Complex Subtraction is Closed

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Theorem

The set of complex numbers is closed under subtraction:

$\forall a, b \in \C: a - b \in \C$


Proof

From the definition of complex subtraction:

$a - b := a + \paren {-b}$

where $-b$ is the inverse for complex number addition.

From Complex Numbers under Addition form Group, it follows that:

$\forall a, b \in \C: a + \paren {-b} \in \C$

Therefore complex number subtraction is closed.

$\blacksquare$


Sources