[Analytic Number Theory] Deriving Machin's Formula using Arguments of Complex Numbers

We can derive Machin's formula by inspecting the complex number relation $(5+i)^4 = 2 (1+i) (239 + i)$ (1).

Essentially, what we are doing is relating each quantity of Machin's formula to the argument of a complex number.

First, we recall that the sum of arguments of complex numbers is the argument of their product, and the argument of a complex number raised to a power is simply that number times the argument. That is, the relations $Arg(z,w) = Arg(z) + Arg(w)$ , and $Arg(z^k) = k( Arg(z))$ hold for all $z \in \mathbb{C}$ .

Taking the argument of both sides of (1), we get $Arg((5+i)^4) = Arg( 2(1+i)(239+i))$ $4(Arg(5+i)) = Arg (2(1+i)) + Arg(239+i)$ $4(Arg(5+i)) -Arg(239+i)= Arg (2(1+i))$ Since $Arg(a+bi) = \arctan (\frac{b}{a})$ , RHS $= \arctan(\frac{2}{2}) = \arctan(1) = \frac{\pi}{4}$ .

So we have the following relation: $4 \arctan(\frac{1}{5}) - \arctan(\frac{1}{239}) = \frac{\pi}{4}$ as required.

It is also possible to derive other Machin-like formulas via the same method using complex numbers. You can try deriving the following formulas as an exercise:

Euler's $\frac{\pi}{4} = \arctan \frac{1}{2} + \arctan \frac{1}{3}.$ Hermann's $\frac{\pi}{4} =2 \arctan \frac{1}{2} - \arctan \frac{1}{7}.$ Hutton's $\frac{\pi}{4} = 2\arctan \frac{1}{3} + \arctan \frac{1}{7}.$

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