Subsection CCN Conjugates of Complex Numbers

Definition CCN Conjugate of a Complex Number

The conjugate of the complex number $\alpha=a+bi\in\complex{\null}$ is the complex number $\conjugate{\alpha}=a-bi$.

Example CSCN Conjugate of some complex numbers

Notice how the conjugate of a real number leaves the number unchanged. The conjugate enjoys some basic properties that are useful when we work with linear expressions involving addition and multiplication.

Theorem CCRA Complex Conjugation Respects Addition

Suppose that $\alpha$ and $\beta$ are complex numbers. Then $\conjugate{\alpha+\beta}=\conjugate{\alpha}+\conjugate{\beta}$.

Proof

Theorem CCRM Complex Conjugation Respects Multiplication

Suppose that $\alpha$ and $\beta$ are complex numbers. Then $\conjugate{\alpha\beta}=\conjugate{\alpha}\conjugate{\beta}$.

Proof

Theorem CCT Complex Conjugation Twice

Suppose that $\alpha$ is a complex number. Then $\conjugate{\conjugate{\alpha}}=\alpha$.

Proof