Machin's Formula for Pi/Proof 1
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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 $x \ge 1$.
Proof
Let $\tan \alpha = \dfrac 1 5$.
Then:
| \(\ds \tan 2 \alpha\) | \(=\) | \(\ds \frac {2 \tan \alpha} {1 - \tan^2 \alpha}\) | Double Angle Formula for Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 / 5} {1 - 1 / 25}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 5 {12}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \tan 4 \alpha\) | \(=\) | \(\ds \frac {2 \tan 2 \alpha} {1 - \tan^2 2 \alpha}\) | Double Angle Formula for Tangent | ||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 \times 5 / 12} {1 - 25 / 144}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {120} {119}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \tan \left({4 \alpha - \frac \pi 4}\right)\) | \(=\) | \(\ds \frac {\tan 4 \alpha - \tan \frac \pi 4} {1 + \tan 4 \alpha \tan \frac \pi 4}\) | Tangent of Difference | ||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\tan 4 \alpha - 1} {1 + \tan 4 \alpha}\) | Tangent of $45^\circ$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {120 / 119 - 1} {120 / 119 + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {239}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 4 \alpha - \frac \pi 4\) | \(=\) | \(\ds \arctan \frac 1 {239}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac \pi 4\) | \(=\) | \(\ds 4 \arctan \frac 1 5 - \arctan \frac 1 {239}\) |
$\blacksquare$
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.