Machin's Formula for Pi

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Theorem

$\dfrac \pi 4 = 4 \arctan \dfrac 1 5 - \arctan \dfrac 1 {239} \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$

This sequence is A003881 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The calculation of $\pi$ (pi) can then proceed using the Gregory Series:

$\arctan \dfrac 1 x = \dfrac 1 x - \dfrac 1 {3 x^3} + \dfrac 1 {5 x^5} - \dfrac 1 {7 x^7} + \dfrac 1 {9 x^9} - \cdots$

which is valid for $x \le 1$.


Proof 1

Let $\tan \alpha = \dfrac 1 5$.

Then:

\(\displaystyle \tan 2 \alpha\) \(=\) \(\displaystyle \frac {2 \tan \alpha} {1 - \tan^2 \alpha}\) Double Angle Formula for Tangent
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 / 5} {1 - 1 / 25}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 5 {12}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \tan 4 \alpha\) \(=\) \(\displaystyle \frac {2 \tan 2 \alpha} {1 - \tan^2 2 \alpha}\) Double Angle Formula for Tangent
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 \times 5 / 12} {1 - 25 / 144}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {120} {119}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \tan \left({4 \alpha - \frac \pi 4}\right)\) \(=\) \(\displaystyle \frac {\tan 4 \alpha - \tan \frac \pi 4} {1 + \tan 4 \alpha \tan \frac \pi 4}\) Tangent of Difference
\(\displaystyle \) \(=\) \(\displaystyle \frac {\tan 4 \alpha - 1} {1 + \tan 4 \alpha}\) Tangent of $45^\circ$
\(\displaystyle \) \(=\) \(\displaystyle \frac {120 / 119 - 1} {120 / 119 + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {239}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 4 \alpha - \frac \pi 4\) \(=\) \(\displaystyle \arctan \frac 1 {239}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac \pi 4\) \(=\) \(\displaystyle 4 \arctan \frac 1 5 - \arctan \frac 1 {239}\)

$\blacksquare$


Proof 2

\(\displaystyle \arg \left({\left({5 + i}\right)^4 \left({239 - i}\right)}\right)\) \(=\) \(\displaystyle \arg \left({\left({5 + i}\right)^4}\right) + \arg \left({239 - i}\right)\) Argument of Product equals Sum of Arguments
\(\displaystyle \) \(=\) \(\displaystyle 4 \arg \left({5 + i}\right) + \arg \left({239 - i}\right)\) De Moivre's Theorem
\(\displaystyle \) \(=\) \(\displaystyle 4 \arctan \frac 1 5 + \arctan \frac {-1} {239}\)
\(\displaystyle \) \(=\) \(\displaystyle 4 \arctan \frac 1 5 - \arctan \frac 1 {239}\) Inverse Tangent is Odd Function


Furthermore, because:

\(\displaystyle \left({5 + i}\right)^4\) \(=\) \(\displaystyle 5^4 + 4 \times 5^3 i - 6 \times 5^2 - 4 \times 5 i + 1\) Binomial Theorem
\(\displaystyle \) \(=\) \(\displaystyle 476 + 480i\) simplifying

we can write:

\(\displaystyle \left({5 + i}\right)^4 \left({-239 + i}\right)\) \(=\) \(\displaystyle \left({476 + 480 i}\right) \left({-239 + i}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle -113764 - 480 - 114720i + 476i\)
\(\displaystyle \) \(=\) \(\displaystyle -114244 - 114244i\)



From there, we substitute:

\(\displaystyle \arg \left({\left({5 + i}\right)^4 \left({-239 + i}\right)}\right)\) \(=\) \(\displaystyle \arg \left({-114244 - 114244 i}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \arctan \frac {-114244} {-114244}\)
\(\displaystyle \) \(=\) \(\displaystyle \arctan 1\)
\(\displaystyle \) \(=\) \(\displaystyle \frac \pi 4\)

By transitivity:

$\dfrac \pi 4 = 4 \arctan \dfrac 1 5 - \arctan \dfrac 1 {239}$

$\blacksquare$


Also known as

Some sources give this as Machin's identity.


Source of Name

This entry was named for John Machin.


Historical Note

John Machin devised his formula for $\pi$ in $1706$.

It allowed him to calculate $\pi$ to over $100$ decimal places.

This greatly surpassed the work of Ludolph van Ceulen.


Sources