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# Dividing Complex Numbers

To divide a complex number by a real number, divide each term by the real number, just as we would divide a binomial by a number. For example,

To understand division by a complex number, we first look at imaginary numbers that have a real product. The product of the two imaginary numbers (3 + i)(3 - i) is a real number, namely (3 + i)(3 - i) = 10

We say that 3 + i and 3 - i are complex conjugates of each other.

Complex Conjugates

The complex numbers a + bi and a - bi are called complex conjugates of one another. Their product is the real number a2 + b2.

Example 1

Products of conjugates

Find the product of the given complex number and its conjugate.

a) 2 + 3i

b) 5 - 4i

Solution

a) The conjugate of 2 + 3i is 2 - 3i.

 (2 + 3i)(2 - 3i) = 4 - 9i2 = 4 + 9 = 13

b) The conjugate of 5 - 4i is 5 + 4i.

 (5 - 4i)(5 + 4i) = 25 + 16 = 41

We use the idea of complex conjugates to divide complex numbers. The process is similar to rationalizing the denominator. Multiply the numerator and denominator of the quotient by the complex conjugate of the denominator.

Example 2

Dividing complex numbers

Find each quotient. Write the answer in the form a + bi.

Solution

a) Multiply the numerator and denominator by 3 + 4i, the conjugate of 3 - 4i:

b) Multiply the numerator and denominator by 2 - i, the conjugate of 2 + i:

c) Multiply the numerator and denominator by -i, the conjugate of i:

The symbolic definition of division of complex numbers follows.

Division of Complex Numbers

We divide the complex number a + bi by the complex number c + di as follows:

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