Gaussian Guesswork .......

.... or Why 1.19814023473559220744 . . .
is Such a Beautiful Number

Article Author: Adrian Rice
Math Horizons, November 2009, pp. 12 — 15.

DOI 10.4169/194762109X476491.



“Before theorems are proved, conjectures must be made, . . . ”



“Before theorems are proved, conjectures must be made, . . . ”

. . . and for that to happen, all kinds of experimentation,

observation, invention and, indeed, imagination

must come into play.”



Carl Friedrich Gauss (1777-1855)

• One of the most creative mathematicians of all time
(alongside Archimedes and Newton).



Carl Friedrich Gauss (1777-1855)

• One of the most creative mathematicians of all time
(alongside Archimedes and Newton).

• Kept a “mathematical diary” for nearly 20 years
(just before his 19th birthday in 1796 until July 1814)



Gauss’ May 30, 1799 Discovery:

Amazing relationship between three particular numbers:

• a sophisticated form of average

• a particular value of an elliptic integral

• the ratio of the circumference of a circle to its diameter (π)



Some of what Gauss knew about π

• π ∼= 3.14159265358979323846etc.



Some of what Gauss knew about π

• π ∼= 3.14159265358979323846etc.

• π = 2
∫ 1
0

dt
√
1 − t2

(arclength of semi-circle)



Some of what Gauss knew about π

• π ∼= 3.14159265358979323846etc.

• π = 2
∫ 1
0

dt
√
1 − t2

(arclength of semi-circle)

• If u =
∫ x
0

dt
√
1 − t2

, then x = sin u . . .



Some of what Gauss knew about π

• π ∼= 3.14159265358979323846etc.

• π = 2
∫ 1
0

dt
√
1 − t2

(arclength of semi-circle)

• If u =
∫ x
0

dt
√
1 − t2

, then x = sin u . . .

. . . and sin(u + 2π) = sin u.



Hmmmmmm . . .

. . . what happens with similar integrals?????

If u =

∫ x
0

dt
√
1 − tn

, then x = . . .



January 1797: Begins to investigate case of n = 4:

∫ x
0

dt
√
1 − t4



January 1797: Begins to investigate case of n = 4:

∫ x
0

dt
√
1 − t4

Cool! Arclength of lemniscate curve: (x2 + y2)2 = x2 − y2



January 1797: Begins to investigate case of n = 4:

∫ x
0

dt
√
1 − t4

Cool! Arclength of lemniscate curve: (x2 + y2)2 = x2 − y2

Follow the sin u pattern to define a leminiscate sine!!!



• If u =
∫ x
0

dt
√
1 − t2

, then x = sin u.

• If u =
∫ x
0

dt
√
1 − t4

, then x = slu.



• If u =
∫ x
0

dt
√
1 − t2

, then x = sin u.

sin(u + 2π) = sin u , where π =

∫ 1
0

dt
√
1 − t2

.

• If u =
∫ x
0

dt
√
1 − t4

, then x = sl u.



• If u =
∫ x
0

dt
√
1 − t2

, then x = sin u.

sin(u + 2π) = sin u , where π =

∫ 1
0

dt
√
1 − t2

.

• If u =
∫ x
0

dt
√
1 − t4

, then x = sl u.

sl (u + 2ω) = sl u , where ω = 2

∫ 1
0

dt
√
1 − t4

.



• If u =
∫ x
0

dt
√
1 − t2

, then x = sin u.

sin(u + 2π) = sin u , where π =

∫ 1
0

dt
√
1 − t2

.

• If u =
∫ x
0

dt
√
1 − t4

, then x = sl u.

sl (u + 2ω) = sl u , where ω = 2

∫ 1
0

dt
√
1 − t4

.

ω = 2.62205755429211981046 etc.



Gauss’ May 30, 1799 Discovery:

Amazing relationship between three particular numbers:

• the ratio of the circumference of a circle to its diameter: π



Gauss’ May 30, 1799 Discovery:

Amazing relationship between three particular numbers:

• the ratio of the circumference of a circle to its diameter: π

• a particular value of an elliptic integral: ω = 2
∫ 1
0

dt
√
1 − t4



Gauss’ May 30, 1799 Discovery:

Amazing relationship between three particular numbers:

• the ratio of the circumference of a circle to its diameter: π

• a particular value of an elliptic integral: ω = 2
∫ 1
0

dt
√
1 − t4

• a sophisticated form of average



The Arithmetic-Geometric Mean of a, b

Define two sequences:

n = 0 : a0 = a b0 = b

n = 1 : a1 =
1
2 (a0 + b0) b1 =

√
a0b0

n ≥ 1 : an = 12 (an−1 + bn−1) bn =
√

an−1bn−1



Facts about (an), (bn) for a, b ≥ 0

• Both sequences converge.



Facts about (an), (bn) for a, b ≥ 0

• Both sequences converge.

• lim
n→∞

an = lim
n→∞

bn



Facts about (an), (bn) for a, b ≥ 0

• Both sequences converge.

• lim
n→∞

an = lim
n→∞

bn

Call this limit the arithmetic-geometric mean of a, b:

M(a, b) = lim
n→∞

an = lim
n→∞

bn



Why 1.19814023473559220744 etc is a Beautiful Number

We have established that

the arithmetic-geometric mean

between 1 and
√
2 is π/ω to 11 places;

the proof of this fact will certainly

open up a new field of analysis.

Gauss’s Diary, May 30, 1799



May 1800 - Gauss completes the proof

Lemma 1. Let a, b > 0, a = a0, b = b0 and set

I(a, b) =

∫ π/2
0

dq√
a2 cos2 q + b2 sin2 q

.

Then I(a, b) = I
(
a+b
2 ,

√
ab
)



May 1800 - Proof Complete!

Lemma 1. Let a, b > 0, a = a0, b = b0 and set

I(a, b) =

∫ π/2
0

dq√
a2 cos2 q + b2 sin2 q

.

Then I(a, b) = I
(
a+b
2 ,

√
ab
)

Lemma 2. I(a, b) =
π/2

M(a, b)
.



May 1800 - Proof Complete!

Lemma 1. Let a, b > 0, a = a0, b = b0 and set

I(a, b) =

∫ π/2
0

dq√
a2 cos2 q + b2 sin2 q

.

Then I(a, b) = I
(
a+b
2 ,

√
ab
)

Lemma 2. I(a, b) =
π/2

M(a, b)
.

Theorem. M(1 ,
√
2) = πω.



Further Reading

J. M. Borwein and P. B. Borwein, Pi and the AGM
(John Wiley & Sons, 1987)

D. Bressoud A Radical Approach to Real Analysis (MAA, 1994).

D. A. Cox, The arithmetic-geometric mean of Gauss,
Enseignement Mathematique, 2nd ser., 30 (1984). 275-330.

G. W. Dunnington, Gauss: Titan of Science,
with additional material by Jeremy Gray (MAA, 2004).

A. Rice. What Makes a Great Teacher? The Case of Augustus Morgan.
The American Mathematical Monthly 106:6 (1999), 534-552.