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a Complex Number?

Numbers that combine real and imaginary parts — extending the number line into a two-dimensional plane to solve equations that real numbers can't.

Simple Definition

A complex number is a number of the form:

$$z = a + bi$$

where:

$a$ is the real part — a real number
$b$ is the imaginary part — also a real number
$i$ is the imaginary unit, defined by $i^2 = -1$

Complex numbers are written as $z = a + bi$. The set of all complex numbers is denoted by $\mathbb{C}$. They live in a two-dimensional plane called the complex plane, where the horizontal axis is the real axis and the vertical axis is the imaginary axis.

Why do we need them? Because equations like $x^2 + 1 = 0$ have no real solutions — but in the complex world, $x = \pm i$ are perfectly valid answers. Complex numbers unlock entire branches of mathematics and physics, from electrical engineering to quantum mechanics.

Real Part

$a$

Complex Number

$a + bi$

Imaginary Part

$bi$

Intuition

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Two dimensions, not one

Real numbers live on a line — you can go left (negative) or right (positive). Complex numbers live in a plane — you can go left/right and up/down. The imaginary unit $i$ is like a "rotator": multiplying by $i$ rotates a number by $90^\circ$ counterclockwise in the complex plane. This rotation is what makes complex numbers so useful for describing waves, oscillations, and rotations.

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Solving the unsolvable

Before complex numbers, equations like $x^2 = -1$ were considered impossible. The invention of $i$ opened up a whole new world: every polynomial equation of degree $n$ has exactly $n$ complex roots (the Fundamental Theorem of Algebra). Complex numbers complete algebra in a way that real numbers alone cannot.

Formal Definition

A complex number is an ordered pair $(a, b)$ of real numbers, written as $a + bi$, where $i = \sqrt{-1}$. The set of complex numbers is defined as:

Set of complex numbers

$$\mathbb{C} = \{ a + bi \mid a, b \in \mathbb{R}, \; i^2 = -1 \}$$

Two complex numbers $a + bi$ and $c + di$ are equal if and only if $a = c$ and $b = d$ — their real and imaginary parts must match independently.

Operations with Complex Numbers

Addition	$(a+bi) + (c+di) = (a+c) + (b+d)i$
Subtraction	$(a+bi) - (c+di) = (a-c) + (b-d)i$
Multiplication	$(a+bi)(c+di) = (ac-bd) + (ad+bc)i$
Division	$\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2+d^2}$
Conjugate	$\overline{a+bi} = a - bi$ — reflect across the real axis
Modulus	$\|a+bi\| = \sqrt{a^2 + b^2}$ — distance from the origin

Key Properties

Powers of $i$	$i^1 = i,\; i^2 = -1,\; i^3 = -i,\; i^4 = 1$ (cycles every 4)
Conjugate product	$z \cdot \overline{z} = \|z\|^2$ — always a non-negative real number
Modulus properties	$\|z_1 z_2\| = \|z_1\|\|z_2\|$, $\|z_1/z_2\| = \|z_1\|/\|z_2\|$
Conjugate properties	$\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$, $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$
Polar form	$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ (Euler's formula)
De Moivre's theorem	$(re^{i\theta})^n = r^n e^{in\theta}$

Quick Examples

Example 1 — Addition

$(3+2i) + (4-3i)$

$= (3+4) + (2-3)i = 7 - i$

Example 2 — Multiplication

$(2+3i)(1-2i)$

$= 2 - 4i + 3i - 6i^2$
$= 2 - i + 6 = 8 - i$

Example 3 — Conjugate

$\overline{5-2i}$

$= 5 + 2i$

Example 4 — Modulus

$|3-4i|$

$= \sqrt{3^2 + (-4)^2} = \sqrt{9+16} = 5$

Example 5 — Solving with $i$

$x^2 + 4 = 0$

$x^2 = -4$
$x = \pm \sqrt{-4} = \pm 2i$
Two complex solutions: $2i$ and $-2i$

Common Mistakes

✗ Thinking $i^2 = i$ instead of $i^2 = -1$

The defining property of $i$ is $i^2 = -1$. This is the source of all complex number operations. Remember: $i$ is not a variable — it's a constant defined by $i^2 = -1$.

✗ Mixing real and imaginary parts

When adding or subtracting complex numbers, combine real parts with real parts, and imaginary parts with imaginary parts. You cannot add a real number to an imaginary number: $a + bi$ is already in simplest form.

✗ Forgetting the conjugate when dividing

To divide by a complex number $c+di$, you must multiply numerator and denominator by the conjugate $c-di$. This makes the denominator real: $(c+di)(c-di) = c^2 + d^2$.

Related Concepts

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Complex numbers appear in algebra, calculus, and physics. Practice with our quizzes.

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Powers of \(i\)	\(i^1 = i,\; i^2 = -1,\; i^3 = -i,\; i^4 = 1\) (cycles every 4)
Conjugate product	\(z \cdot \overline{z} = \|z\|^2\) — always a non-negative real number
Modulus properties	\(\|z_1 z_2\| = \|z_1\|\|z_2\|\), \(\|z_1/z_2\| = \|z_1\|/\|z_2\|\)
Conjugate properties	\(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\), \(\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}\)
Polar form	\(z = r(\cos\theta + i\sin\theta) = re^{i\theta}\) (Euler's formula)
De Moivre's theorem	\((re^{i\theta})^n = r^n e^{in\theta}\)

Addition	\((a+bi) + (c+di) = (a+c) + (b+d)i\)
Subtraction	\((a+bi) - (c+di) = (a-c) + (b-d)i\)
Multiplication	\((a+bi)(c+di) = (ac-bd) + (ad+bc)i\)
Division	\(\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2+d^2}\)
Conjugate	\(\overline{a+bi} = a - bi\) — reflect across the real axis
Modulus	\(\|a+bi\| = \sqrt{a^2 + b^2}\) — distance from the origin