ch1.1.8-properties-of-fractions

Chapter VIII – Of the Properties of Fractions

Summary: Shows that multiplying or dividing both numerator and denominator by the same number leaves a fraction’s value unchanged, and uses this to reduce fractions to lowest terms via common divisors.

Sources: chapter-1.1.8

Last updated: 2026-04-24

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§85–87: Equivalent Fractions

Multiplying both terms of a fraction by the same number $m$ does not change its value: $\frac{a}{b}=\frac{2a}{2b}=\frac{3a}{3b}=\frac{ma}{mb}$

Proof: If $c=a/b$, then $c\times b=a$, so $c\times mb=ma$, giving $c=ma/mb$ (source: chapter-1.1.8).

Examples of equivalent families: $\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{4}{8}=\cdots ,\frac{2}{3}=\frac{4}{6}=\frac{8}{12}=\cdots$

§88–91: Reduction to Lowest Terms

To simplify a fraction, divide both numerator and denominator by a common divisor. Repeat until the only common divisor is 1 — the fraction is then in its lowest terms (simplest form).

Example: $\frac{48}{120}\stackrel{÷2}{\to }\frac{24}{60}\stackrel{÷2}{\to }\frac{12}{30}\stackrel{÷2}{\to }\frac{6}{15}\stackrel{÷3}{\to }\frac{2}{5}$

Since $\gcd (2,5)=1$, the fraction $\frac{2}{5}$ is fully reduced (source: chapter-1.1.8).

§92: Importance for Fraction Arithmetic

The equivalence principle is the foundation of fraction arithmetic: before adding or subtracting fractions with different denominators, one transforms them into equivalent fractions sharing a common denominator (source: chapter-1.1.8).

§93: Integers as Fractions

Every integer $n$ can be expressed as a fraction: $n=\frac{n}{1}=\frac{2n}{2}=\frac{3n}{3}=\cdots$ (source: chapter-1.1.8).

Related pages

• fractions
• ch1.1.7-fractions-in-general
• ch1.1.9-addition-subtraction-fractions
• ch1.1.6-properties-integers-divisors