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 $x \ge 1$.
Proof 1
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 $45^\circ$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {120 / 119 - 1} {120 / 119 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {239}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 4 \alpha - \frac \pi 4\) | \(=\) | \(\ds \arctan \frac 1 {239}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac \pi 4\) | \(=\) | \(\ds 4 \arctan \frac 1 5 - \arctan \frac 1 {239}\) |
$\blacksquare$
Proof 2
\(\ds \map \arg {\paren {5 + i }^4 \paren {239 - i} }\) | \(=\) | \(\ds \map \arg {5 + i}^4 + \map \arg {239 - i}\) | Argument of Product equals Sum of Arguments | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \map \arg {5 + i} + \map \arg {239 - i}\) | Argument of Product equals Sum of Arguments | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \arctan \frac 1 5 + \arctan \frac {-1} {239}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \arctan \frac 1 5 - \arctan \frac 1 {239}\) | Inverse Tangent is Odd Function |
The validity of the material on this page is questionable. In particular: We cannot take it for granted that $\map \arg z = \arctan \dfrac {\Im z} {\Re z}$, we have to be sure we know what quadrant we are in. Clearly here $\Re z > 0$ so complications don't directly arise, but we might want to use the cosine/sine definition to be rigorous You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Furthermore, because:
\(\ds \paren {5 + i}^4\) | \(=\) | \(\ds 5^4 + 4 \times 5^3 i - 6 \times 5^2 - 4 \times 5 i + 1\) | Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds 476 + 480i\) | simplifying |
we can write:
\(\ds \paren {5 + i}^4 \paren {239 - i}\) | \(=\) | \(\ds \paren {476 + 480 i} \paren {239 - i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 113764 + 480 + 114720i - 476i\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 114244 + 114244i\) |
From there, we substitute:
\(\ds \map \arg {\paren {5 + i }^4 \paren {239 - i} }\) | \(=\) | \(\ds \map \arg {114244 + 114244 i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \arctan \frac {114244} {114244}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \arctan 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4\) |
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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
- Weisstein, Eric W. "Machin's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MachinsFormula.html