Complex Conjugation

Definition

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

The complex conjugate of $a + b i$ , where 'a' is the real part and 'b' is the imaginary part, is $a - b i$ . Geometrically, the complex conjugate is the reflection of the complex number about the real axis (X-axis) in the argand plane.

Notation

The complex conjugate of a complex number, $z$ , is denoted by $\bar{z}$ , or $z^{*}$ . The second is preferred in physics, and computer engineering, since the first, the bar notation, could be confused with the Logical Negation Boolean algebra symbol.

Properties

Conjugation is distributive over addition, subtraction, multiplication and division.

Info

Since, a complex number is equal to its complex conjugate if its imaginary part is zero- real numbers are the only fixed points of conjugation.

Conjugation does not change with the modulus operation.
Conjugation is an involution, that is, the conjugate of the conjugate of a complex number $z$ is $z$ .
Conjugation is commutative under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for non-zero arguments.

Definition

Properties

Related Notes