Chapter XII – Of Square Roots, and of Irrational Numbers resulting from them
Summary: Explains how square roots are extracted from perfect squares, shows why non-square numbers such as have no rational square root, and introduces irrational numbers and the radical sign . (source: chapter-1.1.12)
Sources: chapter-1.1.12
Last updated: 2026-04-24
§123–124: Square Roots of Squares, Fractions, and Mixed Numbers
The square root of a given number is the number whose square equals that number. (source: chapter-1.1.12)
Perfect squares have readily identifiable roots, such as . Fractions are treated by extracting the root of numerator and denominator separately, and mixed numbers are first reduced to fractions. (source: chapter-1.1.12)
§125–127: Why Non-squares Have No Rational Root
Euler uses as the main example. Since and , lies between and . (source: chapter-1.1.12)
By testing fractions such as , , , , and , he shows that one can approach the value more closely without ever reaching an exact fractional expression. (source: chapter-1.1.12)
He therefore argues that the square root of is determinate in magnitude, yet not expressible by integers or fractions. (source: chapter-1.1.12)
§128–130: Irrational Numbers and Radical Notation
Numbers such as , , and are called irrational numbers, surd quantities, or incommensurables. (source: chapter-1.1.12)
Euler introduces the radical sign to denote such quantities: (source: chapter-1.1.12)
§131–136: Arithmetic with Square Roots
Euler extends multiplication and division to surds by treating roots multiplicatively: (source: chapter-1.1.12)
He also shows how to simplify radicals when the number under the sign contains a square factor: (source: chapter-1.1.12)
Multiplying a surd by an ordinary number can be absorbed under the radical sign, as in . (source: chapter-1.1.12)
§137–138: Rational and Irrational Numbers
Euler notes that addition and subtraction of unlike surds are written formally, for example or . (source: chapter-1.1.12)
He then distinguishes rational numbers from irrational ones: rational numbers are the ordinary integers and fractions, while irrational numbers arise from roots that cannot be exactly expressed by either. (source: chapter-1.1.12)