Subtraction of Complex Numbers

Theorem

Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers.

The subtraction operation on $z_1$ and $z_2$ is:

$z_1 - z_2 = \paren {a_1 - a_2} + i \paren {b_1 - b_2}$

Proof

\(\ds z_1 - z_2\)

\(=\)

\(\ds z_1 + \paren {- z_2}\)

Definition of Complex Subtraction

\(\ds \)

\(=\)

\(\ds a_1 + \paren {-a_2} + i \paren {b_1 + \paren {-b_2} }\)

Inverse for Complex Addition

\(\ds \)

\(=\)

\(\ds \paren {a_1 - a_2} + i \paren {b_1 - b_2}\)

Definition of Complex Subtraction

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $6.3$: Subtraction of Complex Numbers
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Fundamental Operations with Complex Numbers: $2$. Subtraction