Did Euler make the elementary mistake √-2 √-3 = √6?

Euler did write this, but it was not a mistake! Euler's statement was correct under his own definition of the notation that he was using.

I looked at the PDF version of Elements of Algebra linked to in SCappella's answer.

Reading Section I, Chapter XIII, I found that Euler wrote that most numbers have two square roots, which matches the definition of the phrase "square root" used by today's mathematicians^[1]. He also wrote that the square root sign √ denotes both square roots, which does not match the definition of √ used by today's mathematicians^[2], but which is not actually incorrect.

Here's what he wrote:

150. We have before observed, that the square root of any number has always two values, one positive and the other negative; that √4, for example, is both +2 and -2, and that, in general, we may take -√a as well as +√a for the square root of a. This remark applies also to imaginary numbers; the square root of -a is both +√-a and -√-a; but we must not confound the signs + and -, which are before the radical sign √, with the sign which comes after it.

(Actually, the above does contain an error. Euler claims that every number has two square roots; in fact, every number has two square roots except for 0, which only has one square root, which is 0. Source at ^[3].)

Negative numbers have two square roots, one of which has positive imaginary component and the other of which has negative imaginary component. Nowadays, mathematicians use √ to mean only one or the other according to some rule^[2], but we can see that to Euler, it would have meant either square root.

In particular, Euler considered √6 to mean either the positive or the negative square root of 6.

So, in Euler's notation, the equation (√-2)(√-3) = √6 meant "either square root of -2 times either square root of -3 is a square root of 6", which is completely true^[4].

Some of today's mathematicians would interpret (√-2)(√-3) = √6 as being meaningless, because they decline to give the expression √-2 and the expression √-3 any definition at all^[5].

I think other mathematicians would interpret it as meaning "the square root of -2 with positive imaginary component (i√2), times the square root of -3 with positive imaginary component (i√3), is the positive square root of 6", which is a false statement^[6]—but which is also a misreading of what Euler wrote.

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References and proofs:

• [1]: Weisstein, Eric W. "Square Root." From MathWorld--A Wolfram Web Resource. "A square root of x is a number r such that r^2=x."
• [2]: Ibid. "The principal square root of a [complex] number z is denoted √z [...]." The source does not include a definition of "the principal square root", but does make it clear that it is a function, meaning that it has only one value.
• [3]: Ibid. "Any nonzero complex number z also has two square roots."
• [4]: Proof: Suppose that x is a square root of -2 and y is a square root of -3. Then, by the definition of a square root, x² = -2 and y² = -3. As a consequence, (xy)² = x² y² = (-2) (-3) = 6. This means, by the definition of a square root, that xy is a square root of 6.
• [5]: Denis Nardin's comment on this answer: "[I]n all my (admittedly short) career as a mathematician I never encountered a definition of $sqrt{-2}$: in general it is considered an ill posed symbol (sort of like $0/0$, if you want)."
• [6]: I wasn't able to find a source for the definition of the principal square root of a negative number. However, it would be extraordinarily strange to define √-2 and √-3 as anything besides i√2 and i√3, respectively. (The only alternative would be to define √-2 as -i√2 or to define √-3 as -i√3, which would be inconsistent with the definition of √-1 as i rather than -i.) We have, thus, (√-2)(√-3) = (i√2)(i√3) = i²(√2)(√3) = -(√2)(√3) = -√6, which is negative, whereas √6 is positive.

Euler did argue that √-2 √-3 = √6. Whether this is a mistake depends a lot on context. This appears in Euler's 1770 publication Elements of Algebra in Section I., Chapter XIII. (pdf link).

148. Moreover, as √a multiplied by √b makes √ab, we shall have √6 for the value of √-2 multiplied by √-3; and √4, or 2, for the value of the product of √-1 by √-4. Thus we see that two imaginary numbers, multiplied together, produce a real, or possible one. But, on the contrary, a possible number, multiplied by an impossible number, gives always an imaginary product: thus, √-3 by √+5, gives √-15.

Euler, Elements of Algebra, pages 43-44 (emphasis mine).

Note that Euler correctly multiplied square roots of negative numbers earlier in that chapter.

146. The first idea that occurs on the present subject is, that the square of √-3, for example, or the product of √-3 by √-3, must be -3; that the product of √-1 by √-1, is -1; and in general, that by multiplying √-a by √-a, or by taking the square of √-a, we obtain -a.

Euler, Elements of Algebra, page 43.

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Edit: As noted in another answer, Euler also takes the convention that √a refers to both the positive and negative root, which makes √-2 √-3 = √6 merely misleading, but not wrong. √-1 √-4 = 2 is even more misleading, but still not wrong.

Again, Euler uses complex numbers correctly elsewhere in the same work and abandons the convention of two-valued square roots for the convention of using "±". For example, in Section IV., Chapter XI., we have

[...]this last factor gives x² + 2x = -3; consequently, x = -1 ± √-2;

Euler, Elements of Algebra, page 255.

where the two square roots of -2 are properly distinguished, rather than written as x = -1 + √-2 and using a two-valued square root.

That makes this particular passage where Euler is implicit rather than explicit stick out. Having seen more evidence, I won't argue that this is a mistake, though.