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Complex Conjugates

Two complex numbers of the form a + bi and a - bi are called complex conjugates. Notice that complex conjugates have the same real part. Their imaginary parts are the same, except they have opposite signs.

Here are several pairs of complex conjugates:

 7 + 2i-3 + 6i 12 - 8i andand and 7 - 2i-3 - 6i 12 + 8i

Note:

Here are several pairs of complex numbers that are NOT complex conjugates:

 4 - 7i3 + 5i 7 + 2i andand and 7 - 4i-3 - 5i -7 + 2i

Example 1

Find: (6 + 4i)(6 - 4i)

 Solution Multiply using FOIL. Multiply the factors in each term. Replace i2 with -1. Combine like terms. Note that the i terms cancel. So, (6 + 4i)(6 - 4i) = 52. (6 + 4i)(6 - 4i) = 6 Â· 6 - 6 Â· 4i + 4i Â· 6 - 4i Â· 4i = 36 - 24i + 24i - 16i2 = 36 - 24i + 24i - 16(-1) = 52
Notice that (6 + 4i) and (6 - 4i) are complex conjugates.

Their product is 52, a real number.

The next example demonstrates that the product of two complex conjugates is always a real number.

Example 2

Find: (a + bi)(a - bi)

 Solution Multiply using the FOIL method. Multiply the factors in each term. Replace i2 with -1.  Combine like terms. Note that the i terms add to zero.Thus, (a + bi)(a - bi) = a2 + b2 (a + bi)(a - bi) = a Â· a - a Â· bi + bi Â· a - bi Â· bi = a2 - abi + abi - b2i2 = a2 - abi + abi - b2(-1) = a2 - abi + abi + b2 = a2 + b2

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