In Search of Pi

Michigan State University East Lansing, MI March 14, 2016

PDFfileoftheseslidesavailableatwww.macalester.edu/~bressoud/talks

David Bressoud St. Paul, MN

RhindPapyrus(BriAshMuseum)17thcenturyBCE33cmtall,5.5m.longCopyofmanuscriptfromabout200yearsearlier

Squaringthecircle

8/9diameterdiameter

πr2 = π4d 2 ≈ 0.7854d 2

89

⎛⎝⎜

⎞⎠⎟2

d 2 ≈ 0.7901d 2

Antiphon of Athens (480–411 BC) is credited with finding (and proving?) formula for area of circle

perimeter=π×diameter

radius

Area = ½ × perimeter × radius = ½ × π × diameter × radius = π × radius2

Archimedes of Syracus 287–212 BC

Syracuse

x = r y = 2 33

r

3d < perimeter < 2 3dLet yn be the length of one outer side, xn the length of one inner side of polygons with n sides, d the diameter of the circle

y2n =d yn

d + d 2 + yn2, x2n =

d xn2d 2 + 2d d 2 − xn

2

/2

/2

x = r y = 2 33

r

3d < perimeter < 2 3d

/2

/2

Successively doubling the number of sides to 96:

253448069

d < perimeter < 293769347

d

3.14091d < perimeter < 3.14283d

Liu Hui, late 3rd century AD.

Wrote commentary on Jiuzhang suanshu (the earliest surviving Chinese textbook of mathematics) and an original book: Haidao suanjing (Sea Island Computational Canon)

Rediscovers Archimedes’ approach, goes as far as polygons with 192 sides:

3.141452d < perimeter < 3.141873d

By early 5th century, Chinese mathematicians were using decimal fractions.

Zu Chongzhi, late 5th century, uses 355/113 as approximation to π.

Accurate to within one part in ten million; circle of diameter 50 miles accurate to one inch.

Would be the most accurate approximation to π known anywhere in the world for the next thousand years.

π = 3.14159265…355113

= 3.14159203…

François Viète, 1540–1603

1591: Introduction to the Analytic Art

Viète used Archimedes’ approach with a polygon of 393,216 sides:

#ofsides Inscribedperimeter Circumscribedperimeter

96 3.14103195089… 3.14271459964…768 3.14158389214… 3.14161017660…6144 3.14159251669… 3.14159292738…49152 3.14159265145… 3.14159265786…393216 3.14159265355… 3.14159265365…

Start with a hexagon and double the number of sides 16 times

π2= 2

2⋅ 22 + 2

⋅ 2

2 + 2 + 2

Viète’s formula for π

Uses ratios of areas of consecutive inscribed polygons.

Ludolph van Ceulen (1540–1610)

Around 1600, used a polygon of 262 sides to calculate the first 35 digits of π, showing that π lies between

3.14159265358979323846264338327950288

and

3.14159265358979323846264338327950289

earth

Winter solstice

Spring equinox

Summer solstice

Autumnal equinox

93⅝ days

earth

Winter solstice

Spring equinox

Summer solstice

Autumnal equinox

89 days

89⅞ days

92¾ days

93⅝ days

earth center

Winter solstice

Spring equinox

Summer solstice

Autumnal equinox

89 days

92¾ days

89⅞

days

Basicproblemofastronomy:

Giventhearcofacircle,findthelengthofthechordthatsubtendsthisarc.

To measure an angle made by two line segments, draw a circle with center at their intersection.

The angle is measured by the distance along the arc of the circle from one line segment to the other.

The length of the arc can be represented by a fraction of the full circumference: 43° = 43/360 of the full circumference.

If the radius of the circle is specified, the length of the arc can also be measured in the units in which the radius is measured: Radius = 3438, Circumference = 60×360 = 21600 Radius = 1, Circumference = 2π

92° 17'

earth

center

Winter solstice

Spring equinox

Summer solstice

Autumnal equinox

87° 43'

88° 35'

91° 25'

3° 42'

52'

Find the chord lengths for 3° 42' and 52'.

The distance from the earth to the center of the sun’s orbit is found by taking half of each chord length and using the Pythagorean theorem.

Note that the chord lengths depend on the radius.

FronAspiecefromPtolemy’sAlmagest

PeurbachandRegiomantusediAonof1496

• Constructed table of chords in increments of ½° and provided for linear interpolation in increments of ½ minute.

• Values of chord accurate to 1 part in 603 = 216,000 (approximately 7-digit accuracy).

Ptolemy of Alexandria Circa 85–165 CE

Kushan Empire

1st–3rd centuries CE

Arrived from Central Asia, a successor to the Seleucid Empire

Imported Greek astronomical texts and translated them into Sanskrit

Surya-Siddhanta

Circa 300 CE

Earliest known Indian work in trigonometry, had already made change from chords to half-chords

Ardha-jya = half bowstring

Became jya or jiva

Chord θ = Crd θ = 2 Sin θ/2 = 2R sin θ/2

R

θθ/2

Jiya – SanskritJiba (jyb) – Arabic

jyb jaib, fold or pocketSinus, pocket– LatinSine – English

Keralese astronomers:

Madhava, 1340–1425

Paramesvara, circa 1370–1460

Nilakantha, circa 1444–1544

sin x = x − x3

3!+ x

5

5!− x

7

7!+

arctan x = x − x3

3+ x

5

5− x

7

7+

π4= 1− 1

3+ 15− 17+ Known today as “Leibniz’s

formula”

John Machin (1686–1751), Professor of Astronomy at Gresham College, London Found first 100 digits of Pi

π = 16arctan 15( )− 4arctan 1239( )= 16 1

5− 13⋅53

+ 15 ⋅55

− 17 ⋅57

+!⎛⎝⎜

⎞⎠⎟

− 4 1239

− 13⋅2393

+ 15 ⋅2395

− 17 ⋅2397

+!⎛⎝⎜

⎞⎠⎟

John Wallis, 1616–1703

Was a code-breaker for the Parliamentarians in the Civil War.

1647, read Oughtred’s Clavis Mathematicae (written 1631) on algebra

1649: Savilian chair of Geometry, Oxford

1655 Arithmetica Infinitorum (Arithmetic of Infinities)

1685 Treatise on Algebra

Advocates accepting both negative and complex roots

π4= 1− x2( )12 dx

0

1

∫f p,q( ) = 1− x

1p( )q dx0

1

∫When q is an integer:

f p,q( ) = q!

p +1( ) p + 2( )! p + q( )

π2= 2 ⋅ f 12,

12( ) = 21 ⋅

23⋅ 43⋅ 45⋅ 65⋅ 67!

f p,q( ) = qp + q

f p,q −1( )

f p,q −1( ) > f p,q − 12( ) > f p,q( )

Stirling’s Formula (James Stirling, 1692–1770), published in Methodus Differentialis, 1730.

n!= ne

⎛⎝⎜

⎞⎠⎟n

2πn eE n( ),

where limn→∞

E n( ) = 0.

Isaac Newton (1642–1727)

1/2

1/4 1/2 3/4 1

y = x ! x2

π24

= x − x2( )120

14∫ dx + 3

32

x − x2( )12 = x 12 1− 12 x +1 2( ) −1 2( )

2!x2 −

1 2( ) −1 2( ) −3 2( )3!

x3 +!⎛⎝⎜

⎞⎠⎟

Periodic Functions:

The sine has period 2π:

The exponential function has period 2πi:

Doubly periodic functions have both a real and a complex period.

These are called elliptic functions (also known as Abelian functions).

sin x = sin x + 2π( ) = sin x + 4π( ) =!

ex = ex+2π i = ex+4π i =!

Real axis

Imaginary axis

1

z = x + iy

Given the elliptic function, it is possible to determine the complex period, specified by z.

Imaginary axis

Real axis 1

z = x + iy

C. G. Jacob Jacobi, 1804–51

Given the elliptic function, it is possible to determine the complex period, specified by z.

Given the shape of the parallelogram specified by z, a theta function enables one to find an appropriate elliptic function.

Elliptic functions are also known as modular functions.

Real axis z = 0

Imaginary axis

Srinivasa Ramanujan, 1887–1920

January, 1913: Ramanujan writes to G. H. Hardy

Godfrey Harold Hardy 1877–1947

1−5 12

⎛⎝⎜

⎞⎠⎟

3

+ 9 1⋅32 ⋅4

⎛⎝⎜

⎞⎠⎟

3

−13 1⋅3⋅52 ⋅4 ⋅6

⎛⎝⎜

⎞⎠⎟

3

+!= 2π

dx1+ x2( ) 1+ r 2x2( ) 1+ r 4x2( )!0

∞

∫ = π2 1+ r + r3 + r6 + r10 +!( )

1

1+ e−2π

1+ e−4π

1+!

= e2π /5 5+ 52

⎛

⎝⎜

⎞

⎠⎟ −

5 +12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1

1+ e−2π 5

1+ e−4π 5

1+!

= e2π / 5 5

1+ 53/4 5 −12

⎛

⎝⎜

⎞

⎠⎟

5/2

−15

− 5 +12

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

S. Ramanujan. Modular Equations and Approximations to π. Quarterly Journal of Mathematics, 1914

1π= 8

98011103+ 4!⋅27493

1!⋅3961( )4 +8!⋅53883

2!⋅3962( )4 +12!⋅80273

3!⋅3963( )4 +!⎛

⎝⎜⎜

⎞

⎠⎟⎟

π ≈ 98011103 8

= 3.14159273…

versus π = 3.14159265…

Each additional term adds eight additional digits of accuracy.

In 2014, "houkouonchi” (prefers to remain anonymous) calculated 13.3 trillion digits of π

Took 208 days + 182 hours to verify

PDFfileoftheseslidesavailableatwww.macalester.edu/~bressoud/talks

1π= 12 −1( )k 6k( )! 545140134k +13591409( )

3k( )! k!( )3 6403203( )k+12k=0

∞

∑

1989, Chudnovksy brothers are first to compute one billion digits of π, using a formula based on Ramanujan’s ideas:

This formula will lead to the computation of trillions of digits of π.