Real Division is not Closed
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Theorem
The operation of division on the set of real numbers $\R$ is not closed.
Proof
From Division by Zero, we have that for all $a \in \R$, the operation $\dfrac a 0$ is not defined.
$\blacksquare$
Sources
- 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Example $1$