polar and cartesian coordinates

Since the time that coordinates were taken up as a way to give an algebraic treatment to geometric ideas, there has been an association between pairs of numbers and points in the plane. In the traditional representation, the (real, imaginary) pair (x,y) has been seen as a point with x-coordinate x and y coordinate y; in fact the correspondence is so familiar that it seems redundant to try to describe it.

Besides the rectangular cartesian coordinates, there are many others which can be used to locate points in the plane; one of the most convenient is polar coordinates, wherein

$$\begin{array}{cccc} \begin{array}{ccc} x & = & r \cos \the... ...\\ \theta & = & {\rm arctan}\ y/x \end{array} \end{array}$$

Familiarity with the power series expansions respectively of the exponential, sine, and cosine functions,

$$1 + x + \frac{1}{2}x^2+\frac{1}{3!}x^3 + \frac{1}{4!}x^4 + \cdots$$ e^x = (6)

$$\cos x 1 - \frac{1}{2}x^2 + \frac{1}{4!}x^4 + \cdots$$ = (7)

$$\sin x x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 + \cdots,$$ = (8)

suggests combining the two trigonometric series series to get Euler's formula

$$\cos x + i \sin x$$ e^ix = (9)

or the more general polar form

$$r e^{i\theta}$$ x + i y = (10)

for a complex number.