Complex numbers

A complex number is any number of the form a + bi, where a and b are real numbers. The number a is its real part and b is its imaginary part (note that it is b, not bi). If b = 0 the number is just the real number a; if a = 0 and b ≠ 0 it is pure imaginary. So every real number is also complex: the sets N, Z, Q and R all sit inside the set of complex numbers. Complex numbers let us solve x² = −k for k > 0: x = ±√k × i. For example x² = −16 gives x = ±4i, and you can check (4i)² = 16 × (−1) = −16. Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal: a + bi = c + di means a = c and b = d. This turns one complex equation into two ordinary equations, which is how we find unknown x and y.