Machin's Computation of Pi
Machin's Computation of Pi
Around 1706, John Machin found the arc tangent formula .
π/4=4arctan(1/5)-arctan(1/239)
He expanded it using Gregory's series to compute to over 100 digits. The convergence of this series is much more rapid than that for the simple Gregory–Leibniz series: .
arctan(x)=x-/3+/5-/7+…
3
x
5
x
7
x
π
π/4=arctan(1)=1-1/3+1/5-1/7+…
Let and be the series for and . Manchin's formula is then .
R
1
R
2
arctan(1/5)
arctan(1/239)
π=4(4-)
R
1
R
2
Split =- and =- into four series according to the signs of their coefficients:
R
1
S
1
S
2
R
2
S
3
S
4
S
1
1
5
1
5
5
1
5
S
2
1
3
3
1
5
1
7
7
1
5
S
3
1
230
1
5
5
1
239
S
4
1
3
3
1
239
1
7
7
1
239
Machin used values truncated to 103 decimals, which means that only terms greater than are calculated.
-103
10
To make sure that all terms of and with indices are less than , it is necessary that . For , this means .
S
1
S
2
n>m
-d
10
n>0.25dlog(10)/log(5)
d=103
m=36