the polar and exponential form

There are two equally good ways to name a point in the complex plane. The rectangular form z = x + i y tells you how far right and how far up. The polar form tells you instead how far from the origin and in what direction: z = r (cos theta + i sin theta), where r = |z| is the modulus and theta = arg z is the argument. The two are linked by x = r cos theta and y = r sin theta — the same conversion you know from polar coordinates.

Euler's formula, e^(i theta) = cos theta + i sin theta, lets you write the polar form even more compactly as z = r e^(i theta), called the exponential form. This is not just shorthand: it makes multiplication almost effortless. Because exponents add, (r e^(i theta))(s e^(i phi)) = r s e^(i(theta + phi)) — to multiply two complex numbers you multiply their moduli and add their arguments. Division reverses it: divide the moduli, subtract the arguments. What was a fiddly algebraic expansion becomes one line.

The exponential form is the lens through which most of complex analysis is best seen. Multiplication is now visibly a scaling (by r) plus a rotation (by theta); powers and roots follow instantly from de Moivre's theorem; the roots of unity sit at evenly spaced angles; and the complex exponential and logarithm are built directly on top of it. If you take away one tool from the foundations, let it be z = r e^(i theta).