5

I noticed a large portion of general public (who knows what square root is) has a different concept regarding the surd of a positive number, $\sqrt\cdot$, or the principal square root function.

It seems to me a lot of people would say, for example, $\sqrt 4 = \pm 2$, instead of $\sqrt 4 = 2$. People even would correct a statement of the latter form to one with a $\pm$ sign. Some also claim that, since $2^2 = 4$ and $(-2)^2 = 4$, $\sqrt 4 = \pm 2$. Some people continue to quote other "evidences" like the $y=x^2$ graph. While most people understand there are two square roots for a positive number, some seem to have confused this with the surd notation.

From an educational viewpoint, what might be lacking when teaching students about surd forms? Is a lack of understanding to functions a reason for this misconception?

Now I have noticed another recent question that hinted that poster was confused. Following @AndréNicolas's comment below, might these confusion really come from two different communities using the same symbol?

Community
  • 1
peterwhy
  • 22,256
  • 3
    A fair number of standard North American school texts (at least in the past, I have not kept up) insist that $\sqrt{4}=\pm 2$. And teachers punish(ed) students who left out the $\pm$. So we have (or had) two communities that use $\sqrt{a}$ in different ways. – André Nicolas Sep 08 '13 at 16:59
  • 1
    On very old italian textbooks, there was a different symbol for the so-called algebraic square root, as opposed to the arithmetic square root. It seems there is no LaTeX symbol: it was $\sqrt{x}$ with a little, vertical dash at the end of the horizontal bar. – Siminore Sep 08 '13 at 17:14
  • 1
    @AndréNicolas so according to those texts, does $\sqrt4 + \sqrt4 = (\pm 4$ or $0)$? – peterwhy Sep 08 '13 at 17:18
  • 1
    Presumably that question is not addressed! – André Nicolas Sep 08 '13 at 17:21
  • 1
    It's interesting, that André Nicolas comment, that simply there are two communities is not an answer, and moreover the accepted one. Is the answer of such an elementary question really subject of *discussion* and opinions ? It should be an conclusive entry in wikipedia, if not -for instance- in Abramowitch&Stegun, which definition is authoritative (and should have a remark about the ambiguity in the common usances). – Gottfried Helms Sep 08 '13 at 18:12
  • @Siminore: I learned the version with the vertical line dropping from the end of the vinculum as a synonymous variant of $\sqrt{x}$ that simply made it a little easier to be sure exactly where the vinculum was supposed to end, thereby making it easier to distinguish $\sqrt2 x$ from $\sqrt{2x}$, especially in handwriting. – Brian M. Scott Sep 10 '13 at 08:24
  • @André: I’m not doubting you, but I’m surprised: if I ever encountered that usage in primary or secondary school, I don’t remember it. (Admittedly I might not have noticed, since by the time I’d have enountered the concept and notation in school, I was already thoroughly familiar with both.) But I’m also pretty sure that I never had a university student ask me about it in the last $40$ years or so. I have seen elementary university texts point out that $\sqrt{x^2}=|x|$ as if this were expected to be no surprise. I truly thought it a very well-established convention, in the U.S. at least. – Brian M. Scott Sep 10 '13 at 08:39

5 Answers5

13

They probably just think that $\sqrt{x}$ means any number whose square is $x$, and don't know that the definition is just the positive root. I don't really think it's anything more than that.

JLA
  • 6,442
  • 2
  • 24
  • 40
9

This is entirely context-dependent. First, despite pretenses in U.S. schools, for example, there are no "rules" in mathematics, and certainly no enforcement mechanisms. Further, although it is undeniably a good thing to encourage "careful thinking", to say that this is identical to "logic" is a misrepresentation, as the latter tends to limit its subject to "what can be entirely formalized", while mathematics itself posits no such constraint.

In particular, although pointless ambiguity is not a plus, attempting to "define/control" usage to remove reasonable ambiguities is (I think) at best misguided. If nothing else, rules that have some sense in one context may fail badly in others.

Thus, while there are certainly reasons to sometimes declare $\sqrt{x}$ to be the unique non-negative real square root of non-negative real $x$, there are certainly contexts in which it'd be convenient to allow it to refer to any real square root. And, of course, when taking square roots of complex numbers, there is an inescapable issue of specifying branches, etc. (No, the phrase "principal square root" doesn't really resolve things, because analytic continuation transgresses the declaration that we "always take the principal branch".)

A more vivid example is the age-old discussion of "whether 1 is or is not a prime". First, well into the 19th century many serious mathematicians did refer to it as a prime. The main disadvantage of doing so is that statements of results tend to be messier. Thus, the linguistic or conceptual advantages of saying 1 is prime are outweighed (as it turns out) by disadvantages, so nowadays we say it is not. Nevertheless, one can easily find on-line arguments purporting to "prove" that it is prime, or "should be".

About square roots, in any circumstance, I absolutely do not trust that whoever's writing will conform to whatever rules they or anyone else might claim to prescribe. I myself certainly have no "rules" about this, but would prefer to emphasize explicitly the single-valued-ness or two-valued-ness or complex-variables-ambiguity as context demands.

In fact, attempting to "resolve" the question on grounds of "rules" or "logic" may obfuscate the very real issues about the fact that there are two square roots, branches with complex numbers, and so on, as though those were somehow illicit.

And, e.g., having answers depend on careful attention to the articles "a" or "the" sounds like a trick question. Also, even if we grant that "the" means "just one", it's not the case that "the" means "the unique positive one, if it exists"... This level of fragile formalism isn't really very useful.

paul garrett
  • 52,465
  • I understand your idea not to always control the usage. But for this symbol $\sqrt\cdot$ to convey information, should there be such an ambiguity? For example, consider the square roots of $4i$. (I would avoid writing $\sqrt{4i}$ here) They are $\sqrt 2 + i\sqrt2$ and $-\sqrt2 - i\sqrt2$. Wait, which $\sqrt 2$ for each instance of $\sqrt 2$? Consider golden ratio $\varphi$. Which $\sqrt5$? – peterwhy Sep 08 '13 at 18:50
  • 1
    @peterwhy, I sincerely do not think that these ambiguities can be reliably resolved by notational or linguistic convention. That is, one should expect to take some little trouble to clarify to one's audience what one means. Years ago, I thought there could be a universal "best" usage, but I no longer think so, despite (or because of!?!) my efforts to determine what "best usage" might be. If nothing else, it's easier to add information in any given context than to establish a universal convention. More reliable, too, which is the real point, to my mind. – paul garrett Sep 08 '13 at 18:54
  • The voice of reason. Thank you, thank you, thank you. – Daniel Fischer Sep 08 '13 at 19:29
  • @DanielFischer, :) ... – paul garrett Sep 08 '13 at 19:37
2

A little late in the game but i think i can make a valid point.

The square root is a function, since people did go through all the trouble of finding a derivative for it, and the definition of a function is that is a unique mapping between the an element of the set of inputs and an element of the set of outputs.

Now if the idea proposed by the textbooks would be that square root is not a function because one element of the input say 4 actually maps itself to two elements of the output set which is false. But the square of -2 is actually 4 so we have a small problem.

The answer comes from a more general definition of the square root which is $\sqrt{x^2}$ is the absolute value of x, which works just great. Now the absolute value of -2 is 2 so there is no longer any need for debate because $\sqrt{(-2)^2}=|-2| = 2$ which solves any misconception

ciprianr
  • 121
0

The square root of $x$ is a number which when squared gives $x$. For $16$ there are two such numbers, so there are two square roots of $16$. For $0$, there is one and for any negative number there is none.

Now, simply call the non-negative square root of a number, the principal square root. There is only one such number for all non-negative numbers and thus, the principal square root of a number is unambiguous (unless the number is negative, in which case it is undefined).

We denote the principal square root of $x\geq0$ as $\sqrt{x}$. The other square root is then $-\sqrt x$. So, we can say that the square roots of $2$ are $\sqrt{2}$ and $-\sqrt{2}$ and of $16$ are $\sqrt{16}(=4)$ and $-\sqrt{16}=(-4)$.

I think the reason some people may have confusion with this is that they don't understand/know that $\sqrt{x}$ is used to denote the non-negative square root and nothing else.

Alraxite
  • 5,647
  • 2
    One might object that this viewpoint seems to permanently disallow complex numbers...? – paul garrett Sep 08 '13 at 18:13
  • @paulgarrett I'm exclusively talking about the real numbers here. We may of course, extend these definitions to the complex plane. – Alraxite Sep 08 '13 at 18:31
-1

In my opinion it's lack of logical thinking that generates such confusion.

Also mixing natural language with mathematics gives problems:

$Q_1$: What are the squares roots of $4$?
$A_1$: They are $2$ and $-2$.

Proper interpretation of $Q_1$: find the extension of the set $\{x\in \Bbb R\colon x^2=4\}$.

$Q_2$: What is the square root of $4$, i.e., $\sqrt 4$?
$A_2$: The square root of $4$ is $2$.

Proper interpretation of $Q_2$, once it has been established the truth of statement $(\forall x\in \Bbb R^+)(\exists !y\in \Bbb R^+)(y^2=x)$, what is the only positive number such that its square is $4$?

I find that most students don't have enough logic in them to make the above translations and since mathematicians insist on abusing natural language, confusion is bound to rise.

Once students known that $(\forall x\in \Bbb R^+)(\exists !y\in \Bbb R^+)(y^2=x)$ and that given a positive real number $x$, $\sqrt x=y$, there is absolutely no danger of mistakes. It comes directly from the definition.

Git Gud
  • 31,356
  • I also noticed that, even in Wolfram MathWorld, there is a natural language saying of "the" square root of something, meaning the non-negative square root. – peterwhy Sep 08 '13 at 17:14
  • Yes, 'the' and 'a' also help making the correct interpretation as 'the' implies existence and uniqueness while 'a' doesn't imply uniquess and it might even suggest non-uniqueness. – Git Gud Sep 08 '13 at 17:16
  • 6
    In my opinion you're making this way too complicated. –  Sep 08 '13 at 17:18
  • @RahulNarain I don't think so. An inquisitive and skeptic student will force you to get down to this level of detail, unless you have an alternative rigorous way of achieving the same thing. – Git Gud Sep 08 '13 at 17:20
  • 3
    If someone asks me the square root of $4$, I would say that they are $2$ and $-2$, regardless of whether the phrasing of the question is implying that there are single or multiple roots. – Alraxite Sep 08 '13 at 17:50
  • @Alraxite Then you're making a mistake somewhere. Wether it is mathematical or linguistic, might be discussable, but you're making at least one mistake. 'The' is a definite article. It's semantically senseless to say the square root of $4$ is $2$ and $-2$. – Git Gud Sep 08 '13 at 17:52
  • Then I would also tell the questioner to not use the definite article 'the' (unless they are talking about $0$) as there are multiple solutions (and hence their question is vague). I'm defining the square root of $a$ as the number $y$ which when squared gives $a$ (this is how Wikipedia defines it too). If someone asked me, "What is the natural number which when doubled is less than 6?", I would tell them there are actually multiple solutions as opposed to what their question may imply and then I will list them. – Alraxite Sep 08 '13 at 18:05
  • @Alraxite Wikipedia also says 'a square root' which is something entirely different from 'the square root' and the latter was the term you chose. And for the interaction questioner/answerer, my personal choice would be to say that the question doesn't make sense. I can accept yours, but the problem here is when there is that there os an authorative figure such as a teacher saying wrong things. – Git Gud Sep 08 '13 at 18:10
  • Why would you tell them that the answer is $2$ rather than saying the same thing you would say to my last question: "the question doesn't make sense". Why would you choose the positive root? It's comparable to saying the answer to my last question (about the number whose double is less than $6$) is $2$ because it is the largest of the solutions. But maybe you're implying there is a convention that when the definite article 'the' is used with the square root, the person is talking about the principal square root. That essentially depends on which conventions we like to follow in language. – Alraxite Sep 08 '13 at 18:26
  • @Alraxite Yes, of course, convention. Nothing more. As for the question not making sense, the example questions I used in my answer are meaningful, they make sense. – Git Gud Sep 08 '13 at 18:28