The Conjugate

The conjugate of a + bi is a - bi: the same real part, with the sign of the imaginary part flipped. The conjugate of 2 + 7i is 2 - 7i, and the conjugate of 6 - i is 6 + i.

Multiply a number by its conjugate: (2 + 7i)(2 - 7i) = 4 - 14i + 14i - 49i^2. The two i terms are opposites and cancel, and -49i^2 is +49. The result is 4 + 49, which is 53, a real number.

For real a and b that always happens: (a + bi)(a - bi) = a^2 + b^2. Compare (a + b)(a - b) = a^2 - b^2 from Algebra: the minus turns into a plus because i^2 = -1. This is the tool for the next lesson, since multiplying by the conjugate turns a complex number into a real one.