IMG_8156.HEIC

## Complex Numbers ### Definition of Complex Numbers A complex number is a number of the form $a + bi$, where: - $a$ and $b$ are real numbers. - $i$ is the imaginary unit, defined as $i = \sqrt{-1}$. Thus, $i^2 = -1$. ### Set of Complex Numbers The set of all complex numbers is denote...

## Complex Numbers ### Definition of Complex Numbers A complex number is a number of the form $a + bi$, where: - $a$ and $b$ are real numbers. - $i$ is the imaginary unit, defined as $i = \sqrt{-1}$. Thus, $i^2 = -1$. ### Set of Complex Numbers The set of all complex numbers is denoted by $\mathbb{C}$. $\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\}$ ### Components of a Complex Number For a complex number $z = a + bi$: - $a$ is the real part, denoted as $\operatorname{Re}(z)$. - $b$ is the imaginary part, denoted as $\operatorname{Im}(z)$. ### Examples 1. $z = 3 + 2i$: $\operatorname{Re}(z) = 3$, $\operatorname{Im}(z) = 2$ 2. $z = -2 - i$: $\operatorname{Re}(z) = -2$, $\operatorname{Im}(z) = -1$ 3. $z = 5i$: $\operatorname{Re}(z) = 0$, $\operatorname{Im}(z) = 5$ 4. $z = 7$: $\operatorname{Re}(z) = 7$, $\operatorname{Im}(z) = 0$ ### Equality of Complex Numbers Two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ are equal if and only if their real and imaginary parts are equal: $z_1 = z_2 \Leftrightarrow a = c$ and $b = d$ ### Operations with Complex Numbers Given two complex numbers $z_1 = a + bi$ and $z_2 = c + di$: 1. **Addition:** $z_1 + z_2 = (a + c) + (b + d)i$ 2. **Subtraction:** $z_1 - z_2 = (a - c) + (b - d)i$ 3. **Multiplication:** $z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$ 4. **Division:** $\frac{z_1}{z_2} = \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$, where $z_2 \neq 0$.