Square Roots and Irrational Numbers
Summary: Square roots begin as inverses of squaring, but when the radicand is not a perfect square they produce irrational quantities that Euler represents with the radical sign. (source: chapter-1.1.11, source: chapter-1.1.12)
Sources: chapter-1.1.11, chapter-1.1.12
Last updated: 2026-04-24
Definition
A square root of a number is any quantity whose square equals that number. Every positive square has two roots, one positive and one negative. (source: chapter-1.1.11)
Perfect Squares
If the number is a perfect square, the root is rational and can be read off directly, as with . Fractions and mixed numbers are treated by reducing them to square numerators and denominators. (source: chapter-1.1.12)
Non-square Numbers
If the radicand is not a perfect square, the root is still determinate but cannot be expressed by integers or fractions. Euler uses as his main example. (source: chapter-1.1.12)
These quantities are irrational, also called surds or incommensurables. (source: chapter-1.1.12)
Radical Arithmetic
Euler treats square roots multiplicatively: (source: chapter-1.1.12)
He also simplifies radicals by extracting square factors: (source: chapter-1.1.12)