“Dividing” Complex Numbers with a Conjugate
Mathematicians (that’s you) can add, subtract, and multiply complex numbers. Technically, you can’t divide complex numbers — in the traditional sense. You divide complex numbers by writing the division problem as a fraction and then multiplying the numerator and denominator by a conjugate.
The conjugate of the complex number a + bi is a – bi.
The product of (a + bi)(a – bi) is a^{2} + b^{2}. How does that happen? Where’s the i?
Look at the steps in the multiplication: (a + bi)(a – bi) = a^{2} – abi + abi – b^{2}i^{2} = a^{2} – b^{2}(–1) = a^{2} + b^{2}, which is a real number — with no complex part. So when you need to divide one complex number by another, you multiply the numerator and denominator of the problem by the conjugate of the denominator.
This step creates a real number in the denominator of the answer, which allows you to write the answer in the standard form of a complex number.
Sample question

Divide 10 + 5i by 4 – 3i.
1 + 2i. Express the division as a fraction. Multiply both the numerator and denominator of the fraction by 4 + 3i. You get
Writing the answer in standard form, you get 1 + 2i.
Practice questions

Divide 40 – 20i by 3 + i.

Divide 5 + 10i by 2 – i.

Divide 20 – 10i by –3 + 4i.

Divide 20i by 2 + 6i.
Following are answers to the practice questions:

The answer is 10 – 10i.
Write the problem as a fraction. Then multiply both the numerator and denominator by the conjugate of the denominator, 3 – i. Simplify, and write the final answer in a + bi form:

The answer is 5i.
Write the problem as a fraction. Then multiply both the numerator and denominator by the conjugate of the denominator, 2 + i. Simplify, and write the final answer in a + bi form:

The answer is –4 – 2i.
Write the problem as a fraction. Then multiply both the numerator and denominator by the conjugate of the denominator, –3 – 4i. Simplify, and write the final answer in a + bi form:

The answer is 3 +i.
Write the problem as a fraction. Then multiply both the numerator and denominator by the conjugate of the denominator, 2 – 6i. Simplify, and write the final answer in a + bi form: