Complex Numbers and Operations

Introduction to Complex Numbers

A complex number has the form: $z = a + bi$

Where:

• $a$ is the real part: $\text{Re}(z) = a$
• $b$ is the imaginary part: $\text{Im}(z) = b$
• $i$ is the imaginary unit: $i = \sqrt{-1}$, so $i^2 = -1$

Key Properties of $i$

• $i^1 = i$
• $i^2 = -1$
• $i^3 = i^2 \cdot i = -i$
• $i^4 = i^2 \cdot i^2 = 1$
• Pattern repeats: $i^5 = i$, $i^6 = -1$, etc.

To find $i^n$: divide $n$ by 4 and use the remainder.

Complex Number Operations

Addition and Subtraction

Combine like terms (real with real, imaginary with imaginary): $(a + bi) + (c + di) = (a + c) + (b + d)i$ $(a + bi) - (c + di) = (a - c) + (b - d)i$

Multiplication

Use the distributive property and $i^2 = -1$: $(a + bi)(c + di) = ac + adi + bci + bdi^2$ $= ac + adi + bci - bd$ $= (ac - bd) + (ad + bc)i$

FOIL method works:

• First: $ac$
• Outer: $adi$
• Inner: $bci$
• Last: $bdi^2 = -bd$

Division

To divide complex numbers, multiply by the conjugate of the denominator: $\frac{a + bi}{c + di} = \frac{a + bi}{c + di} \cdot \frac{c - di}{c - di}$

The denominator becomes real: $\frac{(a + bi)(c - di)}{c^2 + d^2}$

Complex Conjugate

The conjugate of $z = a + bi$ is: $\bar{z} = a - bi$

Properties:

• $z \cdot \bar{z} = (a + bi)(a - bi) = a^2 + b^2$ (always real and non-negative)
• $(z + w)^* = \bar{z} + \bar{w}$
• $(zw)^* = \bar{z} \cdot \bar{w}$
• $\overline{\bar{z}} = z$

Absolute Value (Modulus)

The absolute value or modulus of $z = a + bi$ is: $|z| = \sqrt{a^2 + b^2}$

This represents the distance from the origin in the complex plane.

Properties:

• $|z| \geq 0$
• $|z| = 0$ if and only if $z = 0$
• $|zw| = |z| \cdot |w|$
• $|\frac{z}{w}| = \frac{|z|}{|w|}$ (if $w \neq 0$)
• $|z|^2 = z \cdot \bar{z}$

Complex Plane (Argand Diagram)

Complex numbers can be plotted on a coordinate plane:

• Horizontal axis: Real part
• Vertical axis: Imaginary part
• $z = a + bi$ corresponds to point $(a, b)$

Polar Form

A complex number can also be written in polar form: $z = r(\cos\theta + i\sin\theta) = r\text{cis}\theta$

Or using Euler's formula: $z = re^{i\theta}$

Where:

• $r = |z| = \sqrt{a^2 + b^2}$ (modulus)
• $\theta = \arg(z) = \arctan(\frac{b}{a})$ (argument/angle, adjusted for quadrant)

Solving Equations with Complex Numbers

Quadratic Equations

For $ax^2 + bx + c = 0$ with discriminant $b^2 - 4ac < 0$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-b \pm i\sqrt{|b^2 - 4ac|}}{2a}$

Complex Conjugate Root Theorem: If $a + bi$ is a root of a polynomial with real coefficients, then $a - bi$ is also a root.

Higher Degree Equations

Complex solutions come in conjugate pairs for polynomials with real coefficients.

Key Theorems

1. Fundamental Theorem of Algebra: Every polynomial of degree $n \geq 1$ has exactly $n$ complex roots (counting multiplicity).

2. Complex Conjugate Root Theorem: If a polynomial has real coefficients and $a + bi$ is a root, then $a - bi$ is also a root.