#math/linear-algebra The **complex conjugate** of a complex number $z = a + bi$ (with $a,b \in \mathbb{R}$) is denoted $\bar{z}$ and defined by flipping the sign of the imaginary part: $ \bar{z} = \overline{a + bi} = a - bi. $ Multiplying a complex number by its conjugate yields a **non-negative real number** — the square of its [[Absolute Value]]: $ z\bar{z} = (a+bi)(a-bi) = a^2 + b^2. $ This fact that enables [[1A2 - Arithmetic of Complex Numbers|complex division]]: multiplying the numerator and denominator of $\frac{\beta}{\alpha}$ by $\bar{\alpha}$ turns the denominator into a real scalar. With $w, z \in \mathbb{C}$: | Identity | Statement | | ---------------------- | -------------------------------------- | | Conjugate of a sum | $\overline{w + z} = \bar{w} + \bar{z}$ | | Conjugate of a product | $\overline{wz} = \bar{w}\,\bar{z}$ | | Involution | $\bar{\bar{z}} = z$ | | Real part | $z + \bar{z} = 2\,\mathrm{Re}(z)$ | | Imaginary part | $z - \bar{z} = 2\,\mathrm{Im}(z)\,i$ | | Real iff | $z = \bar{z} \iff z \in \mathbb{R}$ | _Notation: $\mathrm{Re}(a+bi)=a$ and $\mathrm{Im}(a+bi)=b$._