Ordering Properties of Real Numbers

Theorem

Trichotomy Law

The real numbers obey the Trichotomy Law.

That is, $\forall a, b \in \R$, exactly one of the following holds:

\((1)\)	$:$		$a$ is greater than $b$:		\(\ds a > b \)
\((2)\)	$:$		$a$ is equal to $b$:		\(\ds a = b \)
\((3)\)	$:$		$a$ is less than $b$:		\(\ds a < b \)

Ordering is Transitive

Let $a, b, c \in \R$ such that $a > b$ and $b > c$.

Then:

$a > c$

Ordering is Compatible with Addition

$\forall a, b, c \in \R: a < b \implies a + c < b + c$

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

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.4$: Inequalities
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order properties (of real numbers)