# Machin's Formula for Pi

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## Theorem

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

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.