Complex Numbers and Quadratic Equations - Algebra of Complex Numbers

Introduction to the Imaginary Unit $i$: The symbol $i$ is defined as $\sqrt{-1}$, such that $i^2 = -1$. Visually, if the real number line is a horizontal axis, multiplying a number by $i$ can be seen as a $90^\circ$ counter-clockwise rotation into a second dimension, called the imaginary axis.

Complex Numbers in Standard Form: A complex number $z$ is expressed in the form $a + bi$, where $a$ is the real part $\text{Re}(z)$ and $b$ is the imaginary part $\text{Im}(z)$. Geometrically, this represents a point $(a, b)$ on a 2D coordinate system known as the Argand Plane or Complex Plane, where the x-axis is the Real axis and the y-axis is the Imaginary axis.

Equality of Complex Numbers: Two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ are equal if and only if their corresponding real and imaginary parts are identical, meaning $a = c$ and $b = d$. This implies that a single equation involving complex numbers is actually a system of two real equations.

Addition and Subtraction: To add or subtract complex numbers, we combine the real parts and imaginary parts separately. Visually, adding two complex numbers $z_1$ and $z_2$ follows the Parallelogram Law of Vector Addition, where the resultant sum is the diagonal of the parallelogram formed by the vectors representing $z_1$ and $z_2$ from the origin.

Multiplication and Rotations: Multiplication of $(a + bi)$ and $(c + di)$ is performed using the distributive law (FOIL), replacing $i^2$ with $-1$. Conceptually, multiplying complex numbers involves scaling their distances from the origin and adding their angles relative to the positive real axis.

The Conjugate of a Complex Number: The conjugate of $z = a + bi$, denoted as $\bar{z}$, is $a - bi$. On the Argand plane, $\bar{z}$ is the mirror image or reflection of $z$ across the horizontal Real axis.

Modulus of a Complex Number: The modulus $|z|$ of $z = a + bi$ is defined as $\sqrt{a^2 + b^2}$. Visually, this represents the absolute distance of the point $(a, b)$ from the origin $(0, 0)$, following the Pythagorean theorem where $|z|$ is the length of the hypotenuse.

Powers of $i$: The powers of $i$ follow a repeating cyclic pattern of four values: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. For any integer $n$, $i^n$ can be simplified by finding the remainder when $n$ is divided by 4.