# What was Euler's motivation for introducing $i$ for $\sqrt{-1}$?

[Mauro Allegranza has answered the question of who introduced the notation $i$ (Euler, followed later by Gauss), so I have changed the title. I have also edited the question in other ways to make it clearer what I am asking.]

A common bit of mathematical folklore states that $i$ was introduced to guard against the fallacy $1=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=-1$. (See for example this question, and this passage from Wikipedia. Note the [citation needed] in the Wikipedia entry.)

This seems plausible to modern mathematicians because of the work of 19th century mathematicians, especially Riemann. We cannot define a (single-valued) square-root function in the entire complex plane satisfying the identity $\sqrt{xy}=\sqrt{x}\sqrt{y}$. (Indeed, the above fallacy, recast, proves that.) So we are naturally suspicious of the expression $\sqrt{-1}$ used to define a value. Which square root is meant, $\pm i$?

However, modern sensibilities are not a reliable guide to the thoughts of 18th century mathematicians. I will try to cast some doubt on the "fallacy-guarding" explanation below, but first let me elucidate my question. An alternate hypothesis is that Euler simply introduced the symbol $i$ for brevity. Perhaps he felt that this fundamental constant deserved a standard name, just as he introduced $e$ for the base of natural logarithms. Is there any documentary evidence to decide between these hypotheses? (For example, if Euler placed a discussion of the above fallacy in close proximity to the introduction $i$, that would be evidence for the "fallacy-guarding" hypothesis.)

I've poked around on the internet, looking for a definitive answer. Searching math.stackexchange with the tags [math-history][complex-numbers] yielded nothing useful. The "Google full text" function applied to Paul J. Nahin's book An Imaginary Tale: The Story of $i$ (with the key "notation") also failed to answer the question.

My reasons for questioning the folklore: First note that introducing $i$ does not unequivocally prevent the fallacy, which can be rewritten $1=\sqrt{(-1)(-1)}=i\cdot i=-1$. A closely related argument is that the notation $\sqrt{-1}$ is inherently ambiguous. However, the notational definition "We will use $\sqrt{-1}$ to denote one of the two square roots of $-1$" is no more ambiguous than "We will use $i$ to denote one of the two square roots of $-1$" . The fallacy lies in the unrestricted use of $\sqrt{xy}=\sqrt{x}\sqrt{y}$, not the use of $\sqrt{-1}$ to stand for an (arbitrarily chosen) square root of $-1$.

There is also a historical reason to question the "fallacy-guarding" hypothesis. The emphasis on single-valued functions was far less prominent in the 18th century. From the Wikipedia article History of the function concept:

A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.

Euler also allowed multi-valued functions whose values are determined by an implicit equation.

Later Euler gave another definition, although this was apparently intended as a generalization of his previous definition. Sorting out the issues with single-valued vs. multi-valued functions was largely a task of the 19th century, with Dirichlet and Riemann playing prominent roles.

Now, $\sqrt{xy}=\sqrt{x}\sqrt{z}$ is fine as an equation between multi-valued functions: it simply says that the set of values on the left is equal to the set of values on the right (given the obvious definition for the product of two sets of values). Unbridled use of multi-valued equations has its pitfalls, but this concern seems to belong more to the 19th century.

Much of Euler's work was more "formula-centric" than we are used to today. His free-wheeling computations with infinite series are well-known. Although they were (mostly) eventually justified, Euler's standards of rigor were not ours.

For these reasons, I find the "fallacy-guarding" argument about as plausible as the "brevity" argument. Is there any contemporaneous historical evidence to help decide the question?

• Notation $\sqrt{a}$ is ambiguous when we speak of complex numbers, because the equation $x^2=a$ has two roots, and there is no way to choose one consistently (that is continously). And the formulas like you wrote make no sense, unless you specify each time which of the two values of the $\sqrt{}$ is used. – Alexandre Eremenko Nov 22 '14 at 21:40
• It is relevant because you are also asking WHY the notation $i$ was introduced. I told you why. – Alexandre Eremenko Nov 22 '14 at 23:45
• Or perhaps you try to explain better what "why" means? You are repeating this "why", I am answering and you say this is irrelevant:-) – Alexandre Eremenko Nov 22 '14 at 23:47
• unfortunately your added explanation only obscures the matter. $i$ is a number, not a function. – Alexandre Eremenko Nov 23 '14 at 3:13
• @AlexandreEremenko -- Thank you for the word "continuously." – Robin Goodfellow Dec 5 '14 at 21:24

According to Florian Cajori, A History of Mathematical Notations (1928 - Dover reprint), Vol II, page 128 :

498. It was Euler who first used the letter $i$ for $\sqrt -1$. He gave it in a memoir presented in 1777 to the Academy at St. Petersburg, and entitled "De formulis differentialibus etc.," but it was not published until 1794 after the death of Euler. [footnote : The article was published in his Institutiones calculi integralis (2d ed.), Vol. IV (St. Petersburg, 1794), p. 184. See W.W.Beman in Bull.Amer.Math.Soc., Vol. IV (1897-98), p. 274. See also Encyclopédie des scien. math., Tom. I, Vol. I (1904), p. 343, n. 60.]

As far as is now known, the symbol $i$ for $\sqrt -1$ did not again appear in print for seven years, until 1801. In that year Gauss [footnote : K.F.Gauss, Disquisitiones arithmeticae (Leipzig, 1801), No. 337; translated by A. Ch. M. Poullet-Delisle, Recherches arithmétiques (Paris, 1807); Gauss, Werke, Vol. I (Gottingen, 1870), p. 414] began to make systematic use of it; the example of Gauss was followed in 1808 by Kramp. [C.Kramp, Eléments d'arithmétique universelle (Cologne, 1808), "Notations."]

See page 129 :

499. [...] in 1847 Cauchy began to use $i$ in a memoir on a new theory of imaginaries [footnote : A.L.Cauchy, Comptes rendus, Vol. XXIV (1847), p.1120=(Oeuvres complètes (1st ser.), Vol. X, p. 313, 317, 318.] where he speaks of " .., le signe symbolique $\sqrt -1$, auquel les geometres allemands substituent la lettre $i$," and continues: "Mais il est évident que la théorie des imaginaires [emphasis added] deviendrait beaucoup plus claire encore et beaucoup facile à saisir, qu'elle pourrait etre mise à la portée de toutes les intelligences, si l'on parvenait à réduire les expressions imaginaires, et la lettre $i$ elle-meme, a n'etre plus que des quantités réelles." Using in this memoir the letter $i$ as a lettre symbolique, substituted for the variable $x$ in the formulas considered, he obtains an équation imaginaire, in which the symbolic letter $i$ should be considered as a real quantity, but indeterminate.

See page 130 :

501. De Morgan's comments on $\sqrt -1$. The great role that $\sqrt -1$ has played in the evolution of algebraic theory is evident from the statements of Augustus de Morgan. In 1849 he spoke of "the introduction of the unexplained symbol [emphasis added] $\sqrt -1$," [footnote : Augustus de Morgan, Trigonometry and Double Algebra (London, 1849), p. 41] and goes on to say: "The use, which ought to have been called experimental, of the symbol $\sqrt -1$, under the name of an impossible quantity, shewed that; come how it might, the intelligible results (when such things occurred) of the experiment were always true, and otherwise demonstrable. I am now going to try some new experiments."

The issue in Cauchy's seems to be : we have here a "name" ($i$) for an "impossible" quantity, that is for something that cannot exists. This is - I think - the context of Cauchy's concerns: to find a "reference", that must be some "real" quantity, for the symbolic expression .

We can see in this post some references to Descartes's mathematical works :

"he names some roots 'true', some 'implicit' (that is, less than nothing), and some 'imaginary' (that is, not explicable at all)".

Regarding Euler and the benefits of a "good" symbolism, see L.Euler, Elements of Algebra (1765; English translation by John Hewlett : 1822) §148-149, page 43 : "as $\sqrt a$ multiplied by $\sqrt b$ makes $\sqrt ab$, we shall have $\sqrt 6$ for the value of $\sqrt −2$ multiplied by $\sqrt −3$".

This passage supports the conjecture that Euler understood the need of a specific symbol in order to avoid the above "fallacy".

For $\pi$, see Vol II, page 9 :
396. First occurrence of the sign $\pi$. The modern notation for 3.14159 . . . . was introduced in 1706. It was in that year that William Jones [footnote : William Jones, Synopsis palmariorum matheseos (London, 1706), p.263.] made himself noted, without being aware that he was doing anything noteworthy, through his designation of the ratio of the length of the circle to its diameter by the letter $\pi$. He took this step without ostentation.
• Is not it clear why? "i" is the first letter of the word "imaginary". I can add that the notation did not catch immediately, even Cauchy used the cumbersome $a+b\sqrt{-1}$. – Alexandre Eremenko Nov 22 '14 at 21:33
• @MauroALLEGRANZA Thanks for the new info on Cauchy. Unfortunately I am having difficulty understanding the end of the second quote. Is Cauchy giving a reason for using $i$, or for not using $i$? How would you translate "a n'être plus que des quantités réelles". – Michael Weiss Nov 23 '14 at 12:35