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Example: Convert the complex number 5 ∠ 53° to rectangular form.
Example: Convert the complex number 5 + 2 i to polar form.
Example: Convert the complex number −5 − 2 i to polar form.
Multiplying and Dividing Complex Numbers in Polar Form
- Multiplication rule: To form the product multiply the magnitudes and add the angles.
x m · x n = x m + n and x m / x n = x m − n . They suggest that perhaps the angles are some kind of exponents. This guess turns out to be correct. We will prove in the next section that: where as usual the base e is the number e = 2.71828… The form r e i θ is called exponential form of a complex number. The next example shows the same complex numbers being multiplied in both forms:
Proof of Euler’s Identity and Proof that r ∠ θ = r e i θ
Now that we have Euler’s identity it is easy to prove that r ∠ θ = r e i θ . Again we will manipulate the left side until it equals the right side. First, convert the left side to rectangular form: Now factor out r and use Euler’s identity on the terms in the brackets. The result is that we have converted the left side to equal the right side. This proves that r ∠ θ = r e i θ
Examples Using Rectangular, Polar and Exponential Form
Example 2: Calculate i n for n = 1, 2, 3, …, plot the numbers on the complex plane, and spot the pattern.
Example 3: Evaluate the exponential (3 + 4 i) (6 + 7 i)
Example 4: DeMoivre’s Theorem states that the equation x n = C (where C is any complex number and n is any positive integer has n complex roots (solutions) and that they are uniformly distributed in a circle about the origin of the complex plane.
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