Real Number Ordering is Compatible with Multiplication/Positive 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.
Corollary
- $\forall a, b, c, d \in \R: 0 < a < b \land 0 < c < d \implies a c < b d$
Proof
From Real Numbers form Ordered Integral Domain, $\struct {\R, +, \times, \le}$ forms an ordered integral domain.
Thus:
\(\ds a\) | \(<\) | \(\ds b\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds b - a\) | \(>\) | \(\ds 0\) | Definition of Positivity Property | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds c \times \paren {b - a}\) | \(>\) | \(\ds 0\) | Definition 1 of Ordered Integral Domain: $(\text P 1)$: Closure under Ring Addition | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds b \times c - a \times c\) | \(>\) | \(\ds 0\) | Ring Axiom $\text D$: Distributivity of Product over Addition | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a \times c\) | \(<\) | \(\ds b \times c\) | 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
- 1957: Tom M. Apostol: Mathematical Analysis ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-3}$ Order properties of real numbers
- 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-2}$: Inequalities
- 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: $\text{(III)}$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order properties (of real numbers): $(4)$