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
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$