Subtraction 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.


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

\(\displaystyle z_1 - z_2\) \(=\) \(\displaystyle z_1 + \paren {- z_2}\) Definition of Complex Subtraction
\(\displaystyle \) \(=\) \(\displaystyle a_1 + \paren {-a_2} + i \paren {b_1 + \paren {-b_2} }\) Inverse for Complex Addition
\(\displaystyle \) \(=\) \(\displaystyle \paren {a_1 - a_2} + i \paren {b_1 - b_2}\) Definition of Complex Subtraction

$\blacksquare$


Sources