ch1.1.7-fractions-in-general

Chapter VII – Of Fractions in General

Summary: Introduces fractions as quotients arising from non-exact division, defines proper/improper fractions and mixed numbers, establishes that unit fractions 1/n decrease toward but never reach zero, and derives the concept of infinity.

Sources: chapter-1.1.7

Last updated: 2026-04-24

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§68–70: Fractions Defined

When a dividend cannot be divided exactly by a divisor, the result is expressed as a fraction $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator. The fraction expresses the quotient of $a$ divided by $b$ (source: chapter-1.1.7).

§71–75: Types of Fractions

• Proper fraction: numerator $<$ denominator; value $<1$. Example: $\frac{2}{3}$.
• Improper fraction: numerator $\ge$ denominator; value $\ge 1$. Example: $\frac{7}{3}=2\frac{1}{3}$.
• Mixed number: integer plus a proper fraction. To convert an improper fraction, divide numerator by denominator and express the remainder: $\frac{43}{12}=3\frac{7}{12}$.

Special case: $\frac{a}{a}=1$ for all $a\ne 0$ (source: chapter-1.1.7).

§76–77: Part-of-a-Whole Interpretation

$\frac{3}{4}$ means: divide unity into 4 equal parts and take 3 of them. The denominator names (denotes) the parts; the numerator counts (numbers) them — hence the terms denominator and numerator (source: chapter-1.1.7).

§78–80: Unit Fractions and Limits

The unit fractions $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ form a strictly decreasing sequence. However large the denominator, the fraction never becomes exactly 0 — each part, however small, still has a definite magnitude (source: chapter-1.1.7).

§81–84: Infinity

Since $\frac{1}{n}\to 0$ only as $n\to \infty$, the denominator must be infinite for the fraction to equal zero. Euler introduces the symbol $\infty$ for an infinitely great number and writes $\frac{1}{\infty }=0$.

Reversing: dividing 1 by 0 gives $\infty$: $\frac{1}{0}=\infty$

Further, $\frac{2}{0}=2\infty$, so infinity admits of degrees — an infinitely great number may still be doubled, tripled, etc. (source: chapter-1.1.7). See infinity.

Related pages

• fractions
• infinity
• ch1.1.5-division-simple-quantities
• ch1.1.8-properties-of-fractions
• ch1.1.9-addition-subtraction-fractions
• ch1.1.10-multiplication-division-fractions