# Polar Form Of A Complex Number

**Polar Form of a Complex Number:**

A consequence of our first definition of the complex exponential— is that if ζ C, |ζ | = 1, then there is a unique number θ, 0 ≤ 2, such that ζ = e^{iθ} (see below Figure). Here θ is the (signed) angle between the positive x axis and the ray Now, if z is any non-zero complex number, then

where ζ = z/|z| has modulus 1. Again letting θ be the angle between the real axis and we see that,

where r = |z|. This form is called the polar representation for the complex number z. (Note that some classical books write the expression z = re^{iθ} = r(cos θ i sin θ) as z = rcis θ . The reader should be aware of this notation, though we shall not use it in this book.) Engineers like the cis notation.

**Example-1
**

The unit-modulus number in parenthesis subtends an angle of π/3 with the positive x-axis. Therefore

It is often convenient to allow angles that are greater than or equal to 2 in the polar representation; when we do so, the polar representation is no longer unique. For if k is an integer, then