Complex numbers as points you can push around the plane

How to use this simulation

1. The horizontal axis is the real axis and the vertical axis is the imaginary axis. A number a + bi sits at the point (a, b). Two numbers start on the plane: z₁ = 2 + i in the primary colour and z₂ = 1 + 2i in the secondary colour. The dashed circle is the unit circle, radius 1.
2. Drag z₁ or z₂ with the pointer, or set their parts with the sliders. Points snap to tenths of a unit. The readings give each number, its modulus (distance from the origin) and its argument (angle from the positive real axis).
3. In “Add” the orange arrow is z₁ + z₂, the diagonal of the shaded parallelogram. Here it is 3 + 3i.
4. In “Multiply” a dashed orange path shows z₁ turning and stretching into z₁z₂. The readings show moduli multiplied and arguments added. The starting pair gives 5i, because 26.57° + 63.43° = 90°.
5. “Conjugate” mirrors z₁ in the real axis. “Power zⁿ” builds z₁, z₁², z₁³ … as a spiral, with n from the slider. “nth roots of z₁” and “nth roots of unity” place n points evenly round a circle.

What is a complex number?

The equation x² = −1 has no answer among ordinary numbers, because no real number squared is negative. So mathematicians invented one: i, defined by i² = −1. Every complex number is written a + bi, where a and b are ordinary real numbers.

The trick that makes them easy to picture: treat a as how far right and b as how far up. Then a + bi is the point (a, b) on a plane. That plane is called the Argand plane (or Argand diagram). The real numbers live on the horizontal axis, and the numbers like 2i live on the vertical one.

Modulus and argument

There are two ways to give an address on the plane. One is “a right, b up”. The other is “r away, at angle θ”. The distance r is the modulus |z| = √(a² + b²), the same Pythagoras step as the length of a vector. The angle θ from the positive real axis is the argument.

Together they give the polar form z = r(cos θ + i sin θ). Some landmark points are in the table; the arguments run anticlockwise and, in this simulation, lie between −180° and 180°.

• The argument of 0 does not exist (it has no direction); the readings say “none”.
• Adding 360° to an argument gives the same point. We usually quote the one between −180° and 180°.

Adding: the parallelogram

Add the real parts and add the imaginary parts. With z₁ = 2 + i and z₂ = 1 + 2i the sum is 3 + 3i.

On the plane this is the vector-addition parallelogram: z₁ and z₂ are two sides, and the sum is the diagonal from the origin. Subtracting z₁ − z₂ is the arrow from z₂ to z₁.

Notice the lengths. |z₁| = 2.24 and |z₂| = 2.24, but the sum is not simply their total. The triangle inequality says |z₁ + z₂| ≤ |z₁| + |z₂|, with equality only when both point the same way.

Multiplying: turn and stretch

Here is where complex numbers earn their keep. Multiply two of them and the moduli multiply while the arguments add: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂.

In the simulation z₁ turns by arg z₂ and is stretched by |z₂|. For the starting pair 26.57° + 63.43° = 90°, and 2.24 × 2.24 = 5, so z₁z₂ = 5i. A tidy right angle, straight up the imaginary axis.

Multiplying by a number of modulus 1 is a pure rotation, because there is no stretching. Multiplying by a positive real number is a pure stretch, because there is no turning.

The conjugate: a mirror in the real axis

The conjugate of a + bi is a − bi. On the plane it is the mirror image in the real axis. The conjugate of 2 + i is 2 − i.

The product z · z̄ is always a real number, equal to |z|². Here it is 5. That is the secret behind dividing complex numbers: multiply top and bottom by the conjugate of the bottom, and the bottom becomes real.

Powers: a spiral

Since multiplying adds arguments and multiplies moduli, repeated multiplication does something regular. Squaring doubles the angle and squares the modulus. Cubing triples the angle. In general zⁿ has modulus rⁿ and argument nθ. This is de Moivre’s theorem.

Plot z, z², z³ … and you get a spiral. If |z| is bigger than 1 the spiral flies outwards. If it is less than 1 it curls into the origin. If |z| = 1 the points stay on the unit circle and just keep turning.

Roots and roots of unity

Taking an nth root is de Moivre backwards: the modulus becomes r^(1/n) and the argument θ/n, but adding a full turn to θ before dividing gives a different answer, so there are n roots in all, evenly spread round a circle at angle 360°/n apart.

The most famous case is the nth roots of unity, the solutions of zⁿ = 1. They sit on the unit circle and form a regular polygon with a corner at 1. Call the first one after 1 ω; the rest are its powers ω², ω³ and so on.

• The roots of unity always add up to 0 (for n ≥ 2), as the readings show.
• Their product is ±1, and each root is a power of the first.
• If the roots of zⁿ = w are known for one w, the rest come from multiplying by the roots of unity.

Try it in the simulation

Six quick experiments, a minute each.

• In “Add” drag z₂ to the opposite side of the origin from z₁. The sum shrinks. When z₂ = −z₁ the sum is 0.
• In “Multiply” set z₂ to 0 + 1i (that is i). The product is z₁ turned a quarter turn, with the same length.
• In “Multiply” put z₂ on the unit circle. The product has the same modulus as z₁ and just turns.
• In “Conjugate” drag z₁ across the real axis and watch z₁ and z̄₁ swap places.
• In “Power zⁿ” put z₁ at about 1.1 + 0i and raise n to 8. See how fast the points run away.
• In “nth roots of unity” step n from 2 to 8. The polygon gains a corner at a time and the sum of the roots stays at 0.

Solved problems

Six common questions. Every number below is computed with the same functions as the simulation.

Mistakes to avoid

These cost marks again and again.

• Forgetting i² = −1, and leaving an i² in the answer.
• Adding the moduli when multiplying. They multiply; the arguments are the ones that add.
• Taking the argument from tan⁻¹(b/a) without checking the quadrant. −1 + i is in the second quadrant, not the fourth.
• Dividing without multiplying by the conjugate first.
• Finding only one nth root. There are n of them, spaced 360°/n apart.
• Writing |z|² as z² . They are not equal unless z is real and non-negative.

Where complex numbers are used

Electrical engineers describe alternating current with complex numbers because a rotating arrow is the natural picture of a wave. Signal processing, radio and Wi-Fi all lean on the same idea, through the Fourier transform.

Computer graphics uses complex multiplication as a quick way to rotate in 2D, and fractals like the Mandelbrot set come from repeatedly squaring a complex number. Quantum physics cannot even be written down without them.

Exam corner

For HSC higher maths and admission tests, remember these.

• z = a + bi sits at (a, b). |z| = √(a² + b²). Check the quadrant when finding the argument.
• Product: multiply moduli, add arguments. Quotient: divide moduli, subtract arguments.
• To divide in rectangular form, multiply top and bottom by the conjugate of the bottom.
• De Moivre: (r cis θ)ⁿ = rⁿ cis nθ. Roots: divide the angle by n and add 360°k ÷ n.
• For z³ = 1: the roots are 1, ω, ω², with 1 + ω + ω² = 0 and ω³ = 1.

Frequently asked questions