Machin's Formula for Pi/Proof 2

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

\(\ds \map \arg {\paren {5 + i }^4 \paren {239 - i} }\)

\(=\)

\(\ds \map \arg {5 + i}^4 + \map \arg {239 - i}\)

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:

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

$\blacksquare$

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.