Imaginary Numbers

Summary: Imaginary numbers arise in Euler’s account as even roots of negative quantities, especially square roots of negative numbers, and serve as algebraic markers of impossible real solutions. (source: chapter-1.1.13, source: chapter-1.1.18, source: chapter-1.1.21)

Sources: chapter-1.1.13, chapter-1.1.18, chapter-1.1.21, chapter-1.4.10, chapter-2.0.11, chapter-2.0.12

Last updated: 2026-05-09

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Origin

No real number squares to a negative quantity, so expressions like $\sqrt{-1}$, $\sqrt{-2}$, and $\sqrt{-4}$ are called imaginary or impossible quantities. (source: chapter-1.1.13)

More generally, Euler says that even roots of negative numbers are imaginary, while odd roots of negative numbers remain real and negative. (source: chapter-1.1.18)

Reduction to $\sqrt{-1}$

Euler reduces any square root of a negative quantity to a real factor times $\sqrt{-1}$: $\sqrt{-a}=\sqrt{a}\sqrt{-1}.$ (source: chapter-1.1.13)

Arithmetic Role

Imaginary quantities can still be manipulated algebraically. Two imaginary factors may give a real product, but a real factor times an imaginary one remains imaginary. (source: chapter-1.1.13)

Euler also remarks that logarithms of negative numbers are impossible and therefore belong to the same class. (source: chapter-1.1.21)

Interpretive Use

When an algebraic problem leads to imaginary answers, Euler treats that as evidence that the requested real solution is impossible. (source: chapter-1.1.13)

Cubic roots and imaginary values

Euler notes that every cube root has three values: one real and two imaginary. For $c=\sqrt[3]{a}$, the other two cube roots are $\frac{-1\pm \sqrt{-3}}{2}\cdot c$ (source: chapter-1.4.10, §713). More generally, an $n$th root has $n$ values total. In ordinary calculation only the real value is used; the imaginary ones are noted for completeness.

This generalises the earlier observation that square roots have two values (one real, one with $\sqrt{-1}$) to the cubic case. See cubic-equations and ch1.4.10-pure-cubic-equations.

Imaginary Quantities Producing Integer Identities

In Part II, Chapters XI–XII, Euler exploits the factorization $x^2+c{y}^2=(x+y\sqrt{-c})(x-y\sqrt{-c})$ to derive identities about integers by manipulations in the imaginary realm. Two key applications:

• brahmagupta-fibonacci-identity: $(p^2+c{q}^2)(r^2+c{s}^2)=(pr-cqs{)}^2+c(ps+qr{)}^2$, derived by multiplying complex factors and reading rational/irrational parts.
• Integer powers of $x^2+c{y}^2$: setting $x+y\sqrt{-c}=(p+q\sqrt{-c}{)}^n$ produces explicit integer parametrizations (cubes, biquadrates, fifth powers).

Euler comments in ch2.0.12-quadratic-form-as-power §191 that this is “remarkable, as we are brought to solutions, which absolutely required numbers rational and integer, by means of irrational, and even imaginary quantities.”

He also notes a subtle caveat: the inference “if a product is a power, each factor is a power” requires no nontrivial common divisors among the factors. For genuine imaginaries this is automatic when $\gcd (x,y)=1$; but for $c=-1$ (real factors $x+y$, $x-y$) the inference can fail. This is an early instance of the failure of unique factorization that would later motivate ring theory.

Related pages

• irrational-numbers
• roots
• positive-negative-numbers
• logarithms
• cubic-equations
• quadratic-equations
• discriminant
• brahmagupta-fibonacci-identity
• ch2.0.11-quadratic-form-factorization
• ch2.0.12-quadratic-form-as-power
• sums-of-two-squares