Complex Numbers Explained: Worked Examples and Practice

TL;DR

A complex number is a number of the form a + bi, where a and b are ordinary real numbers and i is defined by i² = -1. Add and subtract the real and imaginary parts separately. Multiply by expanding like algebra, then replace i² with -1. Divide by multiplying top and bottom by the conjugate. Geometrically, every complex number is a point on a plane, and multiplying two of them rotates and stretches that point.

Table of Contents

• Why mathematicians invented i
• What a complex number is
• The powers of i
• The complex plane
• Adding and subtracting complex numbers
• Multiplying complex numbers
• Conjugates and division
• Modulus: the size of a complex number
• Polar form and Euler’s formula
• Multiplication as rotation
• Where complex numbers are used
• Common mistakes
• Formula cheat sheet
• Practice problems

Why mathematicians invented i

Try to solve x² + 1 = 0. You need a number whose square is -1. Every real number, positive or negative, squares to something zero or greater, so no real solution exists.

Mathematicians didn’t wave that away. They defined a new number, called i, with exactly one property: i² = -1. That single decision gives you a number system where every polynomial equation has a solution.

The idea didn’t arrive politely. In the 1500s, Italian algebraists solving cubic equations kept hitting square roots of negative numbers in the middle of their working, even when the final answer was a plain real number. They used the roots as a trick and distrusted them. The MacTutor history archive at the University of St Andrews traces how that grudging tool became a respected number system over the next three centuries.

The name “imaginary” is a leftover insult. Imaginary numbers are no less real as mathematics than negative numbers are. They describe things you can measure, as the applications section shows.

What a complex number is

A complex number has the form:

z = a + bi

• a is the real part, written Re(z).
• b is the imaginary part, written Im(z). It’s the coefficient of i, so it is itself a real number.

For z = 3 + 4i, the real part is 3 and the imaginary part is 4. Not 4i. This trips up a lot of beginners.

Two special cases are worth knowing. If b = 0, the number is just a real number, so every real number is also a complex number. If a = 0, the number is purely imaginary, like 7i.

Two complex numbers are equal only when both their real parts and their imaginary parts match.

The powers of i

Because i² = -1, the powers of i repeat in a cycle of four:

Power	Value	How it’s built
i¹	i
i²	-1	definition
i³	-i	i² · i = -1 · i
i⁴	1	i² · i² = (-1)(-1)
i⁵	i	back to the start

To compute a large power, divide the exponent by 4 and keep the remainder.

Example: i²⁷. 27 ÷ 4 = 6 remainder 3, so i²⁷ = i³ = -i.

A remainder of 0 gives 1, a remainder of 1 gives i, 2 gives -1, and 3 gives -i.

The complex plane

A real number lives on a line. A complex number needs two coordinates, so it lives on a plane. This is the complex plane, also called the Argand diagram.

• The horizontal axis is the real axis.
• The vertical axis is the imaginary axis.
• The number a + bi sits at the point (a, b).

So 3 + 4i is three units right and four units up. -2 – i is two units left and one down. The number 0 is the origin, and the real numbers sit along the horizontal axis.

This picture is the reason complex numbers are useful. Once you can plot them, arithmetic becomes geometry.

Adding and subtracting complex numbers

Add the real parts together and the imaginary parts together. Treat i like a variable in algebra: you combine like terms.

Example 1: (3 + 4i) + (2 – 5i)

1. Group real parts: 3 + 2 = 5
2. Group imaginary parts: 4i – 5i = -i
3. Result: 5 – i

Example 2: (3 + 4i) – (2 – 5i)

1. Distribute the minus sign: 3 + 4i – 2 + 5i
2. Real parts: 3 – 2 = 1
3. Imaginary parts: 4i + 5i = 9i
4. Result: 1 + 9i

On the plane, addition is the same as the parallelogram rule for vectors. Put the two arrows tail to tip and you land on the sum.

Multiplying complex numbers

Multiply two complex numbers the way you’d multiply two binomials, then use i² = -1 to clean up.

Example: (3 + 2i)(1 + 4i)

1. Expand: 3·1 + 3·4i + 2i·1 + 2i·4i
2. Compute each term: 3 + 12i + 2i + 8i²
3. Replace i² with -1: 3 + 12i + 2i – 8
4. Combine real parts: 3 – 8 = -5
5. Combine imaginary parts: 12i + 2i = 14i
6. Result: -5 + 14i

The step people skip is step 3. If you leave 8i² in the answer, the answer isn’t finished, and it’s wrong if you treat 8i² as an imaginary term.

Conjugates and division

The complex conjugate of z = a + bi is a – bi. You flip the sign of the imaginary part. It’s written z̄ or z*.

z	Conjugate
3 + 4i	3 – 4i
-2 – i	-2 + i
5i	-5i
7	7

On the plane, the conjugate is the mirror image across the real axis.

The conjugate matters because of what happens when you multiply a number by it:

(a + bi)(a – bi) = a² – (bi)² = a² + b²

The imaginary part cancels and you get a real, non-negative number. That’s the tool for division, because you can’t leave i in a denominator.

Example: (5 + 2i) ÷ (1 – 3i)

1. Multiply top and bottom by the conjugate of the denominator, 1 + 3i: (5 + 2i)(1 + 3i) / (1 – 3i)(1 + 3i)

2. Expand the numerator: 5 + 15i + 2i + 6i² = 5 + 17i – 6 = -1 + 17i

3. Expand the denominator: 1² + 3² = 10
4. Divide each part by 10: -1/10 + (17/10)i

This is the same move as rationalizing a denominator that contains a square root. The conjugate removes the troublesome piece.

Modulus: the size of a complex number

The modulus of z = a + bi is its distance from the origin:

|z| = √(a² + b²)

It’s the Pythagorean theorem on the complex plane.

Example: |3 + 4i|

1. Square each part: 3² = 9 and 4² = 16
2. Add: 25
3. Square root: 5

Notice that z · z̄ = |z|². The denominator from the division example, 10, was |1 – 3i|².

The angle of the number, measured counterclockwise from the positive real axis, is called the argument and is written arg(z) or θ. The modulus and the argument together are enough to describe the number completely, and that leads to polar form.

Polar form and Euler’s formula

Instead of giving the point’s horizontal and vertical distances, you can give its distance r from the origin and its angle θ:

z = r(cos θ + i sin θ)

Here r = |z| and θ is the argument.

Example: write 1 + i in polar form

1. Modulus: r = √(1² + 1²) = √2
2. Angle: the point (1, 1) sits on the 45° diagonal, so θ = π/4
3. Result: 1 + i = √2 (cos π/4 + i sin π/4)

One of the most celebrated results in mathematics compresses this form. Euler’s formula states:

e^(iθ) = cos θ + i sin θ

So the polar form becomes z = r·e^(iθ), which is the most compact way to write a complex number. Set θ = π and you get e^(iπ) = -1, or e^(iπ) + 1 = 0, usually called Euler’s identity. It ties together e, i, π, 1 and 0.

Be careful with the angle. If the point is in the second or third quadrant, a calculator’s arctan(b/a) gives the wrong answer, because it only returns angles between -90° and 90°. Plot the point first and add 180° when needed. For -1 + i, arctan(1/-1) gives -45°, but the point is in the second quadrant, so θ = 135°, or 3π/4.

Multiplication as rotation

Polar form explains something the algebra hides. When you multiply two complex numbers:

• the moduli multiply: |z₁z₂| = |z₁|·|z₂|
• the angles add: arg(z₁z₂) = arg(z₁) + arg(z₂)

So multiplying by a complex number rotates a point and scales its distance from the origin.

Multiplying by i has modulus 1 and angle 90°, so it rotates any point 90° counterclockwise without changing its size. Do it four times and you’ve turned a full circle, which is why i⁴ = 1.

Check it with (1 + i)². Algebra first: 1 + 2i + i² = 2i. Now geometry: 1 + i has modulus √2 and angle 45°. Squaring gives modulus (√2)² = 2 and angle 90°. A point at distance 2 and angle 90° is 2i. Both routes agree.

Polar form is also the quick way to raise numbers to large powers. De Moivre’s theorem says [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ), so you raise the modulus to the power and multiply the angle by n.

Where complex numbers are used

AC circuits. Alternating current oscillates, and each component shifts the timing of the voltage relative to the current. Engineers store both the size and the timing shift in one complex number called impedance. They write it with j instead of i, because i already means current. A 3 Ω resistor in series with an inductor whose reactance is 4 Ω has impedance Z = 3 + 4j. The modulus gives the opposition to current, |Z| = 5 Ω, and the angle gives the phase shift, about 53°. No separate trigonometry is needed.

Signal processing. The Fourier transform breaks a signal, such as audio or a radio wave, into rotating complex exponentials e^(iωt). Compression formats, noise filters and the spectrum analyzers in labs all work on that basis.

Quantum mechanics. The state of a quantum system is described by a wave function with complex values. For a long time it was an open question whether the complex numbers were essential or just convenient. A 2021 experiment reported in Nature tested quantum theory built only on real numbers and found it fails to match the predictions of standard quantum theory.

Fractals. The Mandelbrot set is defined by repeating z → z² + c on the complex plane and checking which starting values stay bounded.

Common mistakes

• Forgetting i² = -1. Leaving 8i² in a product, or treating i² as 1, wrecks the answer.
• Writing the imaginary part as bi. In 3 + 4i, the imaginary part is 4.
• Dividing without the conjugate. An answer with i in the denominator is unfinished.
• Conjugating the wrong number. Multiply by the conjugate of the denominator, and multiply the numerator by the same thing.
• Applying √a · √b = √(ab) to negatives. √(-1)·√(-1) is i·i = -1, not √1 = 1. Write √(-n) as i√n before doing anything else.
• Trusting arctan blindly. Check the quadrant before you report an angle.

Formula cheat sheet

Topic	Formula
Definition	z = a + bi, i² = -1
Powers of i	i, -1, -i, 1, repeating every 4
Addition	(a + bi) + (c + di) = (a + c) + (b + d)i
Multiplication	(a + bi)(c + di) = (ac – bd) + (ad + bc)i
Conjugate	conj(a + bi) = a – bi
Product with conjugate	(a + bi)(a – bi) = a² + b²
Division	(a + bi)/(c + di) = (a + bi)(c – di) / (c² + d²)
Modulus	|z| = √(a² + b²)
Polar form	z = r(cos θ + i sin θ)
Euler’s formula	e^(iθ) = cos θ + i sin θ
Product in polar form	multiply moduli, add angles
De Moivre	zⁿ = rⁿ(cos nθ + i sin nθ)

Practice problems

Try each one before you look at the answers.

1. Add: (2 + 3i) + (4 – 7i)
2. Multiply: (1 + 2i)(3 – i)
3. Divide: (4 + i) ÷ (2 – i)
4. Simplify: i²⁷
5. Find the modulus of 5 – 12i.
6. Write -1 + i in polar form.

Answers

1. 6 – 4i. Real parts: 2 + 4 = 6. Imaginary parts: 3i – 7i = -4i.
2. 5 + 5i. Expand: 3 – i + 6i – 2i² = 3 + 5i + 2 = 5 + 5i.
3. 7/5 + (6/5)i. Multiply by (2 + i): numerator 8 + 4i + 2i + i² = 7 + 6i; denominator 2² + 1² = 5.
4. -i. 27 leaves remainder 3 when divided by 4, and i³ = -i.
5. 13. √(25 + 144) = √169 = 13.
6. √2 (cos 3π/4 + i sin 3π/4). The modulus is √2. The point is in the second quadrant, so the angle is 135°, or 3π/4.

If you can do all six without looking back, you have the working toolkit. If you want more problems, the complex numbers notes from Paul’s Online Math Notes at Lamar University have a good set with solutions.