77

The following extract is from Tristan Needham's Visual Complex Analysis,

Even in 1770 the situation was still sufficiently confused that it was possible for so great a mathematician as Euler to mistakenly argue that √-2 √-3 = √6.

I found this to be a bit far fetched. A simple Google search doesn't return anything. Is this claim true?

Oddthinking
  • 140,378
  • 46
  • 548
  • 638
Agile_Eagle
  • 1,441
  • 2
  • 12
  • 16
  • 8
    See Andrew Wiles's [answer](https://youtu.be/0kSYdOwVi4Y?t=4632) to a question from a high-school teacher. Especially at a basic level, people get hung up on what is "the proper definition" rather than understanding that there is some arbitrariness in how we choose definitions. – Robert Furber Jun 15 '20 at 10:42
  • 3
    I don't understand why this is called an "elementary mistake" when you don't get into that kind of math until about the 10th grade! – Daniel R Hicks Jun 15 '20 at 12:55
  • 2
    I'm certain that almost any modern working mathematician would consider any confusion about something like this (or indeed most other 10th grade subjects) quite "elementary". Of course the same isn't true in the historical context of Euler's time! – ManfP Jun 15 '20 at 15:56
  • 2
    @DanielRHicks, it honestly intrigues me when I read something and cannot tell whether the person is joking. Then I look for signs, and I see an exclamation mark. That certainly means something. But what? – Carsten S Jun 16 '20 at 09:24
  • See meta-question about [title formatting](https://skeptics.meta.stackexchange.com/questions/4723/what-is-the-right-way-to-format-a-mathematical-formula) – Oddthinking Mar 02 '21 at 16:14
  • Related https://math.stackexchange.com/questions/2368034/is-i-equal-to-sqrt-1/2368056 – Fizz Mar 04 '21 at 02:08
  • And https://math.stackexchange.com/questions/144364/is-the-square-root-of-a-negative-number-defined/144404 ; the latter q's answer actually touch on this issue more than the other one does. – Fizz Mar 04 '21 at 02:24

2 Answers2

178

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:

  1. 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.

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.
Tanner Swett
  • 1,038
  • 1
  • 6
  • 7
  • 4
    Now I wish we could see the original (pre-translation) version, because it seems like the radical sign is being used in two different ways between the first and second sentences. – Daniel R. Collins Jun 14 '20 at 20:30
  • 11
    Note that it's not possible to take either i or -i as the "principal" square root of -1, in the way it's possible for positive values. Example of problem this causes: 1 = √1 = √-(-1) = i√(-1) = i²√1 = -1 – BlueRaja - Danny Pflughoeft Jun 14 '20 at 21:40
  • 1
    @DanielR.Collins You can see Euler's original publications in German in [1771](https://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10081749_00082.html) and [1802](https://digital.slub-dresden.de/werkansicht/dlf/20160/107/0/) – Henry Jun 14 '20 at 23:31
  • 38
    @BlueRaja-DannyPflughoeft A great example of how silly and unhelpful all this nonsense about the "principal" square root is. Insisting one of the two possible square roots of a number is more special than the other for needlessly pedantic reasons is why some people hate math. – JounceCracklePop Jun 15 '20 at 01:17
  • 13
    "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." - Not necessarily; you can choose to use a number system that includes [signed zero](https://en.wikipedia.org/wiki/Signed_zero) – aroth Jun 15 '20 at 02:59
  • 1
    @aroth True. In the complex numbers, 0 has only one square root (unlike every other complex number, which has two). There are other number systems in which 0 has two square roots; aside from IEEE 754, I don't know of any which have been studied. – Tanner Swett Jun 15 '20 at 03:44
  • 2
    As far as I know there isn't any common convention about which is "the" square root of a negative number and current usage is to use $\sqrt{x}$ only for non-negative $x$ (as you certainly know it's impossible to make a consistent choice of square root for each complex number due to a monodromy obstruction). – Denis Nardin Jun 15 '20 at 07:06
  • 20
    @CarlLeth no, it's a great example of how silly it can be to thoughtlessly extend the identities that work for real numbers to complex domain. And the "nonsense" of choosing the principal branch is the only sane way work with square root as a _function_. Multifunctions like the Euler's square root are a great source of headaches for those who are just trying to calculate something and not get ±f(±g(±x)) as the answer (with the actual signs of individual ± symbols not necessarily correlated). – Ruslan Jun 15 '20 at 10:28
  • Wow! I thought the claim must have been totally imaginary! – Andrew Grimm Jun 15 '20 at 12:26
  • 8
    "Nowadays, mathematicians use √ to mean only the upper square root" --- I wouldn't be so sure. Most mathematicians would simply avoid taking square (or even) roots of negative reals (without explicitly specifying the principal value they are considering). – tomasz Jun 15 '20 at 13:37
  • This is a bit tongue in cheek: the bigger mistake was to write down square root of 2, since he couldnt know that this number exists; he should have waited for Dedekind to come around first. – lalala Jun 15 '20 at 15:22
  • 16
    The square root of zero might be considered a "double root". It has two roots, both of which are 0. See . – PyRulez Jun 15 '20 at 15:48
  • 1
    Actually 0 has 2 square roots as well, the value of the 2 square root happen to be the same thing. That is true in mathematics that if there is no difference between 2 things then there are the same thing, but that is set theory, in programming with arrays you could very well have n items that have the same value. A distinction must be made between value of an object and the actual object itself, there are 2 roots their value is the same, but one also could say there are infinitely many roots that their value is 0. This is a problem similar to 1 being excluded from prime numbers. – jimjim Jun 16 '20 at 06:29
  • 2
    This is a very good answer. Contrary to your last sentence, however, I would not want to use √-2 to denote one single number, though. – Carsten S Jun 16 '20 at 09:17
  • 1
    How can this answer be upvoted if you drew the conclusions yourself instead of citing a credible source? There's no external reference. It's all just in your mind, so it does not count on this site. My answers involving math and logical thinking were deleted. – delete me Jun 16 '20 at 16:49
  • @CarlLeth Yeah, people hate math because someone insist on a preference. Not because math is hard as fuck and they just want to get drunk and have some fun. xD – Red Banana Jun 16 '20 at 16:49
  • @Ruslan But how do you "choose the principal branch" in the general case, in a problem you don't already know how to solve? BlueRaja showed you can't for negative numbers. Nor is the positive number always the right choice. If you do pick one, you at least need to justify rejecting the other one; you can't just ignore it. If you really need √ to be a function, then work in the realm of multisets, so √{4} = {2, -2}. All those different branches in your example might be important! – JounceCracklePop Jun 16 '20 at 18:00
  • @CarlLeth A typical choice of the principal branch of a complex function f is to make sure that f(exp(iα)) is continuous at α=π¯. Having picked it, you don't reject the other solution when solving an equation, you express all the solutions in terms of your newly-defined single-valued function. Thus, the solutions of x²=-1 are now x=±√-1, with the single-valued square root function. Similarly you can get the whole infinite set of solutions of sin(x)=a in terms of the single-valued arcsin function—and then, if needed, pick _one_ of them by specifying the value of the parameter you've introduced. – Ruslan Jun 16 '20 at 18:09
  • @deleteme I could provide a source for any of the statements in my answer (except for the equations in the last two paragraph, which I would have to give a proof for instead). Is there a particular one you'd like to see a source for? By the way, I'm interested in seeing your answer; I might be able to give some insight as to why it was deleted. – Tanner Swett Jun 16 '20 at 18:46
  • @TannerSwett: with 703 reputation, you likely can't see the deleted answer. Anyway. Good luck. – delete me Jun 16 '20 at 20:35
  • @Tanner Swett please [edit] your answer and provide "a source for any of the statements". And maybe a proof for the equations in the last two paragraphs. – danzel Jun 16 '20 at 23:21
  • @danzel I've added a couple of references and proofs. – Tanner Swett Jun 17 '20 at 00:14
  • @TannerSwett At the cost of repeating myself, I have to object to your note 5: "it would be extraordinarily strange to define $\sqrt{-2}$ as anything but $i\sqrt{2}$" is not half of it: in 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). I don't doubt that it is a convention in your subfield, but suggesting that it is widespread is misrepresenting the situation. – Denis Nardin Jun 17 '20 at 09:13
  • It is difficult to see why Euler says the value of √-2 multiplied by √-3 is √6 (not saying it can be -√6) while the product of √-3 by √-3 must be -3 (not saying it can be 3) – Henry Jun 17 '20 at 15:54
  • @DenisNardin If you've ever had a career as a mathematician, then your career as a mathematician has been longer than mine. :) I've edited my answer so that it asserts that some, not all, mathematicians give a definition to √-2. Do you think it looks okay now? – Tanner Swett Jun 17 '20 at 17:43
24

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).

  1. 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.

  1. 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.

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.

SCappella
  • 421
  • 5
  • 10
  • 3
    To be fair, it _is_ sqrt(6) ... up to a factor of -1. – John Dvorak Jun 14 '20 at 15:16
  • 35
    It really is *a* square root of 6, just not *the* "usual" one (i.e. the principal value) – ManfP Jun 14 '20 at 15:36
  • 5
    Yeah, sometimes sqrt(x) is used to really mean +-sqrt(x). And the rest of the text only concerns whether the value is real or imaginary, not whether it is positive or negative. – jpa Jun 14 '20 at 15:54
  • 2
    $\sqrt{-2}$ could just as easily mean the one square-root of -2 as the other one, seeing as Euler doesn't give any explicit remark saying which root he means. So one may take $\sqrt{-2}$ as $-i\sqrt2$ and $\sqrt{-3}$ as $i\sqrt3$, and the product is the (positive) real $\sqrt6$. – Rosie F Jun 14 '20 at 16:27
  • You can see this in Euler's original publications in German in [1771](https://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10081749_00082.html) and [1802](https://digital.slub-dresden.de/werkansicht/dlf/20160/107/0/) – Henry Jun 14 '20 at 23:28
  • 8
    This answer makes it appear as if Euler did make a mistake, when in fact he was perfectly accurate and correct. By not explaining the change of definition during the time since, this answer is just as misleading as the original claim and should be heavily downvoted, not receiving net 18... – Nij Jun 15 '20 at 08:25
  • 4
    With √-1 √-4 = 2, we leave the realm of misleading. As √-1 √-4 = -2 has the same justification, we arrive by transitivity of = at 2 = -2. This can only be solved by redefiniing the equality sign, at least to the same extent as we commonly do in the context of big-Oh notation – Hagen von Eitzen Jun 15 '20 at 12:15
  • 1
    @HagenvonEitzen Hence why mathematics has a distinct difference in definition between *equivalence* (≡) and *equality* (=) – Chronocidal Jun 15 '20 at 21:08