Product of Negative Real Numbers is Positive

Theorem

Let $a, b \in \R_{\le 0}$ be negative real numbers.

Then:

$a \times b \in \R_{\ge 0}$

That is, their product $a \times b$ is a positive real number.

Proof

From Real Numbers form Ring, the set $\R$ of real numbers forms a ring.

The result then follows from Product of Ring Negatives.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems