ch1.1.18-roots-and-powers

Chapter XVIII – Of Roots, with relation to Powers in general

Summary: Generalizes square and cube roots to roots of every order and distinguishes the behavior of even and odd roots for positive and negative numbers. (source: chapter-1.1.18)

Sources: chapter-1.1.18

Last updated: 2026-04-24

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§189–191: Roots of Arbitrary Degree

Euler extends the language of roots beyond square and cube roots to roots of any order: second roots, third roots, fourth roots, fifth roots, and so on. (source: chapter-1.1.18)

He represents them by indexed radical signs and states that the $n$th root of $a$ is the number whose $n$th power equals $a$. (source: chapter-1.1.18)

§192: Size of Roots

If $a=1$, all its roots are $1$. If $a>1$, all its roots are greater than $1$. If $0<a<1$, all its roots are less than $1$. (source: chapter-1.1.18)

§193–194: Even and Odd Roots

When $a$ is positive, all its roots are real. (source: chapter-1.1.18)

When $a$ is negative, even roots such as the second, fourth, and sixth become imaginary, while odd roots such as the third, fifth, and seventh remain negative and real. (source: chapter-1.1.18)

Euler also stresses that whenever the radicand is not an exact power of the required degree, the result is irrational. (source: chapter-1.1.18)

Related pages

• roots
• irrational-numbers
• imaginary-numbers
• powers-and-exponents