Complex conjugate

Let $z \in \mathbb C$. We note $\Re(z)$ the real part of $z$ and $\Im(z)$ the imaginary part of $z$.

Definition

Definition

For all complex $z$, we call $\bar z$ the complex conjugate of the complex number $z$, defined by

$$\bar z = \Re(z) - i \Im(z).$$

Results

Proposition

For all $z \in \mathbb C$, we have : $\qquad \Re(z) = \frac{z + \bar z}{2} \qquad \Im{z} = \frac{z - \bar z}{2i} \quad \text{and} \quad \bar{\bar{z}} = z$.

Proposition

Let $z_1, \dots, z_n \in \mathbb C$. Then we have

$$\overline{\sum_{k = 1}^{n} z_k} = \sum_{k=1}^n \overline{z_k} \qquad \text{and} \qquad \overline{\prod_{k = 1}^{n} z_k} = \prod_{k=1}^n \overline{z_k}$$