The conjugate of a complex number a + bi is the complex number a - bi or a + -bi. The real part of the complex number a + bi is a and the imaginary part is b.
a + bi and a - bi are called complex conjugates.
Notice that to find the complex conjugate, all you need to do is to take the opposite of the imaginary part of the complex number.
Example #1
Find the complex conjugate of 5 + 9i
9 is the imaginary part. The opposite of 9 is -9.
The complex conjugate of 5 + 9i is 5 + -9i or 5 - 9i
Example #2
Find the complex conjugate of -10 - 3i
-10 - 3i = -10 + -3i.
-3 is the imaginary part. The opposite of -3 is 3.
The complex conjugate of -10 - 3i is -10 + 3i
Example #3
Find the complex conjugate of i
i = 1i.
1 is the imaginary part. The opposite of 1 is -1.
The complex conjugate of i is -i
The complex conjugate of a real number is the real number since there is no imaginary part. For example, the complex conjugate of 2 is 2.
Let us see what will happen when we multiply a complex number by its complex conjugate.
(a + bi)(a - bi) = a2 - abi + abi - b2i2
(a + bi)(a - bi) = a2 - b2i2
(a + bi)(a - bi) = a2 - b2(-1)
(a + bi)(a - bi) = a2 + b2
Since i is not here anymore, a2 + b2 is just a real number
Example
(4 + 3i)(4 - 3i) = 42 + 32
(4 + 3i)(4 - 3i) = 16 + 9
(4 + 3i)(4 - 3i) = 25
This property of the complex conjugate can be very useful when you are trying to simplify rational expressions or when you are trying to get rid of the "i" in the denominator of a rational expression.
For example, simplify 5i / (2 - i).
Multiply the numerator and the denominator of 5i / (2 - i) by the conjugate of 2 - i
The conjugate of 2 - i is 2 + i.
[5i(2 + i)] / (2 - i)(2 + i) = (10i + 5i2) / (22 + 12)
[5i(2 + i)] / (2 - i)(2 + i) = [10i + 5(-1)] / ( 4 + 1)
[5i(2 + i)] / (2 - i)(2 + i) = (10i - 5) / 5
[5i(2 + i)] / (2 - i)(2 + i) = 2i - 1
Therefore, 5i / (2 - i) = 2i - 1