Chapter XIII – Of Impossible, or Imaginary Quantities, which arise from the same source
Summary: Introduces imaginary quantities as square roots of negative numbers, reduces every such expression to a real factor times , and argues that imaginary results reveal impossible problems. (source: chapter-1.1.13)
Sources: chapter-1.1.13
Last updated: 2026-04-24
§139–144: Why Is Impossible
Since the square of both and is positive, no real number can have a negative square. (source: chapter-1.1.13)
Euler therefore concludes that quantities such as , , and are neither positive, negative, nor zero, but belong to a separate class of impossible or imaginary quantities. (source: chapter-1.1.13)
§145–150: Calculating with Imaginary Quantities
Although imaginary numbers are impossible as ordinary magnitudes, Euler still treats them as valid algebraic symbols. (source: chapter-1.1.13)
He notes that: (source: chapter-1.1.13)
Every imaginary square root may be reduced to a real factor times : (source: chapter-1.1.13)
Thus , , and . (source: chapter-1.1.13)
Two imaginary quantities may multiply to a real one, as , while a real quantity times an imaginary one remains imaginary. (source: chapter-1.1.13)
Division follows the same pattern, for example: (source: chapter-1.1.13)
Euler also states that imaginary roots have two signs, just as ordinary square roots do: The claim that both signs are roots is explicit in the chapter, though the notation distinguishes signs outside the radical from the negative sign inside it. (source: chapter-1.1.13)
§151: Use of Imaginary Results
Euler argues that imaginary numbers are useful because they expose impossible conditions in algebraic problems. (source: chapter-1.1.13)
His example is the problem of dividing into two parts whose product is . The formal answer is and , so the original problem is impossible in real numbers. (source: chapter-1.1.13)