y^2=x vs y=sqrt(x). Is they is or is they ain't the same thing.

Apparently, they ain’t.

Not homework – just spending some time puttering around with math and trying to work this out in my old, gnarled, withered brain.

The graph of y=sqrt(x) is a half-parabola lying on its side – just the positive numbers.

The graph of y^2 = x is a full parabola lying on its side – both the positive square root and the negative square root

But can’t you just rewrite y^2=x as y=sqrt(x)? So why does one result in both the positive and negative square roots whereas the other is just the positive square root? Is it just a convention that working from y^2=x, you have to include both the positive and negative root, whereas y=sqrt(x) by convention is just the positive square root in the absence of plus/minus sign? Or is there a deeper logic to the situation?

No and yes. Yes, the radical sign denotes the positive square root by convention, so you need to explicitly add a negative sign if you want to talk about the negative square root.

But no, it’s not convention that y^2=x has two solutions. It’s just a fact.

“sqrt(x)”, unqualified, means the positive square root of x. That means if you know that y = sqrt(x), and you know what x is, you know what y is for certain. For example, if x = 4, you know for certain that y = 2.

y^2 = x is an equation with two solutions y = sqrt(x), and y = -sqrt(x). That means that if you know that y^2 = x, and you know what x is, you can only narrow down what y is to two possibilities. For example if x = 4, you only know that either y = 2 or y = -2.

I just want to say that I absolutely love that song, and that specific line in it! I use it all the time, too!

Thank you for the answers. It seems oddly asymmetrical but there it is!

And John Mace: sing it brother! :wink:

TonySinclair and leahcim gave good, simple answers. I’ll just add that this isn’t unique to the equation x = y[sup]2[/sup]; there are plenty of examples of relations between x and y where there’s more than one possible value of y for a given x, so you can’t really solve for y = a function of x. Algebra students learn about the “Vertical Line Test”: if any vertical line can touch the graph of an equation at more than one point, you have the kind of situation you have here: you have more than one y for the same x, so you can’t write y as a function of x.

It depends on whether “sqrt(x)” is defined to mean either square root, or to mean the principal square root.

According to Wikipedia,

If you just say “sqrt(x)” without any qualification, it’s ambiguous which you mean.

No, it’s pretty unambiguously the principal one.

Does this convention never create problems when doing math? Here’s a scenario:

Johnny has the problem:


And he decides say therefore:


And answers the question +/- 5. Is he wrong? Or how in the step from y=sqrt(x) to y^2=x does one denote that one’s only interested in the positive root?

And are there any complex math problems where “phrasing” the question one way, converting it to the other, and then converting back creates logical absurdities?

When I was in school, we learned to use the absolute value markers when solving equations like this.

So if you start with y^2 = x, you would actually want to convert it to |y| = sqrt(x) to avoid losing half of your possible outcomes. Because |-5| = 5, you retain the same graph, with a curve both above and below the x axis

Nobody else has mentioned this, so maybe it was just my teacher making this point.

This is true, but you’re no longer working with functions if you do this. Most of why we assume y=sqrt(x) takes the positive square root is because taking both results in a non-function. |y|=sqrt(x) (or even |y|=x) results in the same “problem”.

Yes. Squaring both sides of an equation can introduce extraneous solutions that don’t apply to the original equation, as your example indicates. y=5 has only one solution, but (y)[sup]2[/sup] = (5)[sup]2[/sup] has two, one of which didn’t apply to the first equation, because -5 and 5 are different numbers that “become” equal when you square them.

Here’s a more sophisticated example of where squaring both sides of an equation can introduce extraneous solutions. If you want more examples, this PDF has some.

Yeah. If you say the words “[a] square root of x,” it’s ambiguous, but if you use the notation “sqrt(x)” or the radical sign, it means, by definition, the prinicipal square root.

Yes, Johnny is wrong. To use an even more blatant example, suppose Johnny has the problem:

y = sqrt(25)

And he decides to multiply both sides of the equation by zero to get:

0*y = 0

Johnny then turns around and says “this equation is true for any y, so y is any real number”.

This is something subtle that is often missed in early algebra courses. When you’re solving an equation by repeatedly applying functions to both sides, you’re reasoning about the variables involved in the equation. Abstractly you are reasoning (perfectly validly):

if a = b,
then f(a) = f(b)

in your example, a = y, b = sqrt(25), and f = u -> u^2 (i.e. f is the function that squares its argument).

What is not necessarily valid, however, is inferring in the other direction. I.e.

if f(a) = f(b),
then a = b

which is what you are doing when you say that y^2 = 25 implies y = 5.

Of course, to muddy the waters, you can make the second inference if f is invertible. (In fact, invertible is defined to mean “functions for which that inference can be made”).

Unfortunately, since most functions used in elementary algebra for the purpose of solving equations (e.g u -> u - b or u -> a * u) just happen to be invertible, teachers tend to gloss over this requirement until bad habits are already formed.

Wasn’t that song by Chicago? No, I remember–that was IF Y = 25 OR 624 THEN GOSUB something.

When I first took Algebra I in 9th grade in 1965, this sort of thing was definitely drilled into us, but I and I think most of the class were perpetually confused by it at the time.

First of all, despite all the contradictory remarks already posted: Yes, there IS an element of pure arbitrary convention here: The notation sqrt(x) or √ is somewhat arbitrarily defined to refer only to the positive square root.

Thus, the statement: y = √25 can ONLY mean the positive root, simply because we’ve all agreed that the symbol √ means just that.

Yet the fact remains that, given the statement: y[sup]2[/sup] = 25, you can plug in y = 5 or y = -5 and it works. In this formulation, the artificial restriction is absent.

The same applies to the absolute value function, which was also a perennial point of confusion.

The convention gives us a way to specify exactly which square root(s) we are interested in using. If we just want the talk about the positive, or principal, root, the √x notation allows us to do that.

If we wanted to talk about the negative root, well, we have the notation -√x to handle that.

AND, if we actually want to talk about BOTH roots (which we sometimes do), we note with much joy and glee that we have the notation ±√x for that! So the notational conventions give us the flexibility to say exactly what we mean and mean exactly what we say!

At the level of beginning algebra classes, this is perhaps most famously seen in the quadratic formula which gives us both solutions to the general quadratic equation ax[sup]2[/sup] + bx + c = 0 as:
y = ( -b ± √(b[sup]2[/sup] - 4ac) ) / 2a

You’ll see a lot of the same thing, only even worse, with trigonometric functions and their “inverses”.

I think he’s talking about the song from Porgy and Bess?

OP, is you is or is you ain’t talking about the song from Porgy and Bess?


When things like this come up, I tend to say:

If you’re confused, that means you’re probably actually paying attention.

If you’re NOT confused, that means you probably aren’t paying attention.

The important thing to be clear on is that the arbitrariness is necessary because it would be impossible to have a function from R to R return both. It is the nature of functions that for each input in their domain they return exactly one output. Even though for any x, there are two possible values of y such that y^2 = x, the sqrt function must pick and return only one of them in order to be a valid function from R to R.

The human conceit that positive numbers are preferable to negative ones makes the choice easy, but basic math demands that a choice must be made.

Nope!. I first came across it from the Joe Jackson cover.