"All limits, especially national ones, are contrary to the nature of mathematics… Mathematics knows no races… For mathematics the whole cultural world is a single country" – David Hilbert, the foremost mathematician of the 20th century, speaking as the Nazis neared power in his native Germany.

Three faces of complex numbers: video clip

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 i^{2}=−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.