How Do You Divide Complex Numbers?

To divide by complex numbers, multiply by the conjugate, distribute to eliminate the parentheses, simplify, combine like terms, and represent the answer in standard form: a + bi. Remember that i^2 = (-1) when sorting out constants and imaginary terms.

  1. Multiply by the conjugate

    Consider the example problem (2+3i) / (4-4i). Reverse the sign in the denominator to get the conjugate (4+4i), and multiply both numerator and denominator by (4+4i). Write the results before factoring. In this example, you get (2+3i)(4+4i)/(4-4i)(4+4i).

  2. Eliminate the parentheses

    Distribute (or use the FOIL method) to get rid of the parentheses. Multiply the first terms of both expressions in the numerator by each other. Then, multiply the first term in the first expression by the last term in the second expression. Next, multiply the second term in the first expression by the first term in the second expression. Finally,multiply the second terms in both expressions by each other. Repeat the process for the denominator. Write the expanded answer in both the numerator and the denominator. In this example, you get (8+12i+8i+12i^2)/(16-16i+16i-16i^2).

  3. Simplify, and combine like terms

    Convert i^2 to (-1), which yields (8+12i+8i-12)/(16-16i+16i+16). Combine like terms by completing addition and subtraction among like real numbers and like imaginary numbers, yielding (-4+20i)/32. Represent in standard form (a+bi), which yields -4/32 + 20i/32. Reduce fractions when possible to leave the answer in the simplest form. In this example, the final answer is -1/8 + 5i/8.