i is the number whose square is −1
No ordinary number squares to a negative, so i is a new one. Once you have it, every number can be written as a + ib: a real part and an imaginary part.
Class 11 · 10 games
Complex numbers, explained with pictures
Numbers that need two directions to write down, not just one.
In everyday life: A position on a map needs two numbers — so many streets across and so many up. A complex number is the same idea used as a number.
Every complex number is a point
Put the real part across and the imaginary part up, and a + ib becomes a point on a plane — the Argand plane. Adding two complex numbers is then the same tip-to-tail walk you use for arrows.
The conjugate is a reflection
The conjugate of a + ib is a − ib. On the plane the point flips across the real axis: straight down if it was up, straight up if it was down. It never moves sideways, because the real part does not change.
A worked example
z = 3 + 2i. Where is i × z on the Argand plane?
−2 + 3i
i(3 + 2i) = 3i + 2i², and i² is −1, so that is −2 + 3i. On the plane the point has turned a quarter turn anticlockwise about 0 — multiplying by i always does exactly that.
Words to know
- real part
- The a in a + ib — how far across the point is.
- imaginary part
- The b in a + ib — how far up the point is.
- conjugate
- a − ib: the same point reflected across the real axis.
- modulus
- How far the point is from 0, straight line.
In the games
You will place complex numbers on the Argand plane and work out conjugates, sums and products.
Skills
- Complex numbers and the Argand plane · Class 11
By class
Practise complex numbers in Tivumo
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