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 |
and and
and |
7 - 2i -3 - 6i
12 + 8i |
Note:
Here are several pairs of complex numbers
that are NOT complex conjugates:
| 4 - 7i 3 + 5i
7 + 2i |
and and
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 · 6 - 6
· 4i + 4i
· 6 - 4i · 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 · a - a
· bi + bi
· a - bi · bi
= a2 - abi + abi - b2i2
= a2 - abi + abi - b2(-1)
= a2 - abi + abi + b2
= a2 + b2
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