Equality of Complex Numbers

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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:

$z_1 = \tuple {a_1, b_1}$
$z_1 = \tuple {a_2, b_2}$

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

The result follows from Equality of Ordered Pairs.


Sources