## sqrt(16) – how many answers?

[21 Sep 2005]

I had a heated debate with some colleagues yesterday when I claimed that √16 = 4. They were strongly advocating that there are 2 answers, ±4.

I pointed out that there is a difference between this question:

Solve for x: x2 = 16

and this question:

Evaluate x: x = √16.

The first has 2 solutions, the second has one answer.

1. Casey says:

http://mathforum.org/library/drmath/view/52613.html

2. Murray says:

Thanks Casey, but your link points to an article on the square root of a negative number. That is not what we are talking about here.

Anyway, at the top it says “-4 [is] the negative square root of 16.”

Yep, I agree:
√16 = 4
-√16 = -4

When you do √16 on a calculator, what does it say?

3. w says:

Ya i agree

plotting y = sqrt{x} is not the same as plotting y = +/- sqrt{x}

While the first equation is a 1 to 1 function with its inverse y=x^2 for x>0, the second is not a 1 to 1 function.

Using graphical argument, hence sqrt{16} has only one root and not two.

The “two root” arises when we have

(+/-) sqrt{16} = +/- 4

4. Casey says:

(Sorry my link was the wrong one… I’ll give that to you.)

Are you saying that square and root are not opposite opperations?

——

This is all basic algebra. Square root is nothing but raising to the (1/2) power. If x^2 = +-4 … the opposite opperation 16^(1/2) must equal +-4.

Here is tons of background information on where the theory comes from (link no longer available)

When you can root both sides, you always include the +- … otherwise they wouldnt be equal.

5. Murray says:

No, I fully agree that square and square root are opposite operations.

What I am talking about is the fact that the notation √16 means the positive value 4 only.

There is a difference between the function defined by y = √x and the relation y = ±√x. The first has one y-value for each x-value, the second has two (one positive and one negative).

Consider the formulas for inverse trigonometric differentiation. If the √ means 2 values, then the derivative of arcsin x will be ±(1/√(1 – x^2)) and the derivative of arccos x will be ±(-1/√(1 – x^2)), which is the same result. This does not make sense.

Summary: √ notation gives us one positive value only.

BTW, your sentence “x^2 = +-4 means the opposite operation 16^(1/2) must equal +-4.” has errors. I think you meant

x^2 = 16 means the opposite operation 16^(1/2) must equal +-4.

What I wrote at the outset is that yes, x^2 = 16 has 2 solutions, but √16 has only value.

You may also wish to see the later post on this topic.

6. Alan says:

I agree that your original post is consistent with the generally accepted usage in most of the world today, and that for non-negative a, √a is defined as the non-negative solution of x^2 = a.

But I think the first sentence in your last comment is misleading (perhaps because of your adoption, without a precise definition, of Casey’s phrase “opposite operations”). I think most people think of an “operation” as having a well defined result – ie in mathematical terms being a function.

In fact the square root is not the inverse function of the square but rather of the slightly different function defined by restricting the square to just non-negative arguments, and as you point out, the inverse relation of y=x^2 is y = ±√x and not just y =√x

If it was true that “square and square root are opposite operations” then the square root of the square would always give back the number you started with, and this is not in fact the case. (In fact the square root of the square of x is the absolute value of x which only agrees with x if x is non-negative.)

7. Paul says:

Hi I am new at this but I agree in that the square and square root in this circumstance you are correct. As we all know when you multiply two negative numbers it then becomes a positive or if you myltiply two positive numbers it stays a positive number but if you square root a positive number it usually stays as a positive number. If I am wrong I will accept it but please tell me why as I said I am still learning this type of maths.

Cheers,

Paul.

8. Murray says:

Hi Paul. Yes, your summary is quite correct. Good luck with your studies!

9. Qlumbo says:

On a calculator, it comes out as positive 4. The calculator that comes with windows, that is. I am not sure what happens if you are programming and you use the sqrt() function, and I am not sure what happens with maybe other calculators.

My opinion on it, anyways, is simple:
This is similar to the way you can graph 2 lines, representing 2 equations; With a normal solvable equation, there is a single intersection between the lines, proving to be the answer. With parallel lines, there is no answer because there is no intersection. With 2 lines on top of each other exactly, there is an infinite amount of answers because the line is constantly intersecting.

If something can have 1 answer, no answers, or an infinite amount of answers, why can it not have 2 answers? There is no rule in math that is violated by having 2 answers. The answer is plain and simple: the square root of 16 is both 4 and -4.

And Paul, maybe the fact that when squaring something the solution will always be positive has something to do with the constant positivity of absolute value. I have no idea.

10. Murray says:

I am not sure what happens if you are programming and you use the sqrt() function…

You get one positive answer, as it should be.

I am not sure what happens with maybe other calculators…

All calculators give you one positive answer, as they should.

There’s another example where it is assumed (correctly) that the square root symbol, √, gives a positive answer only.

The quadratic equation, which everyone manages to learn sometime in their math career, says:

The solution for ax2 + bx + c = 0 is

x = [-b ± √(b2 - 4ac)]/2a

If the √ sign meant plus and minus, why then why would we need the ± sign in the formula?

√16 is 4, only.

11. johnson ojofu says:

hi
as for me i strongly believe that the answer is four, because when u take the square to the other side, it then becomes square sixteen. therefore the square root of sixteen is four

12. Qlumbo says:

Hmm…that’s weird. Getting a solid definition of square root would entirely answer the question though.

We should also remember that square root is not some super-special operation…you’re just taking the exponent of 0.5.

I’m starting to wonder about the cube root of 8. Cube rooting isn’t special either and belongs in the same group with any other exponent, but just since it isn’t divisible by 2 we come out with one answer without argument.

Definition of square root:
A number that when multiplied by itself equals a given
number.
-4, when multiplied by itself gives 16 and so does 4. And once again, there is nothing wrong with having 2 answers.
Is there anything against this?

13. Murray says:

In my post, I did not use the words “square root”. I used the radical symbol, √.

I agree with the Wikipedia author who wrote:

Every positive number x has two square roots. One of them is √x, which is positive, and the other −√x, which is negative.

This is a notation issue, as well as a semantic one.

14. Qlumbo says:

Ah. Thanks for the answer. Good point.

15. favor says:

yes, that’s true.
the first equation has two solutions ±√16 which are +4 and -4, but in the second one, the solution is already limited to just one of the two which is +√16 = +4. Period

16. John says:

Murray is correct. √16 = 4 , not ±4. If the latter were true, then the quadratic formula would not need a ± symbol in it.

Furthermore, the definition of the absolute value function is

|x|=√(x^2)

This would not work were √(x^2)=±x

17. Murray says:

Thanks John for your input – and great to hear from you!

18. Nabin K. Neupane says:

Thanks for the answer. I agree with u

19. Khurram says:

there is a difference between x^2 = 16 and x = √16
as when we add a root in a question by ourselves than and than only will the answer be ± .If it is given already in the question that x equals to the root of something than the answer will only be the positive one

20. Tammy says:

http://www.mathsisfun.com/algebra/square-root.html

If you scroll about halfway down to the section labeled Principal Square Root, there is an easy-to-understand (to me, anyway) explanation about the fact that a number has both a negative and a positive square root, but the positive one is the Principal Square Root. There is even a graph of an equation showing why this must be the case.

Thoughts?

21. Murray says:

Hi Tammy. Yes, most of the time the square root sign is taken to mean the “principal” square root (that is, the positive case only).

And that’s my point.

In a follow-up post I give the Oxford Concise Dictionary of Mathematics definition of square root.

22. Suresh says:

I think that defination has to be reviewed.
lf we look at the square root as an mathematic operation such as +,-,X & divisions. All operations behaves regardless of the choice of a sign.
Similar concept applies to square root if we say square root 4 is two. Meaning it is not positive or negative either, it is a natural number.

But if we say its positive 2 than it is partial answer that we are seeking from the operation. It contridicts with multiplication rule. Product of positive is positive and product of negatives are positive. Thus in mathematic we have contridicting operations. We have predetermine the sign of choice for square root. Thus the square root symbol its self does not indicate as a complete mathematical operation which is aline with +/-/X etc operations.
All operation has to be logically explainable or its not maths.
Eventhough the defination says square root has one answer but i strongly disagree. I am comfortable with +/- 2.

23. Dave Marain says:

The √ symbol is used for the square root ‘function’ which is why there can be only one value for each nonneg real. It helps some students to associate √ with a FUNCTION key on their calculators. Yes, it’s a MathNotations issue, hence the name of my blog!

24. huhu says:

By definition the square root of a number A is the POSITIVE number B such as B^2 = A. It’s simple high school maths.

p.s. That’s why the solution to x^2 = c is x = +- sqrt(c) where x is a real number.

25. Kenneth says:

The answer is really 2 if you are dealing with square roots.

x^2 = sqrt 16 is really : x^2 = positive and negative sqrt of 16 . It is like: -4 X -4 is 16 and 4 X 4 is also 16. So -4 and +4 are viable square roots of +16.

26. Murray says:

@Kenneth:You said “x^2 = sqrt 16 is really : x^2 = positive and negative sqrt of 16 “. No, it should be:

x^2 = sqrt 16 is really : x = positive and negative sqrt of 16

So sqrt(16) has one, positive value, otherwise you don’t need to say “positive and negative”. Like I said.

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