Complex Numbers — Mathematics Knowledge System

Overview

A complex number extends the real numbers by introducing i, a formal square root of -1. Every complex number can be written a + bi with a, b real, and can be plotted as a point on a 2D plane (the Argand plane) — turning algebra involving square roots of negative numbers into geometry involving rotation and scaling.

Intuition

You can't take the square root of a negative real number and stay on the real number line — there's no point on that line that, squared, gives a negative. So mathematicians added a second dimension: i, defined so that i² = -1, sitting perpendicular to the familiar real axis. Every complex number a+bi is then just a point (a,b) in this 2D plane, and multiplying by i turns out to be a 90° rotation — arithmetic with i is secretly geometry.

Interactive Graph

Formal Definition

Definition

The imaginary unit and the standard (rectangular) form of a complex number:

i^2 = -1

Defining property

z = a + bi,\quad a, b \in \mathbb{R}

a is the real part, b is the imaginary part

Notation

Notation	Meaning
$i$	The imaginary unit, i² = -1
$\operatorname{Re}(z)$	The real part of z
$\operatorname{Im}(z)$	The imaginary part of z
$\bar{z} = a - bi$	The complex conjugate of z = a+bi
$\|z\| = \sqrt{a^2+b^2}$	The modulus (distance from origin)

Properties

Addition

(a+bi) + (c+di) = (a+c) + (b+d)i

Multiplication

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

Conjugate product is real

z\bar{z} = a^2+b^2 = |z|^2

Euler's formula

e^{i\theta} = \cos\theta + i\sin\theta

Example: Connects complex exponentials to rotation

Theorems

Theorem 1: Fundamental Theorem of Algebra

\text{Every non-constant polynomial with complex coefficients has at least one complex root.}

Applications

AC circuit analysis represents voltage and current as complex phasors, turning differential equations into algebra.

Worked Examples

Add real and imaginary parts separately.
$(3+1) + (2-4)i = 4 - 2i$

Answer: 4 - 2i

Practice Problems

Difficulty 3/10

Find the modulus of z = 3 + 4i.

Common Mistakes

Common Mistake

Treating i like a normal variable and simplifying i² to i·i without reducing to -1.

Always substitute i² = -1 immediately — leaving i² unreduced is the single most common complex-arithmetic error.

Quiz

What is i² ?

1-1i0

Flashcards

1 / 2

Historical Background

Complex numbers first appeared not from square roots of negatives directly, but from Gerolamo Cardano's 1545 work on cubic equations, where intermediate steps required manipulating square roots of negative numbers even when the final answer was real. Rafael Bombelli developed formal rules for this arithmetic in 1572. The term 'imaginary' (coined dismissively by Descartes in 1637) stuck despite Euler and Gauss showing by the early 1800s that these numbers were perfectly rigorous and geometrically meaningful — Gauss himself popularized the Argand plane representation.

1545
Cardano encounters square roots of negatives solving cubics
Gerolamo Cardano
1572
Bombelli gives formal arithmetic rules for these quantities
Rafael Bombelli
1637
Descartes coins the (dismissive) term 'imaginary'
René Descartes
1799
Gauss's doctoral thesis uses complex numbers rigorously (Fundamental Theorem of Algebra)
Carl Friedrich Gauss

Summary

Complex numbers extend ℝ with i, where i² = -1.
z = a + bi has real part a and imaginary part b; plots as a point on the Argand plane.
Modulus |z| = √(a²+b²); conjugate z̄ = a - bi; z·z̄ = |z|².
Euler's formula e^(iθ) = cos θ + i sin θ links complex exponentials to rotation.
Fundamental Theorem of Algebra: every non-constant polynomial has a complex root — this is why ℂ, unlike ℝ, is 'algebraically closed'.

References

WebsiteWikipedia — Complex number