Complex Numbers

The square root of −1 unlocks equations like x² = −64. The solutions of x² = a are ±√a when a is positive and ±i√|a| when a is negative: x² = −64 gives ±8i, and x² = −18 gives ±3√2 i, since √18 = √9 · √2 = 3√2. Numbers like 7 and √5 are real numbers; a real number times i, like 7i or √5 i, is an imaginary number.

Solve (x − 6)² = −49 in two steps: x − 6 = ±7i, so x = 6 ± 7i. Numbers like 6 + 7i are complex numbers, a + bi: a is the real part and b is the imaginary part. When a = 0 the number is imaginary as well as complex; when b = 0 it is real as well as complex.

Complex numbers live in the complex plane. Real numbers sit on the horizontal axis, the real axis; imaginary numbers sit on the vertical axis, the imaginary axis. The number a + bi is the point a along and b up (or down): 4 + 2i is 4 to the right and 2 up. Every point of the plane is a complex number, on an axis or not.

Read a point of the complex plane as a number: its real coordinate is the real part a, its imaginary coordinate is the imaginary part b, and the number is a + bi. A point on the real axis has imaginary part 0, so it is a real number like 2; a point on the imaginary axis has real part 0, like 4i.

Fifteen questions from Level 1: solve x² = −a and (x − h)² = −k², simplify i² terms to a + bi, read and plot points of the complex plane, and five look-back problems on real squares.