Section 6.5 Complex Numbers in Polar Form

Overview Recall that a complex number is written in the form a + bi, where a and b are real numbers and While it is not possible to graph complex numbers on a real number plane, a similar setup can be used.

The Complex Plane

Graph Each of the Following z = 3i z = i z = 3 – 4i

Absolute value of a complex number The absolute value of a complex number z is the distance from the origin to the point z in the complex plane:

Polar form of a complex number When a complex number is in a + bi form, it is said to be in rectangular form. Just as we superimposed the polar plane onto the rectangular coordinate plane, we can do the same thing with the complex plane.

Continued r is called the modulus and “theta” is called the argument.

Examples Graph each of the following, then write the complex number in polar form:

Now, the Other Way Write each complex number in rectangular form:

Products and Quotients Given, two complex numbers in polar form. Their product and quotient can be found by the following:

In Other Words… When multiplying, multiply the moduli and add the arguments. When dividing, divide the moduli and subtract the arguments. Keep in mind that you may have to re- name your argument so that is an angle between 0 and 360° or 0 and 2π radians.

Raising to a Power When raising a complex number to a power, use DeMoivre’s Theorem: In other words, raise the modulus to the nth power and multiply the argument by n (again, be prepared to rename your argument).

A Final Word Before the Examples Pay particular attention to the form your final answer should take (complex polar or complex rectangular).

Find the Product (Answer in Polar Form)

Find the Quotient z 1 /z 2 (Answer in Polar Form)

Use the French Guy’s Theorem (write answers in rectangular form)

Finding Complex Roots Let w = r(cos θ + i sin θ) be a complex number in polar form. w has n distinct complex n th roots given by

Examples Find all the complex cube roots of 8. Write your answers in rectangular form. Find all the complex fourth roots of 16(cos120° + I sin120°). Write your answers in polar form.