Complex Numbers

Complex numbers extend the real numbers by introducing the imaginary unit $i$, where $i^2 = -1$.

Definition

A complex number $z$ is expressed in standard (Cartesian) form as:

$$z = a + bi$$

where:

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

Number System Hierarchy

$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$

  • $\mathbb{N}$ — Natural numbers
  • $\mathbb{Z}$ — Integers
  • $\mathbb{Q}$ — Rational numbers
  • $\mathbb{R}$ — Real numbers (including irrational numbers $\mathbb{I}$)
  • $\mathbb{C}$ — Complex numbers (encompassing all of the above plus imaginary numbers)

Real and Imaginary Parts: Examples

$z = a + bi$ $2 + 3i$ $-1 - i\pi$ $10i$ $3$ $0$
$\text{Re}(z)$ $2$ $-1$ $0$ $3$ $0$
$\text{Im}(z)$ $3$ $-\pi$ $10$ $0$ $0$

Note: For real numbers like $3$, $\text{Im}(z) = 0$. For purely imaginary numbers like $10i$, $\text{Re}(z) = 0$.

Argand Diagram

The Argand diagram (or complex plane) provides a geometric representation of complex numbers:

  • Horizontal axis: Real axis
  • Vertical axis: Imaginary axis
  • A complex number $z = a + ib$ corresponds to the point $(a, b)$
  • The modulus $r = |z| = \sqrt{a^2 + b^2}$ is the distance from the origin to the point
  • The argument $\theta = \arg(z)$ is the angle measured from the positive real axis to the line joining the origin to $z$
  • The complex conjugate $\overline{z} = a - ib$ is the reflection of $z$ across the real axis

Forms of Complex Numbers

Cartesian Form

$$z = a + bi$$

Polar Form

$$z = r(\cos\theta + i\sin\theta) = r[\cos\theta + i\sin\theta]$$

where:

  • $r = |z| = \sqrt{a^2 + b^2}$ (modulus)
  • $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ (argument)

Conversion Procedure (4 Steps)

Given $z = a + bi$:

  1. Identify $a$ and $b$
  2. Find the radius: $r = \sqrt{a^2 + b^2}$
  3. Find the angle: $\theta = \tan^{-1}\left(\frac{b}{a}\right)$
    • Note: $-\pi < \theta \leq \pi$ (Principal Argument)
  4. Write: $z = r[\cos\theta + i\sin\theta]$

Examples from Lecture

  • Write $z = -4 + 4i$ in polar form
  • Write $z = \sqrt{3} - i$ in polar form

Exponential Form (Euler's Formula)

$$z = re^{i\theta} = r(\cos\theta + i\sin\theta)$$

Forms Comparison

graph TB
    Z["Complex Number z"]

    subgraph cartesian["Cartesian Form"]
        C1["z = a + bi"]
        C2["Components: Real a, Imaginary b"]
        C3["Use: Addition and Subtraction"]
    end

    subgraph polar["Polar Form"]
        P1["z = r[cosθ + isinθ]"]
        P2["Components: Modulus r, Argument θ"]
        P3["Use: Multiplication and Division"]
    end

    subgraph euler["Exponential Form"]
        E1["z = re^{iθ}"]
        E2["Components: Modulus r, Argument θ"]
        E3["Use: Powers, Roots, De Moivre"]
    end

    Z --> cartesian
    Z --> polar
    Z --> euler

    C1 -.->|"r = sqrt(a^2 + b^2)<br/>θ = tan^-1(b/a)"| P1
    P1 -.->|"Euler's Formula"| E1
    E1 -.->|"a = rcosθ<br/>b = rsinθ"| C1

    style Z fill:#e7f5ff,stroke:#1971c2
    style cartesian fill:#c5f6fa,stroke:#0c8599
    style polar fill:#fff4e6,stroke:#e67700
    style euler fill:#d3f9d8,stroke:#2f9e44

Powers of $i$

The imaginary unit cycles every 4 powers:

$i^0$ $i^1$ $i^2$ $i^3$
$1$ $i$ $-1$ $-i$
$i^4$ $i^5$ $i^6$ $i^7$
$1$ $i$ $-1$ $-i$

General rule: to simplify $i^n$, divide $n$ by 4 and use the remainder.

Operations

Addition/Subtraction

$$(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i$$

Example: Given $z_1 = 2 + 4i$ and $z_2 = 1 - 3i$: $$z_1 - z_2 = (2 + 4i) - (1 - 3i) = 1 + 7i$$

Multiplication

$$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$

Example: $(2 + 4i)(1 - 3i) = 2 - 6i + 4i - 12i^2 = 2 - 2i + 12 = 14 - 2i$

Division

To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator: $$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}$$

Example: $\frac{2 + 4i}{1 - 3i} = \frac{(2 + 4i)(1 + 3i)}{1^2 + 3^2} = \frac{2 + 6i + 4i + 12i^2}{10} = \frac{-10 + 10i}{10} = -1 + i$

Operations Flowchart

graph TD
    Start([Operation on<br/>Complex Numbers])
    Start --> ChooseOp{"Which Operation?"}

    ChooseOp -->|"Addition / Subtraction"| CartAS["Use Cartesian Form"]
    CartAS --> ASRule["(a+bi) ± (c+di)<br/>= (a±c) + (b±d)i"]

    ChooseOp -->|"Multiplication /<br/>Division"| ChooseForm{"Preferred Form?"}

    ChooseForm -->|"Cartesian"| CartMD["Expand and use i^2 = -1"]
    CartMD --> MDRule["(ac-bd) + (ad+bc)i"]

    ChooseForm -->|"Polar / Euler"| PolMD["Multiply moduli and add args<br/>OR divide moduli and subtract args"]
    PolMD --> PolRule["z1*z2 = r1r2[cos(θ1+θ2) + isin(θ1+θ2)]"]

    ChooseOp -->|"Powers / Roots"| EulerPower["Use Polar / Euler Form"]
    EulerPower --> DeMoivre["De Moivre's Theorem:<br/>[cosθ + isinθ]^n = cos(nθ) + isin(nθ)"]

    ChooseOp -->|"Conjugate"| ConjRule["conjugate = a - bi"]

    style Start fill:#e7f5ff,stroke:#1971c2
    style ChooseOp fill:#ffe8cc,stroke:#d9480f
    style ChooseForm fill:#ffe8cc,stroke:#d9480f

Complex Conjugate

$$\overline{z} = a - bi$$

Properties:

  • $z \cdot \overline{z} = |z|^2 = a^2 + b^2$
  • $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
  • $\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}$

Multiplication and Division in Polar Form

For $z_1 = r_1[\cos\theta_1 + i\sin\theta_1]$ and $z_2 = r_2[\cos\theta_2 + i\sin\theta_2]$:

Multiplication Rule

$$z_1 z_2 = r_1 r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$$

When multiplying in polar form: multiply the moduli and add the arguments.

Division Rule

$$\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$$

When dividing in polar form: divide the moduli and subtract the arguments.

Example: Find the product and division for $z_1 = -1 + i$ and $z_2 = \sqrt{3} + i$.

Roots of Complex Numbers

To find the square root of a complex number $z = a + bi$, let $\sqrt{z} = x + yi$ and solve: $$(x + yi)^2 = a + bi$$ Equate real and imaginary parts to form a system of equations in $x$ and $y$.

Example: Find $\sqrt{5 + 12i}$. Let $z_1 = a + bi$ such that $(z_1)^2 = 5 + 12i$. $$a^2 - b^2 = 5 \quad \text{and} \quad 2ab = 12$$ Solving gives $a = 3, b = 2$ or $a = -3, b = -2$. Thus $\sqrt{5 + 12i} = \pm(3 + 2i)$.

De Moivre's Theorem

$$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$$

n-th Roots

The n-th roots of a complex number $z = r(\cos\theta + i\sin\theta)$ are:

$$z^{1/n} = r^{1/n}\left[\cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right)\right]$$

for $k = 0, 1, 2, ..., n-1$

Geometric Interpretations

Circle

$$|z - z_0| = r$$ Circle with center $z_0$ and radius $r$

Perpendicular Bisector

$$|z - z_1| = |z - z_2|$$ Perpendicular bisector of line segment joining $z_1$ and $z_2$

Half-line

$$\arg(z - z_0) = \theta$$ Half-line (ray) from $z_0$ at angle $\theta$

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