Further Maths · Beta

Complex numbers and Argand diagrams

Drag complex numbers to explore addition, multiplication, division and conjugates.

Drag Z and W on the Argand plane. The green point shows the result.

z\timesw=1+3iz\timesw=1+3i
Drag the labelled rings or use their arrow keys.-6-4-2246-6-4-2246ReIm0ResultZW

Tab to a handle, then use the arrow keys. Use the sliders for precise values.

Horizontal coordinates are real parts; vertical coordinates are imaginary parts.

z
2+i2+i
Modulus of z
∣z∣=5≈2.2361|z|=\sqrt{5}\approx 2.2361
Principal argument of z
arg⁡z≈0.4636 radians\arg z\approx 0.4636\text{ radians}

Multiplication multiplies moduli and adds arguments, modulo 2π. The result coordinates use exact fractions.

Predict, test and explain

Step 1 of 3

Predict

What is (2 + i)i?

InvestigateMultiply by i. What happens to a point on the Argand plane?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

What is (2 + i)i?

(2+i)i(2+i)i

Hint

The horizontal coordinate is the real part; the vertical coordinate is the imaginary part.

Reveal answer

2i + i² = 2i − 1. This agrees with a 90° anticlockwise rotation.

−1+2i-1+2i

Build an explanation

Hint 1

The horizontal coordinate is the real part; the vertical coordinate is the imaginary part.

Hint 2

Use i² = −1 when multiplying.

Hint 3

Multiplying by i rotates a nonzero point 90° anticlockwise.

Worked example

This example uses fixed values, separate from the diagram controls.

Multiply (2 + i)(1 + i).

  1. Expand to 2 + 2i + i + i².
  2. Replace i² with −1.
  3. Collect the real and imaginary parts.

Answer: 1 + 3i.

Connect this idea

Watch out: i² equals −1, not 1. The argument of zero and division by zero are undefined.