General misconception about $\sqrt x$

Question

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?

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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?

Answer 1

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.

Answer 2

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 $4$?

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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.

Answer 3

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.

Answer 4

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.