# Complex Numbers

## Polar Form Terminology

Recall that in addition to rectangular form, $$a+bi$$, we can also represent complex numbers in polar form:

$$z=r(cos\theta+isin\theta)$$,

where $$r$$ is called the modulus and $$\theta$$ is called the argument of the complex number.

Sometimes, you might see $$z=r(cos\theta+isin\theta)$$ abbreviated as $$z=rcis\theta$$.

1. Convert all of the complex numbers from polar form to rectangular form (see the Rectangular/Polar Form Conversion page).

2. Perform addition/subtraction on the complex numbers in rectangular form (see the Operations in Rectangular Form page).

3. After evaluating the sum/difference, convert it back into polar form for your answer (see the Rectangular/Polar Form Conversion page).

The following example will demonstrate how to apply these steps to find the sum/difference of complex numbers.

Example: Evaluate $$4\sqrt{2}(cos(\frac{3\pi}{4})+isin(\frac{3\pi}{4})) + 3(cos(\frac{\pi}{2})+isin(\frac{\pi}{2}))$$ in polar form (in radians), rounding to two decimal places.

See this video for the solution:

## Multiplication/Division

To multiply/divide complex numbers in polar form, multiply/divide the two moduli and add/subtract the arguments. More specifically, for any two complex numbers, $$z_1=r_1(cos(\theta_1)+isin(\theta_1))$$ and $$z_2=r_2(cos(\theta_2)+isin(\theta_2))$$, we have:

$$z_1 z_2=r_1r_2[cos(\theta_1+\theta_2)+isin(\theta_1+\theta_2)]$$

$$\frac{z_1}{z_2}=\frac{r_1}{r_2}[cos(\theta_1-\theta_2)+isin(\theta_1-\theta_2)], z_2 \ne 0$$

Example: Determine the product and quotient of $$z_1=5cis(\frac{5\pi}{2})$$ and $$z_2=7cis(\frac{3\pi}{4})$$.

Solution:

1. Identify the moduli and arguments. In this example, we have:

$$r_1=5,r_2=7,\theta_1=\frac{5\pi}{2},\theta_2=\frac{3\pi}{4}$$

2. Substitute these values into the above produce and quotient formulas:

$$z_1 z_2=(5)(7)[cos(\frac{5\pi}{2}+\frac{3\pi}{4})+isin(\frac{5\pi}{2}+\frac{3\pi}{4})]$$

$$\frac{z_1}{z_2}=\frac{5}{7}[cos(\frac{5\pi}{2}-\frac{3\pi}{4})+isin(\frac{5\pi}{2}-\frac{3\pi}{4})]$$

3. Evaluate the product and quotient by simplifying both expressions:

$$z_1 z_2=35[cos(\frac{13\pi}{4})+isin(\frac{13\pi}{4})]$$

$$\frac{z_1}{z_2}=\frac{5}{7}[cos(\frac{7\pi}{4})+isin(\frac{7\pi}{4})]$$