Fractions

Summary: Numbers of the form $a/b$ (numerator over denominator) that arise when a dividend is not exactly divisible by a divisor; the principal extension of the integers within Part I of the algebra.

Sources: chapter-1.1.7, chapter-1.1.8, chapter-1.1.9, chapter-1.1.10

Last updated: 2026-04-24

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Motivation

Division of integers is not always exact: $7÷3$ has no integer answer, but the operation still makes geometric sense (divide a 7-foot line into 3 equal parts). The result is expressed as $\frac{7}{3}$ (source: chapter-1.1.7).

Notation

$\frac{a}{b}:a=\text{numerator},b=\text{denominator}$

The denominator names the equal parts into which unity is divided; the numerator counts how many are taken.

Types

Type	Condition	Value	Example
Proper	$a<b$	$<1$	$\frac{2}{3}$
Improper	$a\ge b$	$\ge 1$	$\frac{7}{3}=2\frac{1}{3}$
Mixed number	integer $+$ proper fraction	—	$3\frac{7}{12}$

Special: $\frac{a}{a}=1$ for all $a\ne 0$.

Key Properties

Equivalence (Ch. VIII): $\frac{a}{b}=\frac{ma}{mb}$ for any $m\ne 0$. Multiply or divide both terms by the same number without changing value.

Lowest terms (Ch. VIII): divide numerator and denominator by their greatest common divisor until only 1 remains. Example: $\frac{48}{120}=\frac{2}{5}$.

Arithmetic Summary

Operation	Rule
Same-denom. addition	$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$
General addition	$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$
General subtraction	$\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$
Multiplication	$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}$
Division	$\frac{a}{b}÷\frac{c}{d}=\frac{ad}{bc}$

Sign rules are identical to those for integers (source: chapter-1.1.10).

Limiting Behaviour

Unit fractions $\frac{1}{n}$ decrease as $n$ grows but never reach 0. Only an infinite denominator yields zero: $\frac{1}{\infty }=0$. See infinity (source: chapter-1.1.7).

Fractions as Generators of Infinite Series

When a fraction cannot be reduced or divided exactly, its numerator can still be divided by its denominator term by term, producing an infinite power series whose value equals the fraction. For example $\frac{1}{1-a}=1+a+a^2+\cdots$ (source: chapter-1.2.5). See infinite-series and ch1.2.5-infinite-series.

Related pages

• ch1.1.7-fractions-in-general
• ch1.1.8-properties-of-fractions
• ch1.1.9-addition-subtraction-fractions
• ch1.1.10-multiplication-division-fractions
• infinity
• positive-negative-numbers
• infinite-series
• ch1.2.5-infinite-series