How to Divide Complex Numbers

Example Let's divide the following 2 complex numbers

Step 1) Determine the conjugate of the denominator	The conjugate of (7+4i) is (7-4i)
Step 2) Multiply the numerator and denominator by the conjugate.
Step 3) Simplify. (Remember that i2 =-1)

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Video Tutorial on Dividing Complex Numbers

Problem 1) Divide the complex numbers below::

Step 1

Step 1) Determine the conjugate of the denominator

The conjugate of (2 + 6i ) is (2 - 6i)

Step 2

Step 2) Multiply the numerator and denominator by the conjugate.

Step 3

Step 3) Simplify. (Remember that i2 =-1)

Problem 2)Find the quotient

Step 1

Step 1) Determine the conjugate of the denominator

The conjugate of (5 + 7i ) is (5 - 7i)

Step 2

Step 2) Multiply the numerator and denominator by the conjugate.

Step 3

Step 3) Simplify

Are there any Interesting Complex Quotients?

The next two problems will example the following two quotients

Problem 3)

Problem 4)

Make a Prediction: Do you think that there will be anything special or interesting about either of the following quotients? Scroll down the page to see the answer.

Problem 3)Find the quotient

Step 1

Step 1) Determine the conjugate of the denominator

The conjugate of (3 + 2i ) is (3 - 2i)

Step 2

Step 2) Multiply the numerator and denominator by the conjugate.

Step 3

Step 3) Simplify

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Problem 4)Find the quotient

Step 1

Step 1) Determine the conjugate of the denominator

The conjugate of (5 + 7i ) is (5 - 7i)

Step 2

Step 2) Multiply the numerator and denominator by the conjugate.

Step 3

Step 3) Simplify

A Parting Question

After looking at problem 4, do you think that all complex quotients of the form are equivalent to -1?

Answer

The answer is yes.
Any rational-expression in the form is equivalent to -1(with the caveat that y −x ≠0)