Ordering Properties of Real Numbers

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


Ordering is Transitive

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

Then:

$a > c$


Ordering is Compatible with Addition

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


Ordering is Compatible with Multiplication

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