Irrational Numbers

Summary: In Euler’s presentation, irrational numbers are determinate magnitudes that cannot be exactly expressed by integers or fractions and arise chiefly from inexact roots. (source: chapter-1.1.12, source: chapter-1.1.15, source: chapter-1.1.18)

Sources: chapter-1.1.12, chapter-1.1.15, chapter-1.1.18, chapter-1.1.19

Last updated: 2026-04-24

---

Origin

Euler first introduces irrational numbers when discussing square roots of non-squares such as $\sqrt{12}$. He later extends the same idea to cube roots and higher roots when the radicand is not an exact power of the required order. (source: chapter-1.1.12, source: chapter-1.1.15, source: chapter-1.1.18)

Character

These quantities are not indeterminate. Euler insists that they have definite magnitude even though they cannot be written as integers or fractions. (source: chapter-1.1.12)

He also calls them surd quantities or incommensurables. (source: chapter-1.1.12)

Notation

Irrational roots may be written either with radicals such as $\sqrt{a}$, $\sqrt[3]{a}$, and $\sqrt[4]{a}$, or with fractional exponents such as $a^{1/2}$, $a^{1/3}$, and $a^{1/4}$. (source: chapter-1.1.12, source: chapter-1.1.19)

Operations

Euler extends the ordinary rules of multiplication and division to irrational roots by combining radicands or, equivalently, by adding and subtracting fractional exponents. (source: chapter-1.1.12, source: chapter-1.1.15, source: chapter-1.1.19)

Related pages

• ch1.1.12-square-roots-irrational-numbers
• square-roots-and-irrational-numbers
• roots
• powers-and-exponents
• imaginary-numbers