Background Material

Before we start to think about $\pi$ we will need some information about complex numbers. In 1797, the Norwegian surveyor Caspar Wessel was the first to discover a geometric interpretation of complex numbers that greatly simplifies the computations when multiplying complex numbers. For Wessel (and now us), a complex number was simply a point $a+bi$ in the complex plane. Associated with this point is a directed line segment from the origin.

This was a major breakthrough, but an even bigger contribution was his idea about how to multiply these directed line segments in the complex plane. Wessel decided (using analogies to real numbers) that the length of the product should be equal to the product of the two lengths. But what about the direction of the product? Wessel said ``the line segment direction should differ in direction from each line segment factor by the same angular amount as the other line segment factor differs in direction when compared to the unit direction line segment.'' What this means is that we simply add the angles of the line segment factors to come up with the new direction. For example, if one line segment had an angle $\theta=\pi/3$ and the other line segment had an angle $\phi=\pi/4$, the angle of the directed line segment arising from their product would be $\alpha=\pi/3+\pi/4=7\pi/12$.

A complex number in the form $a+bi$ is said to be in Cartesian coordinates. An enormously useful alternative is the polar form where each point in the complex plane is represented by its magnitude and its angle measured off the positive x-axis. (It is also convenient to represent points in the standard $(x,y)$-plane in polar coordinates as well).

$$a+bi=\sqrt{a^2+b^2}\angle\arctan\left(b/a\right)\;$$when$$\;a>0$$

If the complex number is in the second or third quadrants we have

$$a+bi=\sqrt{a^2+b^2}\angle\;\pi+\arctan\left(b/a\right)\;$$   when$$\;a<0$$

Let's use the ideas above to determine the angle resulting from the multiplication $(5+i)^4(-239+i)$. The angle of the line segment $(5+i)$ is $\arctan(1/5)$. Therefore the angle of the line segment $(5+1)^4$ is $4\arctan(1/5)$. Next, the angle of the complex number $(-239+i)$ is $\pi+\arctan(1/(-239))=\pi-\arctan(1/239)$. So we get

$$(5+i)^4(-239+i)=r\angle\;4\tan^{-1}(1/5)+\pi-\tan^{-1}(1/239)$$

where $r$ is the magnitude of the vector. Just doing the calculations by hand we get

$$(5+i)^4 = (476+480i)$$

$$\Downarrow$$

$$(5+i)^4(-239+i) = (476+480i)(-239+i)$$

$$= -110244-110244i$$

$$= -110244(1+i)$$

This is a line segment in the third quadrant with angle $5\pi/4$. So we now have the following equality

$$4\tan^{-1}(1/5)+\pi-\tan^{-1}(1/239) = \displaystyle \frac{5\pi}{4}$$

$$\Downarrow$$

$$4\tan^{-1}(1/5)-\tan^{-1}(1/239) = \displaystyle \frac{\pi}{4}$$

This is called Machan's formula which we will now use to compute $\pi$ to as many digits as we please.