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 $b$:    \(\ds a < b \)      


Transitive Law

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

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


Sources

  • 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.4$: Inequalities
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): order properties (of real numbers)
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order properties (of real numbers)
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