First watch this video on complex numbers by David Eisenbud.
Then my video explaining how complex numbers can be seen three ways. Mine is a bit long (30 minutes), so maybe watch it in bits, pausing from time to time.
Complex numbers can be seen in three ways:
- As points in a plane, with rules for adding and multiplying them (in the same way as real numbers are points on a number line, with rules for adding and multiplying them)
- As what you get when you add, to those you have from the number line, one new number, i, with i2=−1, plus all those you get from combining that new number with the old ones in additions and multiplication. Then the number 2+3i, for example, corresponds to the point 2 units across, 3 units up, in the number plane.
- In polar form, i.e. describing the points by distance from the origin plus angle of anticlockwise rotation from the positive number line.