pi to a trillion digits : How and Why?

What is pi? How to calculate the value of π? Computing π in C Why a trillion digits?

Pi to a trillion digits: How and why

V N Krishnachandran

V N Krishnachandran


Outline

1 What is pi?

2 How to calculate the value of π?

3 Computing π in C

4 Why a trillion digits?

V N Krishnachandran


1. What is pi?

V N Krishnachandran


What is pi?

‘pi ’ is the sixteenth letter of the Greek alphabet!

V N Krishnachandran


What is π?

π denotes a very special number in
mathematics.

V N Krishnachandran


Definition of π

Definition
The ratio of the circumference of a circle to its
diameter is denoted by π. (The symbol π was first
proposed by the Welsh mathematician William
Jones in 1706.)

V N Krishnachandran


Deﬁnition of π

C

d Circumference of a circle (C )
et e
r π=
dia
m
Diameter of the circle (d)

V N Krishnachandran


On the value of π

Which of the following is true?

π = 22/7
π = 3.14
√
π = 10
π = 335/113

None!!

V N Krishnachandran


On the value of π

The numbers 22/7, 3.14, ... are only
approximate values of π.

V N Krishnachandran


On the value of π

π is not an integer.
π is not a rational number, that is, not the
ratio of one integer to another integer.
π is not an algebraic number, that is, not a
solution of a polynomial equation with integer
coeﬃcients.

V N Krishnachandran


On the value of π

The decimal representation of π has inﬁnite
number of digits appearing apparently without
any pattern.

V N Krishnachandran


On the value of π

First 200 digits in the value of π:
3.
1415926535 8979323846 2643383279 5028841971
6939937510 5820974944 5923078164 0628620899
8628034825 3421170679 8214808651 3282306647
0938446095 5058223172 5359408128 4811174502
8410270193 8521105559 6446229489 5493038196
Fine! How can one obtain all these digits?

V N Krishnachandran


2. How to calculate pi?

V N Krishnachandran


Geometrical procedure to ﬁnd π

Draw a circle.
Measure the circumference of the circle.
Measure the diameter of the circle.
Divide the circumference by the diameter.
The number obtained is the value of π.

V N Krishnachandran



The geometrical procedure will not give the
exact value of π.
This is because of the limitations of the
drawing instruments.
For example, we cannot draw a line with
thickness 0.0000000001 mm.

V N Krishnachandran



Pi cannot be calculated by measuring
circumferences and radii of circles!
We need mathematical machinery!!

V N Krishnachandran


Mathematical machinery to ﬁnd π

Pieces of the machinery required:
Radian measure of angle
Arctangent function
Gregory series
Machin formula
Or
Ramanujan’s formula

V N Krishnachandran



V N Krishnachandran



180 degrees = π radians
1 degree = π/180 radians
D degrees = (π/180) D radians

Special case:
45 degrees = π/4 radians

V N Krishnachandran


Arctangent function

a
arctan = Angle A.
b
V N Krishnachandran


Arctangent function

Deﬁnition
arctan(x) is an angle in radian measure whose
trigonometric tangent is x.

V N Krishnachandran


Arctangent function

A formula for π

arctan(1) = π/4

V N Krishnachandran


James Gregory (1638 - 1675)

James Gregory, a Scottish mathematician,
developed a series expansion for arctan(x).
V N Krishnachandran


Gregory series

Gregory series

x3 x5 x7
arctan(x) = x − + − + ···
3 5 7

V N Krishnachandran


John Machin (1686 - 1751)

Machin, English astronomer, discovered a formula
for faster computation of π.
V N Krishnachandran


Machin’s formula

Machin’s formula

1 1
4 arctan − arctan = arctan(1)
5 239

V N Krishnachandran


Numerical expression for π

Numerical expression for π using Machin’s formula

1 1 1 1 1
π = 16 5 − 3 53 + 5 55 + ···
1 1 1 1 1
−4 239 − 3 2393 + 5 2395 + ···

V N Krishnachandran


Srinivasa Ramanujan (1887 1920)

A formula due to Srinivasa Ramanujan, Indian
mathematical genius, is also used to compute π.

V N Krishnachandran



√ ∞
1 8 (4n)! 26390n + 1103
= ×
π 9801 n=0
(n!)4 3964n

V N Krishnachandran


Some other methods

Chudnovsky formula: The fastest converging
series to compute π
Iteration scheme due to Borwein and Borwein

V N Krishnachandran


3. Computing π in C

V N Krishnachandran


Basic idea

Consider the ﬁrst 20 digits in the decimal
representation of the proper fraction 4 :
7

4
7 = .57142857142857142857

We split this into blocks of two digits each:

57 14 28 57 14 28 57 14 28 57

V N Krishnachandran


Basic idea

These blocks, treated as integers, are assigned to
the cells in a one-dimensional array:
x[0] = 57 x[1] = 14 x[2] = 28 x[3] = 57

x[4] = 14 x[5] = 28 x[6] = 57 x[7] = 14

x[8] = 28 x[9] = 57

V N Krishnachandran


Program concepts: Notations

a, b positive proper fractions
k positive integer
a+b <1
a−b >0
ka < 1

V N Krishnachandran


Program concepts: Notations

Digits of Stored in
a x[i]
b y[i]
a+b s[i]
a−b d[i]
a/k q[i]

V N Krishnachandran


Program concepts: Computation of x[i]
Let a = N/D (N and D positive integers).

Initialize:
R = N;

Repeat the following for i=0 to 20:
x[i] = floor( ( R * 100 ) / D ) ;
R = R - x[i] * D ;

Computation of y[i] is similar.

V N Krishnachandran


Program concepts: Computation of q[i]

Initialize:
R = 0;
T = R * 100 + x[0] ;
q[0] = floor (T/k) ;

R = x[i] - q[i] * k ;
T = q[i] * 100 + x[i] ;
q[i+1] = floor(T/k) ;

V N Krishnachandran


Program concepts: Computation of s[i]

Initialize:
carry = 0 ;

t = x[i] + y[i] + carry ;
if( t < 100 ) s[i] = t ;
else { s [i] = x[i] + y[i] - 100;
carry = 1 ; }

V N Krishnachandran


Program concepts: Computation of d[i]

Initialize:
borrow = 0 ;

x[i] = x[i] - borrow ;
if( x[i] >= y[i] ) d[i] = x[i] - y[i] ;
else { d[i] = 100 + x[i] - y[i];
borrow =1 ; }

V N Krishnachandran


Program to compute π

Put the pieces together to construct a program to
compute π!

V N Krishnachandran


Another program to compute π
The following programme due to Dik T. Winter of
Netherland, computes pi to 800 digits:
long int a=10000,b,c=2800,d,e,f[2801],g;
main(){ for(;b-c;)f[b++]=a/5;
for(;d=0,g=c*2;c-=14,
printf("%.4d",e+d/a),e=d%a)
for(b=c;d+=f[b]*a,f[b]=d%--g,d/=g--,--b;d*=b);
}
Check the claim!
V N Krishnachandran


4. Why a trillion digits?

V N Krishnachandran


Why a trillion digits of π?

To test hardware reliability
To help develop eﬃcient algorithms
To help generate random digits

V N Krishnachandran


Known record

T2K Open Supercomputer, at University of
Tsukuba, Japan, calculated
2, 576, 980, 370, 000 decimal digits
in the value of π (news dated 17 August 2009).
V N Krishnachandran


The hunt for the digits of π continues ...

V N Krishnachandran


Department of Computer Applications
Vidya Academy of Science and Technology
Thrissur - 680501, Kerala

(October 2009)

V N Krishnachandran

pi to a trillion digits : How and Why?

Recommandé

Recommandé

Contenu connexe

Plus de PlusOrMinusZero

Plus de PlusOrMinusZero (20)

Dernier

Dernier (20)

pi to a trillion digits : How and Why?