Can it be defended that $\sqrt 4$ is both $2$ and $-2$ (and likewise for general square roots)?

Question

Over the past one or two years, at least two different teachers have told my children that $\sqrt 4$ is $2$ or $-2.$ I don't think this is useful, but if you want to define $\sqrt x$ as the set of solutions of $y^2 = x$, I guess you could. Of course you get into all kinds of confusion with very common constructions and notation, for example the graph of $x\mapsto \sqrt x$ (should we write $\max\sqrt x$?), Pythagoras' theorem or the root formula for quadratic equations, but let's say we accept that, and the common notation is just sloppy. In fact, the teacher is consistent, and says that the root formula should be written

$$x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$$

rather than

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$

Now the teacher went one step further, saying that

$$x + \sqrt{7 - 3x} = 1$$

has two solutions: $x = -3$ (obviously), but also $x = 2$. This is found by writing

$$\sqrt{7 - 3x} = 1 - x$$

and squaring. Since it is assumed $\sqrt{7 - 3\cdot 3} = \pm 1$, we select the $-1$, and in this way this also is a solution to the original equation.

This seems obviously wrong to me, but I'm not sure how obvious that really is. If we really define the square root as a set of values (of 0, 1 or 2 elements) rather than as a (single-valued) function on non-negative real numbers, we could say that the solutions of an equation involving square roots are those values of $x$ so that from each of these sets we can select a value satisfying the equation. That would have strange consequences like

$$(1 + \sqrt x)^2 = -3$$

having the solution $x = 4$. Also you cannot factorize in terms of square roots anymore: $a - b$ is a single value, but $(\sqrt a + \sqrt b)(\sqrt a - \sqrt b)$ can have 8 different values.

Of course these last two are less defensible than the original example, for the teacher's example at least for each of the solutions a branch of the square root can be defined for which it is a solution, but they are not taught that you have to fix whether you want $\sqrt 4$ to be $2$ or $-2,$ but rather that it is both.

After this long introduction, my question: could it be reasonably defended that $x = 2$ is a valid solution of $x + \sqrt{7 - 3x} = 1$ alongside $x = -3$? In other words, if at an exam a student only produces $x = -3$, could they protest if full points are not awarded, even though the teacher taught them that both are solutions? After all, even though mathematical truth is non-negotiable, definitions and conventions are, and to a certain extent the teacher is free to fix these.

Answer 1

This comes up over and over again on the MSE site: see for example here and here. However, the question you ask is slightly different than the ones asked at those sites; rather than ask "Is this correct?", you want to know "Is there a way to define things consistently so that this is correct?" That's a more subtle question.

The real problem, in my opinion, with the convention introduced by this teacher is that it is incompatible with the way we normally use the equals sign. If the teacher is comfortable with saying both $5 + \sqrt{4} = 7$ and $5 + \sqrt{4} = 3$, does it not then follow that $7 = 3$? Or are we now going to decide that $=$ is not a transitive relation?

It is possible to use this convention in a consistent way. Let $\sqrt{a}$ to denote the set of solutions to the equation $x^2 = a$. Given a set of real numbers $S$ and a real number $r$, let $r + S$ denote the set $\{ r + s \vert s \in S \}$. Then we can write: $$\sqrt{4} = \{-2, 2 \}$$ $$5 + \sqrt{4} = \{3, 7\}$$ Then instead of using the equals sign, we can use the symbol $\ni$, as follows: $$5 + \sqrt 4 \ni 3$$ $$5 + \sqrt 4 \ni 7$$

Because $\ni$ is not a transitive relation, we don't end up with the same problems we would get from using the $=$ sign.

But all of this creates an unnecessarily complicated scheme. It makes much more sense to say, as is universally done at the secondary level, that $x^2 = 4$ does indeed have two square roots, positive and one negative, but that $\sqrt{4}$ denotes the positive square root. The standard convention is standard precisely because it makes things simpler, and other approaches require more complicated machinery.

Answer 2

Sure, you can define $\sqrt{4}$ to be $\pm 2$ if you like. The math you teach this way will be internally consistent.

You can also teach students that the symbol "@" means multiplication, and you can teach students to write fractions using ordered pairs bracketed by semicolons. $;4,5; @ ;1,2; = ;2,5;$ "can be defended" in the same way as $\sqrt{4} = \pm 2$; this nonstandard notation is internally consistent.

However, the damage in both cases is the same; the students are not being taught the standard conventions and will end up harmed in the end when they are unable to transfer their knowledge to other courses.

Answer 3

There has been a change in the conventions people use in math, and I think the change probably resulted from the introduction of cheap electronic calculators. In the before times, it was common to talk about "multi-valued functions," and to use phrases like "a square root of 4" or "an arctangent of 1." When lots of people started to have calculators with square root keys, then inverse trig functions, and so on, it became necessary for the manufacturers to pick branches of the functions to use. In many cases there is no natural choice of the branch cut, but they just picked one. People who wrote math textbooks then had to adjust to the fact that to a student, the inverse tangent function was nothing more or less than that button on their calculator. If the teacher says that an inverse tangent of 1 is 225 degrees, and the calculator says it's 45, then the teacher must be wrong. Get him fired! So it became necessary to start putting standard branch cuts in all the math textbooks, as a side-effect of which students were forced to mindlessly memorize those choices.

None of this does the world any good. For instance, when a student is adding two vectors in the cartesian plane, gets (-1,-1), and is then trying to find the angle of the vector, they have been conditioned to think that "the" answer has to be 45 degrees. (I used to suggest to my students that they get some nail polish and put a dab on their arctangent key as a reminder of this pitfall. I don't think any of them ever did.)

So it's not quite right to say that the OP's kid's teacher is just violating well-established conventions. What has happened is that both this teacher and the current high school textbooks are oversimplifying the well-established conventions, which are still understood by working mathematicians. The conventions are actually kind of complicated and ambiguous, but that's OK. That reflects the fact that human thought is complicated and ambiguous. We have notations like $\pm$ and $\mp$. Sometimes we write $x=+2$ to emphasize the sign. These conventions make sense to humans, but they don't map nicely onto some sort of literalistic idiot-savant calculator mindset.

The OP complains that havoc will ensue if a student is asked to graph $\sqrt{x}$. Well, not really. Actually what will happen is that the student will be encouraged to think about graphing it as a nice, complete parabola, rather than a crippled, ugly half-parabola.

I would agree that what this teacher is doing is a bit out of step with the rest of the world, but what the rest of the K-12 world is doing is also a bit out of step with what mathematicians do. And in any case the really important thing is to get kids to think. The ambiguity in sign of a square root isn't something that can be handled by habit. It has to be handled by thinking about what the ambiguity means in whatever real-world application we're talking about.

Answer 4

The defensibility/correctness depends on context. And, in the first place, the idea that $\sqrt{t}$ for positive real $t$ means just the positive real square root is just a convention, not any sort of mathematical assertion. And the possible attraction of having a single-valued square root function cannot be sustained when considering square roots of complex numbers.

So, while the current convention in K-12 and lower division undergrad math seems to be that $\sqrt{t}$ is the positive square root, no such convention persists beyond that, except in context, where it's not a mere convention that that's what $\sqrt{t}$ means, but is a tangible answer to some question.

So, for current K-12, no, it is not a good idea to break with K-12 convention.

In other situations, pretending to insist that $\sqrt{t}$ is somehow magically only the positive square root can lead to pointless ambiguities and misunderstandings.

Answer 5

Here is a defense of the notation for students trying to problem-solve. Let's say I have the equation, $x^2 = 4$. This has two solutions, $-2$ and $+2$. However, what operation should I do to result in these numbers? If you say that $\sqrt{x^2}$ only yields the principle square root, then which operation yields both solutions? I can't take the square root of both sides, because, according to the understanding of the square root presented, it only yields one value.

It would make more sense for $\sqrt{x^2}$ to be only the positive square root if there were another operator which yields all of them. Then, I could distinguish between them. But, when $\sqrt{x^2}$ just means the positive side, this is very confusing for students because it leaves them without a concrete operation to apply to an equation to actually get both answers.

You could tell them that there is an exception here. However, I've always found when teaching that students get really hung up on exceptions. They have enough trouble just following the rules. My point is that, if you teach that $\sqrt{x^2}$ only means a positive value, don't be surprised if they later tell you that $x^2 = 4$ only has one solution. If you teach them that $x^2 = 4$ has two solutions, and that square rooting is the inverse operation of squaring, then don't be surprised if they give two answers for square roots.

Answer 6

If you want to defend $\sqrt{4}$ being both $-2$ and $+2$, you'll need to accept that $\sqrt{1} + \sqrt{9} + \sqrt{25}$ can have all of the following values:

\begin{cases} \\ -9 = (-1) + (-3) + (-5) \\ -7 = (+1) + (-3) + (-5) \\ -3 = (-1) + (+3) + (-5) \\ -1 = (+1) + (+3) + (-5) \\ +1 = (-1) + (-3) + (+5) \\ +3 = (+1) + (-3) + (+5) \\ +7 = (-1) + (+3) + (+5) \\ +9 = (+1) + (+3) + (+5) \end{cases}

In other words, not a good idea :-)

Answer 7

$$ \frac{-b\pm\sqrt{b^2-4ac}}{2a} = \frac{2c}{-b\mp\sqrt{b^2-4ac}} $$ How would those two teachers write something like the above efficiently?

Answer 8

$x^2-4=0$

Is a quadratic equation that gives the 2 answers, a positive and negative 2.

The simple question “what is the square root of this number?” is answered with just the positive result. In High School, we see this mistake all the time, left over from middle school teachers who are missing the distinction between the 2 circumstances.

Answer 9

If you apply the same method ( I don't go into detailed answer assuming you know how to find square roots of a complex number) of finding square root of a complex number 5 + 12i you can not avoid getting two values for √4 . Since real numbers are part of complex numbers you need to say square roots of 4 are ± 2 and principal square root of 4 is 2.

Answer 10

Can it be defended that $\sqrt 4$ is both $2$ and $-2$ (and likewise for general square roots)?

I will try to make plausible arguments defending this, although my arguments have no scientific backing, and are also obviously somewhat in jest, although there is nothing completely obviously outlandish about them.

By doing this, it is clear that it will:

1. Generally cause confusion in students learning maths.
2. Waste unnecessary amounts of students' time learning, at best, superfluous notation.

$1$ might benefit weaker students in that they are more likely to give up entirely on maths more quickly; thus, they can spend more time focusing on their actual career paths.

$2$ might benefit stronger maths students because they are likely to take the task of learning the new notation seriously, so they are less likely to waste their time on their iPhones and PS5's, the result being a more focused and hard-working maths elite.