Real Number Ordering is Compatible with Multiplication/Negative Factor

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Theorem

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


where $\R$ is the set of real numbers.


Proof

From Real Numbers form Ordered Integral Domain, $\struct {\R, +, \times, \le}$ forms an ordered integral domain.

Thus:

\(\displaystyle a\) \(<\) \(\displaystyle b\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle b - a\) \(>\) \(\displaystyle 0\) Definition of Positivity Property
\(\displaystyle \leadsto \ \ \) \(\displaystyle c \times \paren {b - a}\) \(<\) \(\displaystyle 0\) Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative
\(\displaystyle \leadsto \ \ \) \(\displaystyle b \times c - a \times c\) \(<\) \(\displaystyle 0\) Ring Axioms: Product is Distributive over Addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle a \times c\) \(>\) \(\displaystyle b \times c\) Definition of Positivity Property

$\blacksquare$


Examples

$15 \times 3$ Greater than $12 \times 3$

We have that:

$15 > 12$

so by Real Number Ordering is Compatible with Multiplication:

$15 \times 3 > 12 \times 3$

That is:

$45 > 36$


$\dfrac {15} 3$ Greater than $\dfrac {12} 3$

We have that:

$15 > 12$

so by Real Number Ordering is Compatible with Multiplication:

$\dfrac {15} 3 > \dfrac {12} 3$

That is:

$5 > 4$


$15 \times \paren {-3}$ Less than $12 \times \paren {-3}$

We have that:

$15 > 12$

so by Real Number Ordering is Compatible with Multiplication:

$15 \times \paren {-3} < 12 \times \paren {-3}$

That is:

$-45 < -36$


$\dfrac {15} {-3}$ Less than $\dfrac {12} {-3}$

We have that:

$15 > 12$

so by Real Number Ordering is Compatible with Multiplication:

$\dfrac {15} {-3} < \dfrac {12} {-3}$

That is:

$-5 < -4$


Sources