# Real Number Ordering is Compatible with Multiplication/Negative Factor

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## Contents

## 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

- 1972: Murray R. Spiegel and R.W. Boxer:
*Theory and Problems of Statistics*(SI ed.) ... (previous) ... (next): Chapter $1$: Inequalities - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.4$: Inequalities: $\text{(IV)}$