Euler's Investigations on the Roots of Equations - Imaginary Quantities

Todd Doucet, translator

This flows into the second half of the article, where Euler discusses the ways a quantity can be imaginary. For Euler, imaginary means "neither greater than zero, nor less than zero, nor equal to zero," and he gives as an example \( \sqrt{-1} \), or more generally a+b\( \sqrt{-1} \). So Euler addresses the question of whether there might be other ways a quantity can be imaginary, perhaps without being reducible to this form.

To this end, Euler shows that when you apply the common operations of analysis-addition, subtraction, multiplication, and division-to quantities of the canonical form, the result can be reduced to the canonical form.

He then goes further and considers whether perhaps transcendental operations might yield quantities that are imaginary in some distinct way, and shows that the known transcendental operations-those involving logarithms, angles, and the like-all yield quantities that can be reduced to the canonical form. This involves a remarkably clear and insightful explanation of raising real and imaginary quantities to real and imaginary powers, of the trigonometric functions applied to imaginary arguments, and of logarithms and antilogarithms of imaginary quantities.

Euler's purpose is to demonstrate that all of these operations yield quantities, often an infinity of quantities, that can be written in the canonical form. But the results themselves and Euler's explanation of them offer much to the contemporary reader who is interested in what might be meant by these kinds of expressions. As just one example, Euler observes that the expression \( \sqrt{-1} \) raised to \( \sqrt{-1} \) has an infinity of values, and that, surprisingly, they are all real. More generally, the periodic aspect of the transcendental operations when applied to imaginary quantities is explained quite clearly. Many of the familiar identities involving e and the trigonometric functions are laid out.

The ideas in this article are for the most part algebraic, but not exclusively so. Intuitive use of the intermediate value theorem enters into his proof early on, when he argues that every odd-degree equation has at least one real root, a result which he uses repeatedly. And in the second part of the article, when Euler shows how to find all the roots of an imaginary quantity M + Ni his exposition is tantalizingly close to what we think of as the complex plane, an idea that was articulated in modern form about a half century later.