Complex conjugates are two complex numbers of the form a + bi and a - bi where a and b are real numbers.^{[3]} Complex conjugates are denoted using a bar over the variable: If z = a + bi then z = a - bi Complex conjugates have the property that, when added together, the result is a real number.
Complex conjugates also have the property that, when multiplied together, the result is a real number.
Complex conjugates are usually used when dividing complex numbers.
Equation | Description |
---|---|
(z + w) = z + w | The conjugate of a sum is equal to the sum of the conjugates. |
(z - w) = z - w | The conjugate of a difference is equal to the difference of the conjugates. |
(z · w) = z · w | The conjugate of a |
(z / w) = z / w | |
|z| = |z| | The magnitude of a complex number is the same as the magnitude of its complement. |
|z^{2}| = z·z = z·z | The magnitude of the square of a complex number equals the complex number multiplied by its conjugate. |
The conjugate of the conjugate of a complex number equals the original complex number. | |
e^{z} = (e^{z}) | Euler's number (e) raised to the conjugate of a complex number is equal to the conjugate of Euler's number raised to the complex number. |
Table 1 - Properties of complex conjugates |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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