ch1.1.11-square-numbers

Chapter XI – Of Square Numbers

Summary: Defines square numbers as products of a number by itself, extends squaring to fractions and products, and observes that every positive square has two roots, one positive and one negative. (source: chapter-1.1.11)

Sources: chapter-1.1.11

Last updated: 2026-04-24

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§115–117: Definition and Table of Squares

A square number is the product of a number multiplied by itself, and the original number is called a square root. (source: chapter-1.1.11)

Euler lists the squares of the natural numbers $1$ through $13$, such as $2^2=4$, $3^2=9$, and $12^2=144$. (source: chapter-1.1.11)

He also observes that consecutive squares differ by odd numbers: $4-1=3,9-4=5,16-9=7,$ so the successive differences form the series $3,5,7,9,\dots$. (source: chapter-1.1.11)

§118–119: Squares of Fractions and Mixed Numbers

Fractions are squared by squaring numerator and denominator separately: ${\left(\frac{a}{b}\right)}^2=\frac{a^2}{b^2}.$ (source: chapter-1.1.11)

Mixed numbers must first be reduced to improper fractions, after which the same rule applies. Euler uses examples such as $2\frac{1}{2}=\frac{5}{2}$, whose square is $\frac{25}{4}=6\frac{1}{4}$. (source: chapter-1.1.11)

If the root contains a fractional part, its square also contains one. (source: chapter-1.1.11)

§120–121: General Products

If the root is $a$, then its square is $a^2$; if the root is $2a$, the square is $4a^2$; if the root is $3a$, the square is $9a^2$. Doubling the root therefore makes the square four times as large, and tripling it makes the square nine times as large. (source: chapter-1.1.11)

If the root is a product, the square is the product of the squares: ${\left(ab\right)}^2=a^2{b}^2,{\left(abc\right)}^2=a^2{b}^2{c}^2.$ (source: chapter-1.1.11)

Conversely, if a square is factored into square factors, its root is found by multiplying the roots of those factors. Euler gives $2304=4\cdot 16\cdot 36$, so $\sqrt{2304}=2\cdot 4\cdot 6=48$. (source: chapter-1.1.11)

§122: Signs and Double Roots

The square of a positive number is positive, and the square of a negative number is also positive: ${\left(+a\right)}^2=+a^2,{\left(-a\right)}^2=+a^2.$ (source: chapter-1.1.11)

Euler concludes that every positive square has two roots, one positive and one negative. Thus $25$ has roots $+5$ and $-5$. (source: chapter-1.1.11)

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