Complex Numbers — Chapter Summary

The imaginary unit is defined by

The imaginary unit is defined by i = sqrt(-1), so i^2 = -1. It is a solution of x^2 + 1 = 0. Powers of i repeat in the cycle i, -1, -i, 1.

An imaginary number is a number

An imaginary number is a number whose square is less than zero. For every real number b, ib is an imaginary number. For any non-negative a, sqrt(-a) =

A complex number has the form

A complex number has the form a + ib, where a and b are real numbers. The set of all complex numbers is C. Here, a is the real part and b is the imagi

Two complex numbers a1 + ib1

Two complex numbers a1 + ib1 and a2 + ib2 are equal exactly when a1 = a2 and b1 = b2. If Im z = 0, the number is purely real. If Re z = 0, it is purel

If z1 = a1 + ib1

If z1 = a1 + ib1 and z2 = a2 + ib2, then z1 + z2 = (a1 + a2) + i(b1 + b2), z1 - z2 = (a1 - a2) + i(b1 - b2), and z1 z2 = (a1a2 - b1b2) + i(a1b2 + a2b1