Complex numbers
i² = −1, adding, multiplying and the modulus.
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Mathematicians invented i, a number whose square is −1.
Look: 3 + 4i is the point 3 across, 4 up. Its distance from 0 is 5.
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1i³ = ?
2(1 + i) + (2 + 3i) = ?
3Solve x² = −9
! Careful!
✗i² = 1.
✓i² = −1.
✗Adding 3 + 4i to get 7i.
✓Plain and i parts stay separate.
See the whole lesson on one page
Mathematicians invented i, a number whose square is −1.
A complex number like 3 + 4i means 3 across and 4 up. Add them part by part. When you multiply, swap every i² for −1.
Think of it like thisNormal numbers live on a line going left and right. i opens up a new direction: up. So complex numbers live on a whole flat map.
Solved together
Work out (3 + 4i) + (1 − 2i), then (2 + i)(2 − i).
Add the plain parts, then the i parts.
(3 + 1) + (4 − 2)i = 4 + 2iMultiply like brackets.
4 − 2i + 2i − i²Swap i² for −1.
4 − (−1) = 5
Answer4 + 2i and 5.
For older learners and grown-ups: a little more
Modulus |a + bi| = √(a² + b²). To divide, multiply top and bottom by the conjugate a − bi: (3 + 4i)/(1 − 2i) = −1 + 2i.
Now practise it
Kanakku makes new questions on complex numbers every time. It shows the working and gives a hint when you are stuck.