Infinity

Summary: Euler introduces infinity as the limiting denominator that reduces a fraction to zero, symbolised by $\infty$; dividing 1 by 0 yields infinity, and infinity itself admits of degrees.

Sources: chapter-1.1.7, chapter-1.1.10

Last updated: 2026-04-24

---

Origin: Limits of Unit Fractions

The sequence $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ decreases without bound but never reaches 0. Euler reasons that for the fraction to equal zero, the denominator must be infinite — a number greater than any assignable quantity. He introduces the symbol $\infty$ (source: chapter-1.1.7):

$\frac{1}{\infty }=0$

Division by Zero

Reversing: since $\frac{1}{\infty }=0$, dividing 1 by 0 must give $\infty$: $\frac{1}{0}=\infty$

Confirmed in ch1.1.10-multiplication-division-fractions: as a divisor shrinks toward 0, the quotient grows without limit. Dividing 1 by $\frac{1}{1,000,000,000}$ gives $1,000,000,000$ (source: chapter-1.1.10).

Degrees of Infinity

Euler argues that infinity is not a fixed ceiling: $\frac{2}{0}=2\times \frac{1}{0}=2\infty$

“A number, though infinitely great, may still become twice, thrice, or any number of times greater” (source: chapter-1.1.7).

Note

Euler’s treatment is intuitive and pre-rigorous by modern standards. The symbol $\infty$ is introduced operationally, not axiomatically. Rigorous foundations (limits, extended real line) were developed in the 19th century.

Infinite Series Connection

The geometric series $1+a+a^2+\cdots =\frac{1}{1-a}$ confirms the identification $\frac{1}{0}=\infty$ when $a=1$: the infinite sum $1+1+1+\cdots$ equals $\frac{1}{1-1}=\frac{1}{0}$ (source: chapter-1.2.5). See infinite-series and ch1.2.5-infinite-series.

Related pages

• ch1.1.7-fractions-in-general
• ch1.1.10-multiplication-division-fractions
• fractions
• infinite-series
• ch1.2.5-infinite-series