Complex numbers

The real numbers have a gap: x² + 1 = 0 has no solution among them. Complex numbers close it by declaring a number i with i² = −1 and then carrying on with ordinary algebra. Every complex number is then a + bi, with a the real part and b the imaginary part, and addition and multiplication follow the usual rules with i² replaced by −1 wherever it appears.

The word "imaginary" is a historical accident and has done real damage. It suggests these numbers are fictitious in a way that, say, negative numbers are not — but negative numbers were once resisted on exactly the same grounds. The practical answer is that complex numbers describe physical things routinely: alternating-current impedance, wave amplitude and phase together, and the quantum wavefunction are all naturally complex, and the arithmetic is what makes those calculations tractable.

Two genuine differences from the reals are worth knowing. There is no ordering: asking whether 2 + 3i is greater than 3 + 2i has no answer consistent with arithmetic, and the question simply does not apply. What you can compare is the modulus |a + bi| = √(a² + b²), the distance from the origin when the number is drawn as the point (a, b) — which is why it is √(3² + 4²) = 5 for 3 + 4i, not 3 + 4. That picture, the complex plane, turns complex arithmetic into plane geometry and is where the polar form comes from.