Complex numbers as rotations

TL;DR:

• The number i is defined by i² = −1. Multiplying by i turns a point by 90° anticlockwise.

• A complex number a + bi is the point (a, b) on the complex plane.

• Multiplying complex numbers multiplies their sizes (modulus) and adds their angles (argument).

No real number squares to −1, because squares of real numbers are never negative. Mathematicians decided to invent a new number, called i, with exactly that property: i² = −1. It looks like a trick, but it fits together so well with rotations that it became a standard tool in engineering and physics.

The complex plane

A complex number has the form a + bi, with real numbers a and b. Plot it as the point (a, b): a on the horizontal axis (the real part) and b on the vertical axis (the imaginary part). So 3 + i is the point (3, 1).

The modulus is the distance from the origin, √(a² + b²). The argument is the angle from the positive real axis. For 3 + i the modulus is √10 ≈ 3.16 and the argument is tan⁻¹(1/3) ≈ 18.4°.

Worked example: multiplying by i rotates a point by 90 degrees

1. Start with 3 + i, the point (3, 1).

2. Multiply by i: i(3 + i) = 3i + i².

3. Use i² = −1: 3i − 1 = −1 + 3i.

4. Read it as a point: (−1, 3).

The point moved from (3, 1) to (−1, 3). That is a 90° anticlockwise turn about the origin. Check: the modulus of −1 + 3i is √(1 + 9) = √10, unchanged. The dot product of (3, 1) and (−1, 3) is −3 + 3 = 0, so the two arrows are perpendicular. And the earlier matrix [[0, −1], [1, 0]] gave exactly the same result for this point.

Why i² = −1 means two quarter turns

Multiplying by i rotates by 90°. Doing it twice rotates by 180°, which flips a point to the opposite side of the origin, exactly what multiplying by −1 does. So i · i = −1 is not strange at all: two quarter turns make a half turn. Four turns bring you home: i⁴ = 1. The powers cycle: i, −1, −i, 1, i, ….

Multiplying any two complex numbers

In general, multiplying complex numbers multiplies the moduli and adds the arguments. Take (1 + i)(1 + i). Expanding: 1 + i + i + i² = 1 + 2i − 1 = 2i. Now the geometric view: 1 + i has modulus √2 and argument 45°. Squaring gives modulus √2 × √2 = 2 and argument 45° + 45° = 90°. The point at distance 2 and angle 90° is 2i. The algebra and the geometry agree.

Addition and conjugate

Adding complex numbers works like adding vectors: add the real parts and the imaginary parts. (3 + i) + (1 + 2i) = 4 + 3i. The conjugate of a + bi is a − bi, its mirror image across the real axis. Multiplying a number by its conjugate gives a real number: (3 + i)(3 − i) = 9 − i² = 9 + 1 = 10, which is the modulus squared. Check: the modulus is √10, and (√10)² = 10.

Division using the conjugate

To compute (1 + 2i) / (3 + i), multiply the top and bottom by the conjugate of the bottom: (1 + 2i)(3 − i) / ((3 + i)(3 − i)). The top is 3 − i + 6i − 2i² = 3 + 5i + 2 = 5 + 5i. The bottom is 10. So the result is (5 + 5i)/10 = 0.5 + 0.5i. Check by multiplying back: (3 + i)(0.5 + 0.5i) = 1.5 + 1.5i + 0.5i + 0.5i² = 1.5 + 2i − 0.5 = 1 + 2i. Correct.

Try the powers of i yourself

The powers of i repeat every four steps: i¹ = i, i² = −1, i³ = i² · i = −i, i⁴ = 1. To find i¹⁰, divide the exponent by 4 and keep the remainder: 10 = 2 × 4 + 2, so i¹⁰ = i² = −1. For i⁷, 7 = 1 × 4 + 3, so i⁷ = i³ = −i. Geometrically, i¹⁰ is 10 quarter turns, which is two full turns plus a half turn, so it lands on −1.

To turn 3 + i by 180°, multiply by −1 and get −3 − i. To turn it by 270° anticlockwise, multiply by i³ = −i: −i(3 + i) = −3i − i² = 1 − 3i, the point (1, −3). Check the length: √(1 + 9) = √10. Right again.

How to check your answer

• Replace i² by −1 every time it appears.

• Modulus. Under a rotation by i, the modulus should not change.

• Picture. Plot the points and see whether the turn looks right.

• Multiply back after dividing.

Common mistakes

• Writing i² = 1.

• Using √a × √b = √(ab) for negative numbers. For example, √−1 × √−1 = i × i = −1, but √(−1 × −1) = √1 = 1. Convert to i first.

• Forgetting to change the sign of the imaginary part in a conjugate.

• Treating i as a variable that can be dropped.

• Adding the real and imaginary parts together as if they were like terms.

Where you will use it next

Complex numbers connect to the rotation matrices in matrices: what multiplying them means and to arrows in vectors: adding forces and velocities. Wave and circuit problems in engineering often use them for rotating quantities, building on the angles from radians and sine graphs.

Frequently asked questions

Is i a real thing?

It is a defined number, just like negative numbers once were new. It is useful because calculations with it describe rotations and waves correctly.

Why is the imaginary axis vertical?

By convention, so that multiplying by i turns the real axis into the vertical one, which gives a quarter turn.