Mathematics Dictionary

Sin(x) = x - (x^3)/(3!) + (x^5)/(5!)- (x^7)/(7!) + ....
If sin(x) = 0, the roots are r0 = 0, r1 = pi, r2 = 2*pi, r3 = 3*pi, ....
Hence r1, r2, r3, ... are roots of 1 - (x^2)/6 + (x^4)/120 - ..... = 0
Let x^2 = u,
- The roots of equation : u1 = (pi)^2, u2 = (2*pi)^2, u3 = (3*pi)^2
Equation theory
- Equation : a*x^n + b*x^(n-1) + ..... + p*x + q = 0
- Roots are x1, x2, x3, ....
- Coefficient of x and roots : 1/x1 + 1/x2 + 1/x3 + ..... = p/q
Hence coefficient and roots u
- 1/u1 + 1/u2 + 1/u3 + ..... = -(-1/6)/1
- 1/(pi^2) + 1/((2*pi)^2) + 1/((3*pi)^2) + 1/((4*pi)^2) + .... = 1/6
- Multiply (pi)^2 on both sides
- 1 + 1/(2^2) + 1/(3^2) + ..... = ((pi)^2)/6

cos(x) = 1 - (x^2)/(2!) + (x^4)/(4!)- (x^6)/(6!) + ....
If cos(x) = 0, the roots are r1 = pi/2, r2 = 3*pi/2, r3 = 5*pi/2, ....
Hence r1, r2, r3, ... are roots of 1 - (x^2)/2 + (x^4)/24 - ..... = 0
Let x^2 = u,
- The roots of equation : u1 = (pi/2)^2, u2 = (3*pi/2)^2, u3 = (5*pi/2)^2
Equation theory
- Equation : a*x^n + b*x^(n-1) + ..... + p*x + q = 0
- Roots are x1, x2, x3, ....
- Coefficient of x and roots : 1/x1 + 1/x2 + 1/x3 + ..... = p/q
Hence coefficient and roots u
- 1/u1 + 1/u2 + 1/u3 + ..... = -(-1/2)/1
- 1/((pi/2)^2) + 1/((3*pi/2)^2) + 1/((5*pi/2)^2) + .... = 1/2
- Multiply (pi/2)^2 on both sides
- 1 + 1/(3^2) + 1/(5^2) + ..... = ((pi/2)^2)/2 = ((pi)^2)/8

pi = ((2^5)/2)*sin(360/32)
= 16*Sqr((1 - cos(360/16))/2)
= (16/Sqr(2))*Sqr(1 - cos(360/16))
= (16/Sqr(2))*Sqr(1 - Sqr(1 + cos(360/8))/2))
= 8*Sqr(2 - Sqr(2 + Sqr(2))) = 3.121445

(1) To 1000 decimal place : see Computer Math p47
(2) Story of Pi (See Computer Math Chapter 6)
(3) New record : 106 billion decimal places (Genume 2002) by Janpanese scientists

1949 ENIAC was used to compute Pi to 2037 decimal places.
1961 IBM 7090 computed Pi to 100000 decimal places by Shanks and Wrench.
1966 IBM Stretch supercomputer computed pi to 250000 decimal places.
1983 Pi is computed to 16 million decimal places.
1987 Pi is calculated to 134 million decimal places
- by Yasumasa Kanada of the University of Tokoyo
- using Nippon Electronic SX-2 supercomputer
1989 Pi calculated to 1;011;196;691 decimal places
- by David and Gregory Chudnovsky at the Columbia University; New York
- The calculation was performed twice on an IBM
- 3090 mainframe and on a Cray-2 supercomputer; and the
- results matched.
1991 Brothers David and Gregory Chudnovsky
- use a formula of their discovery and low budget supercomputer
- to compute pi to 2.1 billion decimal places

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