Square of Complex Number

Theorem

Let $z = a + i b$ be a complex number.

Then its square is given by:

$z^2 = a^2 - b^2 + i \paren {2 a b}$

Proof

\(\ds z^2\)

\(=\)

\(\ds \paren {a + i b}^2\)

by hypothesis

\(\ds \)

\(=\)

\(\ds \paren {a + i b} \paren {a + i b}\)

Definition of Square Function

\(\ds \)

\(=\)

\(\ds \paren {a \times a - b \times b} + i \paren {a \times b + b \times a}\)

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds a^2 - b^2 + i \paren {2 a b}\)

simplification

$\blacksquare$

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Powers: $3.7.18$