Complex numbers on the plane

See a + bi as a point, add and multiply them, and watch roots spread evenly round a circle. About 8 minutes, A-level Further Maths.

What you will do

1. A number is a point. A complex number a + bi has a real part a and an imaginary part b, so it lives at the point (a, b) on the Argand diagram. Here is z₁ = re1 + im1i in red and z₂ in green.
2. Move it. Drag the red point and watch its real and imaginary parts change in the Controls tab. Put z₁ at -1 + 2i.
3. Modulus and argument. The same point can be described by its distance from the origin, the modulus |z|, and the angle it makes with the real axis, the argument. Look at 3 + 4i.
4. Adding is vector addition. Adding complex numbers adds real parts and imaginary parts: the sum is the far corner of the parallelogram built on the two arrows.
5. Multiplying rotates and scales. Multiplying is different: the moduli multiply and the arguments add. Here 1+i has argument 45°, so multiplying it by itself should turn it through another 45°.
6. Make it turn. Drag z₂ so that z₁ × z₂ points straight along the imaginary axis (real part about 0). Hint: you need the arguments to add up to 90°.
7. Roots spread evenly. The fifth roots of 1 are five numbers that give 1 when raised to the power 5. They sit on the unit circle, equally spaced.
8. Your turn. Everything is unlocked. Move the points, type exact values in the Controls tab, try division, show the conjugate, or take the roots of any number.