Equality of Complex Numbers

Theorem

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

Then $z_1 = z_2$ if and only if $a_1 = a_2$ and $b_1 = b_2$.

Proof

By definition of a complex number, $z_1$ and $z_2$ can be expressed in the form:

$z_1 = \tuple {a_1, b_1}$

$z_2 = \tuple {a_2, b_2}$

where $\tuple {a, b}$ denotes an ordered pair.

The result follows from Equality of Ordered Pairs.

Sources

• 1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers: $\text {(i)}$
• 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): $6.1$: Equality of Complex Numbers
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Complex Number System
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $\text {A}$. Equality
• 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.1$ Complex numbers and their representation