John Machin
Approximating
⇡ with Machin’s
Formula
Approximating
⇡ with Machin’s
Formula
Dr. Bill Slough
Approximating ⇡ with Machin’s Formula
Dr. Bill Slough
Mathematics and Computer Science Department
Eastern Illinois University
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1680–1751
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English mathematician and
astronomer
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Private tutor to Brook Taylor
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Best known for formulas he
invented for calculating ⇡
March 19, 2014
Line drawing from MacTutor History of Mathematics archive
Mathematical underpinnings
Partial sums
Approximating
⇡ with Machin’s
Formula
Approximating
⇡ with Machin’s
Formula
arctan x =
Taylor’s series for arctangent
arctan x =
=
j=0
x
x3
x5
x7
+
+ ...
1
3
5
7
1
X
( 1)j x 2j+1
j=0
2j + 1
Machin’s formula
⇡ = 16 arctan
1
X
( 1)j x 2j+1
2j + 1
1
5
4 arctan
1
239
x
1
x
a2 =
1
x
a3 =
1
x
a4 =
1
...
a1 =
ak+1 =
j runs from 0 to 0
x3
3
x3
x5
+
3
5
x3
x5
+
3
5
x
1
j runs from 0 to 1
j runs from 0 to 2
7
x
7
x3
x5
+
3
5
j runs from 0 to 3
x7
( 1)k x 2k+1
+ ··· +
7
2k + 1
Computing partial sums with a recurrence
Approximating
⇡ with Machin’s
Formula
MATLAB code
Approximating
⇡ with Machin’s
Formula
a1 =
x
1
% Total number of desired approximations
n = 10;
x3
3
x5
a3 = a2 +
5
x7
a4 = a3
7
...
a2 = a1
% atan approximations for xA and xB using just one term
a(1) = xA;
b(1) = xB;
% ...and the corresponding approximation for pi
p(1) = 16*a(1) - 4*b(1);
% Improve the approximation by increasing the number of terms used
for k = 1:n-1
a(k + 1) = a(k) + (-1)^k * xA^(2*k+1)/(2*k+1);
b(k + 1) = b(k) + (-1)^k * xB^(2*k+1)/(2*k+1);
p(k + 1) = 16*a(k + 1) - 4*b(k + 1);
end
k 2k+1
ak+1 = ak +
( 1) x
2k + 1
% Arguments for atan()
xA = 1/5;
xB = 1/239;
,k
1
Some numerical results
Approximating
⇡ with Machin’s
Formula
Summary
Approximating
⇡ with Machin’s
Formula
n
1
2
3
4
5
6
7
8
9
10
p(n)
3.18326359832636
3.14059702932606
3.14162102932503
3.14159177218218
3.14159268240440
3.14159265261531
3.14159265362355
3.14159265358860
3.14159265358984
3.14159265358979
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For centuries, mankind has been fascinated with ⇡.
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How can we compute accurate approximations of ⇡?
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We have observed Machin’s formula leads to fast convergence.