Real Number Ordering is Compatible with Multiplication

Theorem

Let $\R$ denote the set of real numbers.

Then:

Positive Factor

$\forall a, b, c \in \R: a < b \land c > 0 \implies a c < b c$

Negative Factor

$\forall a, b, c \in \R: a < b \land c < 0 \implies a c > b c$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inequality
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inequality: 2.