 
 error by the same portion of the divisor that was 
 used in finding the units figure of the quotient, we 
 Have, as the limit of error of the resujt, 6 feet, 
 which only differs by a tenth of a foot from the limit 
 of error when the computation was made by log- 
 arithms, while the result itself is the same. It is 
 thus seen that natural functions with five decimals 
 give nearly the same precision as logarithmic func- 
 tions with six decimals. 
 EXAMPLE 63. Compute in the same way the side 
 c of the same triangle, and compare with the result 
 of example 61. 
 
 
 
COMPLEX COMPUTATIONS. 
 
 32. Having explained the elementary processes 
 of approximate computations, we will now consider 
 the casein which several of these processes are to be 
 employed in a single problem. We will first take ex- 
 amples in which the quantities proposed may be found 
 as accurately as we please, and afterwards give a few 
 in which some of the quantities are known with only 
 a limited degree of accuracy. 
 
 PROBLEM 24. Any complex monomial being proposed, 
 whose value is required with an absolute error not exceeding 
 an assigned limit, and whose factors may each be taken as ac- 
 curately as we please, it is required to assign an allowable limit 
 to the error of each factor, and to compute the value of the 
 monomial. 
 
 RULE 24. Make a rough calculation of a superior 
 limut of the result, and from this determine, by Rule 3, 
 the allowable retative error of the result. Then 
 count the number of single factors whose values are ta 
 be taken approximately, both in the numerator and 
 denominator of the monomial, and add to the number 
 of such factors the number of operations that are to be 
 performed on them after their values are obtained 
 divide the allowable relative error of the final result 
 by the sum thus made, and the quotient will be an al- 
 lowable relative error for each single factor. Com- 
 pute each single factor until its error does not exceed 
 this limit, and then perform the remaining operations 
 an such a way that the new error introduced by each 
 abridged process shall be very small compared with 
 that due to the errors of the factors. Indicate, as the 
 work proceeds, the limit of error of each partial result, 
 and lastly of the final result. 
 
COMPLEX COMPUTATIONS. 87 
 
 EXAMPLE 64. Compute the value of 
 
 
 
 
 
 Tee Vi0 xz — 
 v 5.27963289.... 
 0.4318965021.... 
 
 so that the absolute error shall not exceed 0.001. 
 As approximation for assigning the allowable 
 relative error of the result, we may substitute 
 
 i 
 raxV2 as 13X12 
 
 0.4 0.4 
 limit of the final result. Hence, the allowable rela- 
 tive error of the result may be taken at 74,5. There 
 are 5 single factors to be taken approximately, and 
 there are 5 operations to perform on them after their 
 values are found ; we therefore divide the allowable 
 relative error of the result by 10, which gives the 
 limit of the relative error for each factor, goto. By 
 Rule 4 we may then assume the allowable absolute 
 
 error of V2 at gyhg7=0. 000025, that of W1oat a305 
 —0.00005, and that of 2 at z5#y9=0.000075. 
 
 Taking then V10 with four decimals, = 2.1544 + 
 and multiplying by 7=3.1416—, we have 
 2.1544 + 
 I—00141.3 
 O40 3202 rs 
 21544 + 
 8618 — 
 BN ESO 
 kee irre! 
 ~~ 6.76826 + 16 
 Dividing this result by 5.27963 + 
 
 
 
 
 
 <4; that is, 4 is a superior 
 
 
 
 
 
88 APPROXIMATE COMPUTATIONS, 
 
 6.76826 + 16 |5.279°3 + 
 527903 + 1.28196 + 4 
 148863 
 105593 — 
 43270 
 42237 — 
 1033 
 528 — 
 505 
 StI 
 30 + 78 
 Extracting the square root of the quotient, 
 1.28196+4 (1.132244+3 
 I 
 21) 28 
 ue 
 223) 719 
 OGG 
 2262) 5060 
 4524 
 536 
 452 
 84t 47 
 Taking 4/2 with five decimals=1.25992+, and 
 multiplying, 
 
 
 
 T;1 3224; tee 
 + 29952.1 
 1.132240 + 39 
 226448 + 
 S001 25-- aan 
 IOI9QO + 
 IOIQ + 
 23 05 
 
 1.426532 + 44 
 
 
 
COMPLEX COMPUTATIONS, 89 
 
 Finally, dividing this result by 0.4318965+, 
 
 1.426532 + 44 | 0.431896,5+ 
 BAe RREY f | 3.3029 + 2 
 130842 
 129569 
 1273 
 864 
 409 
 389 
 20 +46 
 
 We may then take for the final result, with three 
 exact decimals, 3.303+, the absolute error being then 
 less than 0.0003, or less than 0.001, the assigned 
 limit. 
 
 The reason for Rule 24 may be given in a few words. 
 If a given monomial were composed of any number 
 of approximate factors, either for multiplication or 
 division, and if there were to be no new error made 
 by abridged processes of computation, we might take, 
 for the allowable limit to the relative error of each 
 factor, the allowable relative error of the final result 
 divided by the number of such factors. But since the 
 abridged processes are liable to produce further 
 errors, we must allow for them in assigning the limit 
 of error of the factors, which Rule 24 evidently does. 
 But since, as the operation proceeds, the error in- 
 creases, it would not even with this allowance be safe 
 to let the new error, produced by each abridged pro- 
 _cess, be equal to that due to the error of the two fac- 
 tors on which we are operating. How much less 
 than this the error of each operation ought to be 
 made will depend on the degree of complexity of the 
 expression which we have to compute. For all ordi- 
 nary cases, if nine-tenths of the error of each partial 
 result is due to the errors of the two factors which 
 
 
 
gO APPROXIMATE COMPUTATIONS. 
 
 give it, there will evidently be a sufficient margin of 
 safety. 
 
 If in the statement of a problem the limit of relative 
 error of the result be assigned in advance, we may 
 evidently employ the method of Rule 24, except that 
 we are saved the necessity of making the rough cal- 
 culation of a limit of the result to begin with. 
 
 EXAMPLE 65. Compute 
 V857 x 4/9847.27 5 
 
 1 
 
 with a relative error in the result less than zp45>. 
 There are 3 factors, and 2 operations after they are 
 
 found, hence the allowable absolute error of 857 
 
 may be taken at 35,5 = 0.00058, that of 9847.27 
 at ay4°57 = 0.0004, and that of mat sa3a9 = 0.00000. 
 
 Take, therefore, “857 with 3 exact decimals, 
 9847.27 with 4, and a also with 4; make the mul- 
 tiplication so as to have 3 decimals in the product, 
 and the division so as to have 2 decimals in the quo- 
 tient, and the result will be 199.73 = 7. 
 
 EXAMPLE 66. Compute within 0.0001 the radius 
 of a circle whose area shall be equal to that of a regu- 
 lar hexagon, one side of which equals I. 
 
 If R be the radius, we may easily obtain the form- 
 
 ala, R =y/3¥3, 
 
 We see that the result will be less than 1, hence the 
 allowable relative error of the result is zo94y,. And, 
 since we may regard the relative error of a sq. root as 
 equal to half that of the number, this expression not 
 
 lowable relative error of the quantity under the large 
 radical Further, since multiplying and dividing by 
 
\ 
 
 COMPLEX COMPUTATIONS, OI 
 
 exact factors does not alter the relative error, we need 
 to allow for the error of only one operation on the two 
 approximate factors, before taking the final square 
 root. We have then for the allowable relative error 
 of each of these factors zghoq. It is just as well to 
 
 Wilt, Wee and we may then make the abso- 
 atta. 
 
 lute error of V27 equal to gy$oa, Say 0.0001, and that 
 of 27 equal to sp$sy, Say 0.0002. Take, therefore, 
 4 decimals each in V 27 and 22, make the division 
 so as to retain 4 exact decimals in the quotient, and 
 extract the root so that the error of it shall be mostly 
 due to that of the quantity. (Compare Art. 41). 
 The answer, with 4 exact decimals, is 0.9093 +. 
 
 EXAMPLE 67. Compute within 0.001 the radius of 
 a sphere, whose volume shall be equal to that of a 
 truncated pyramid, the altitude of which is 0.752, the 
 bases being *regular hexagons, whose sides are re- 
 spectively 1.42 and 0.843. 
 
 The formula will be 
 
 ae Ve * [0-732 (1-42)’+ (1.42) (0.843) + (0.843)? 
 2 7 
 
 It is not difficult to see that the result will be less 
 than 1. Then the allowable relative error of the 
 result will be zj4,. There are 3 factors that will be 
 taken approximately, viz., WY 3, 7, and |(1-42)’+ 
 (1°42) (0°843)+(0°843)’], and we allow for 3 opera- 
 tions, besides those with the exact factors, so that 
 the relative error of each of the approximate factors 
 may be made 75,5. The factor just written in brack- 
 ets will be greater than 3, hence its allowable abso- 
 lute error will be 5 = 0°0005. The multiplications 
 and additions to obtain this factor may then as well 
 be made exactly, but only 3 exact decimals need 
 
 
 
 
 
92 APPROXIMATE COMPUTATIONS, 
 
 be retained in the value of it when found. The 
 answer within 0.001 is R=0.839-+. 
 
 EXAMPLE 68. Find within 0.0001 the radius of a 
 sphere inscribed in a cone whose altitude is equal to 
 the diameter of its base, and whose volume is equal 
 loos 
 
 fe AS, 
 
 The formula will be 
 
 en 
 = V9 
 iis / 27 3 
 
 EXAMPLE 69. ‘The volume of a gas at a tempera- 
 ture 7°, and pressure 7™, Being ”, the volumeme 
 temperature 0°, and pressure 760" is given by the 
 formula. 
 
 x A. 
 760(1 + ¢ X 0:0036650) 
 Supposing 7. = 1056.7..... , Af = 202-0) 
 150.1..., determine how many more decimals would 
 
 have to be given in each of these quantities, so that 
 V could be calculated with an absolute error not 
 exceeding 0-002. (Proceed by finding the allowable 
 limits of error of the factors, that of the result being 
 0.002. ) 
 
 EXAMPLE 70. Compute within one cent the 
 amount of eighty-nine dollars and thirty-seven cents 
 at compound interest for three years and three hun- 
 dred and forty-seven days, at the rate of seven and 
 one-half per cent. per annum. 
 
 Denoting the amount by A, we shall have 
 
 Al i= 780'37 (1-075)"(1 TELE) 
 | 305 
 os. If a polynomial be proposed, whose terms 
 may be found as accurately-as we please, and if it is 
 to be calculated with a relative error in the result 
 not exceeding an assigned limit, then, before we can 
 
COMP EX COMPUTATIONS, 93 
 
 ¢ ssign the allowable limit to the error of each term, 
 we have to determine a rough inferior limit of the 
 final result, from which the allowable absolute error 
 of this result may be found; and from this, as in ad- 
 ‘ dition and subtraction, the allowable absolute error 
 of each term, and then, if necessary, the allowable 
 relative error of each term, and so that of each factor 
 in each term. 
 
 EXAMPLE 71. Compute the length of one side of 
 a regular pentedecagon inscribed in a circle whose 
 radius s I, with a relative error in the result less 
 Chet a eS ‘ 
 
 000 
 The formula will be 
 
 i 7V 1042 in 7¥3 (V5—1). 
 
 We can see by a little trial that the first of these 
 two terms will be greater than 0.8, and that the sec- 
 ond will be less than 0.6; hence, the result will be 
 greater than 0.2, which is, therefore, an inferior limit 
 of the result. We may then, by rule 4, assume the 
 allowable absolute error of the result at ;92, = 0.0002. 
 The absolute error of each of the two terms may then 
 be 0.0001. The allowable errors of the approximate 
 factors may now be assigned, and the result com- 
 puted. 
 
 EXAMPLE 72. Compute the expression : 
 
 7 G3 ™ + wae ree an fe 
 
 ean ev to) + 732—V10 
 
 4/2 V7 V6 a i) 
 with a relative error in the at less than = et. 
 
 In solving such a problem as this, it is well to ar- 
 range the work on paper in such a way that after 
 having carried the computation of each term far 
 enough to serve for assigning an inferior limit of the 
 result, from which to determine the allowable errors, 
 
94 APPROXIMATE COMPUTATIONS. 
 
 the computations may be resumed again at the same 
 points, without having to repeat any of the work. 
 
 24. If it be required to compute as accurately as 
 possible the value of a complex expression which 
 contains factors whose values are only known with 
 a limited degree of approximation, it is semetimes 
 convenient to determine in advance the degree of 
 precision which we may expect in the result ; while, 
 in other cases, the form of the expression is such 
 that the most convenient way is to proceed directly 
 with the computation, taking care that the errors in- 
 troduced by abridged processes are made small in 
 comparison with those due to the errors of the quan- 
 tities, and indicating the possible error of each partial 
 result when found, and lastly that of the final re- 
 sult. If a monomial contains a factor whose rela- 
 tive error is very much greater than those of the 
 other factors, it will generally be useless to retain in 
 these latter all the figures that might be taken, al- 
 though to obtain the result as closely as may be it is 
 best to work with one or two redundant figures. Or, 
 if, in a series of terms for addition or subtraction, one 
 of them would have a much greater absolute error 
 than the others, then it will, of course, avail little to 
 compute these latter to a much lower order of units 
 than can be found in the term having the greatest 
 absolute error. 
 
 EXAMPLE 73. An iron cylinder, weighing 6 kilo- 
 grammes, is found to be lengthened 2 millimetres by 
 passing from the temperature of melting ice to that 
 of boiling water. Considering the specific gravity of 
 iron, and the coefficient of dilatation, the radius ¢ 
 the cylinder in decimetres. yay, be computed from 
 
 the formula : 
 ; 6. 
 7.8 X 16.920 X 7 
 
 
 
COMPLEX COMPUTATIONS. ape fe 
 
 Assuming that the factors 7.8 and 16.920 are liable 
 to an absolute error of half a unit of the lowest order 
 in each, determine what degree of relative and ab- 
 solute approximation may be expected in the value 
 of RX, and then make the computation. 
 
 The error of the result will evidently be almost 
 entirely due to that of the factor 7.8. If the com- 
 putation were to be made with the greatest possible 
 precision, we might then expect the limit of relative 
 error of the result to be about 1. 7385 = shy. We 
 can see that the quantity under the radical will not 
 differ much from ;4;, hence the result will be about 
 0.12, and the absolute error, therefore, approxi- 
 mately 442 = 0.0004. We can probably employ the 
 abridged modes of computation without increasing 
 the error to more than 0.0005, which will enable us 
 to get the result within half a unit of thousandths 
 place. 
 
 Taking mw = 3.142—, the denominator may be 
 found=414.6+4 30, the quotient of 6 by this = 0.01447 
 +7z and the square root of this = 0.120345. 
 
 EXAMPLE 74. Determine the limit of relative and 
 absolute error of the value of the following expres- 
 sion, supposing the computations were to be exactly 
 made; then make the computations by abridged 
 processes, having regard to Rules 11 and 16, and de- 
 termine by inspection or otherwise the limit of error 
 of the result as obtained. 
 
 (67.34547) (6 326147) (1.7253-42) 
 (4.2785 47) (0.627344 /) 
 
 Taking the sum of. limits. of. the relative errors of 
 all the factors, we have THbo otsubc0 t ts000 t co000 
 + erbo0=ie0b00- But tetas <azoo7 3 «hence the rel- 
 ative error of the result, if no considerable new 
 error is introduced in the computation, will not 
 
 
 
 
 
 
 
96 APPROXIMATE COMPUTATIONS, 
 
 exceed p45; and since the result will be less than 
 280, the absolute error need not exceed 78% = 0.04. 
 
 EXAMPLE 75. If Zand & denote the length and 
 breadth of one base of a prismoid, 7 and 4 the length 
 and breadth of the opposite base, and % the distance 
 between the bases, the volume of the prismoid is 
 given by the following formula : 
 
 V = xh (2BL+ 261+ Bl+6L). 
 
 Assume 410.2022, B=7.844, L=0:0240ee 
 6.4342, and /=8.495+ ; compute the value of V 
 with what accuracy the data allow, and exhibit the 
 possible error. 
 
 EXAMPLE 76. The weight P of a volume of air 
 V, saturated with aqueous vapor, at temperature 7 
 and pressure /7, the tension of the vapor being /, 
 and the coefficient of expansion of air being a, is 
 given by the formula 
 
 Lao eee ‘ 
 Prep aty7G0 Nt aah 
 
 Let. V=1000j F=773-71, 1 =1500;, G01 eee 
 and ¢==100.5, these being supposed liable to error of 
 half a unit of the lowest order in each, but the other 
 numbers in the formula being supposed exact. Tind 
 the value of /, and its limit of error. 
 
 Ans. 360.7+5. 
 
 EXAMPLE 77. The formula for computing heights 
 by the barometer, in feet, is 
 
 
 
 i 
 a, (1-5) 
 
 1000 Pees 
 Let A= 30.025, #’=28.230;-f=17.32, and’ =i 
 the limit of error of these quantities being 5 units 
 
 x = 60346(1+0.002560 cos 2¢) (1+ 
 
COMPLEX COMPUTATIONS. Q7 
 
 of the lowest order in each, as also of the factors 
 60346 and 0.002560, Find the value of x when d= 
 65°, and show the limit of error of the result by ob- 
 serving the possible error at each step of the work. 
 AMS RIOTA TLL: 
 
 EXAMPLE 78. The interior diameter of a copper 
 ring being 18 centimetres at the temperature of 10°, 
 and the diameter of an iron sphere being 18.05 at the 
 same temperature, to what common _ temperature 
 must they be heated before the latter will pass 
 through the former ? 
 
 From a consideration of the coefficients of expan- . 
 sion of copper and iron, the following formula is 
 given : 
 
 18.05 X 1.000170— 18 x 1.000126 
 ~ 18 X 1.000126 X 0.0000170— 18.05 X 1.000170 XO. 0000126 
 
 Required to compute the value of x, under the 
 supposition that all the factors except 18 and 18.05 
 are liable to error of 1 of their lowest units, and to 
 exhibit the limit of error of the result. 
 
 Mins O40 gh u 
 EXAMPLE 79. The density D,, of the air in a cer- 
 
 tain air-pump, after four strokes of the piston, is 
 given by the formula 
 
 Bs BS) es | (a0 85) 
 Bie ( 3 0.885 E 3 
 Compute D,, supposing that where the quantity 
 0.885 occurs in the formula it is liable to an error 
 equal to 0.0005, and exhibit the limit of error of the 
 result. 
 
 EXAMPLE 80. The index of refraction of a prism 
 being given by the formula 
 
 __ sin $ (¢+a) 
 
 sind ¢ : 
 
 
 
 
 
 
 
 
 
 
 
98 APPROXIMATE COMPUTATIONS. 
 
 suppose ¢=60° 00’ 46430, a=47° 41’ 50”430”, 
 compute z by natural or logarithmic functions, and 
 exhibit the limit of error. 
 
 EXAMPLE 81. Having given the three sides of a 
 spherical triangle, Aa=7A° 23%, §0=35°% 40st eee 
 C22 100820 625. the formula for deterninine the angle 
 A, is 
 
 Lisle 
 coed 4 = 4/sin gs ys sin (} s—a@) 
 
 sin 6 sinc 4 
 
 in which s=a+6+c. Supposing the given sides to 
 be each liabie to an error of 30”, compute the angles 
 A, £, and C by natural and by logarithmic functions, 
 and exhibit their limits of error. 
 

 
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