Complex numbers and operations on them

The equation x² + 4 = 0 has no real solution, since the square of every real number is non-negative. So we introduce the symbol i with i² = −1 (the imaginary unit). A number a + bi with real a, b is a complex number; a is its real part Re(z) and b its imaginary part Im(z). Two complex numbers are equal when their real parts and their imaginary parts are equal separately; they cannot be compared with “<” or “>”. Addition and subtraction act separately on the real and imaginary parts; multiplication expands the brackets and uses i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The conjugate of z = a + bi is z̄ = a − bi, and z · z̄ = a² + b² is real. For division we multiply numerator and denominator by the conjugate of the denominator. The number z = a + bi is shown in the plane as the point (a, b) (or the vector to it): Ox is the real axis and Oy is the imaginary axis.