1.1Complex Numbers and the Complex Plane

A complex number is an expression of the form $z = x + iy$, where $x, y \in \mathbb{R}$ and $i^2 = -1$.

1. We write $x = \text{Re}(z)$ for the real part of $z$, and $y = \text{Im}(z)$ for the imaginary part of $z$.

2. If $\text{Re}(z) = 0$, then $z$ is purely imaginary.

3. The set of all complex numbers is denoted by $\mathbb{C}$.

$\mathbb{C}$ is isomorphic to the Euclidean plane $\mathbb{R}^2$ as a real vector space via the canonical bijection:

Under this identification, the real numbers correspond to the $x$-axis (the real axis) and the purely imaginary numbers to the $y$-axis (the imaginary axis).

Let $z = x + iy \in \mathbb{C}$.

1. The absolute value (or modulus) of $z$ is the Euclidean length of its corresponding vector in $\mathbb{R}^2$:

   $$|z| = (x^2 + y^2)^{1/2}$$

2. The complex conjugate of $z$ is its reflection across the real axis:

   $$\overline{z} = x - iy$$

Let $z, z_1, z_2 \in \mathbb{C}$. The following properties hold:

1. $\text{Re}(z) = \frac{z + \overline{z}}{2}$ and $\text{Im}(z) = \frac{z - \overline{z}}{2i}$.

2. $|z|^2 = z \overline{z}$.

3. If $z \neq 0$, then $\frac{1}{z} = \frac{\overline{z}}{|z|^2}$.

4. Triangle Inequality: $|z_1 + z_2| \leq |z_1| + |z_2|$.

5. Reverse Triangle Inequality: $||z_1| - |z_2|| \leq |z_1 - z_2|$.

Any non-zero complex number $z \in \mathbb{C}$ can be written in polar form:

where $r = |z| > 0$ is the modulus, and $\theta \in \mathbb{R}$ is the argument of $z$ (defined uniquely up to a multiple of $2\pi$), denoted by $\arg z$. By Euler's formula:

If $z = r e^{i\theta}$ and $w = s e^{i\phi}$, then $zw = rs e^{i(\theta + \phi)}$. Thus, multiplication by a complex number corresponds geometrically to a homothety in $\mathbb{R}^2$ (a rotation by $\arg z$ composed with a dilation by $|z|$).