Real Numbers under Multiplication do not form Group

From ProofWiki
Jump to navigation Jump to search

Theorem

The algebraic structure $\struct {\R, \times}$ consisting of the set of real numbers $\R$ under multiplication $\times$ is not a group.


Proof

Aiming for a contradiction, suppose that $\struct {\R, \times}$ is a group.

By the definition of the number $0 \in \R$:

$\forall x \in \R: x \times 0 = 0 = 0 \times x$

Thus $0$ is a zero in the abstract algebraic sense.

From Group with Zero Element is Trivial, $\struct {\R, \times}$ is the trivial group.

But $\R$ contains other elements besides $0$.

From this contradiction it follows that $\struct {\R, \times}$ is not a group.

$\blacksquare$


Also see


Sources