# sqrt(16) - how many answers?

By Murray Bourne, 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.

See the 39 Comments below.

### 39 Comments on “sqrt(16) - how many answers?”

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 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.

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:

Thanks for your opinion, Qlumbo.

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.

According to the Oxford Concise Dictionary of Mathematics,

The notation √a is used to denote quite specifically the non-negative root of a

Yet another example of how math notation is our enemy...

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.

27. Suresh says:

We can't define squareroot with what we get in calculators. Calculator is unable to do parallel calculation if there's 2 answers.

28. Kara says:

OK I get that 4 is the square root of 16, would -4 also be the square root of 16 since a negative times a negative is a positive, (for some odd ball reason, that's math for ya) but a negative times a positive you take the sign of the higher number.

29. a german dude says:

Sqrt(16) has two solutions, but our calculators ignore the negative ones.

http://www.wolframalpha.com/input/?i=sqrt%2816%29

As you can see in wolframalpha the 2nd root of sqrt(16) is -4, but i think this is just relevant if you are calculating with complex numbers

30. Murray says:

No, the Wolfram|Alpha solution says there is one square root of 16, 4.

There are 2 "roots" (for the equation x2 = 16), one principle root (4) and the other real root is −4.

These are not imaginary solutions at all.

31. Jim Siple says:

These quadratic equations like x^2 = 16 definitely have 2 solutions BUT there's only 1 results for (4)^2. There's only 1 result for (-4)^2. In the same manner, there's only 1 result for the square root of 16 ... 4! Equations can have multiple solutions. That's okay. But these single operational questions have just 1 answer.

This question of inverse operations extended to trig functions is even more fascinating. The sin(pi/6) is ONLY 1/2. The sin(5pi/6) is ONLY 1/2. The sin(13pi/6) is ONLY 1/2. But the equation sin(x) = 1/2 has, not 1, not 2, not 10, but an infinite number of solutions (any angle coterminal to pi/6 and 5pi/6). But, for the inverse operation arcsin(1/2), there's only 1 answer ... and that is pi/6. NO ... NOT even 5pi/6! All of this is to say, equations and single operations are different.

32. Steve says:

I will vote that a square root symbol indicates a number without a sign. A number of absolute value is indicated. Another argument that has not been discussed is the notation for "imaginary numbers". The square root of a negative number is the signless square root times "the square root of negative one". There is no possible way to say, "the square root of negative one can be either one or negative one". It is only the value for "the square root of negative one" whatever that is imagined to be.

33. cuthbert twillie says:

With regard to the square root of 16, it depends on whether you are using practical or theoretical mathematics. There are no "negative" goods; they are a theoretical confection only. The negative square root of a number of things in the real world (lawn mowers, cabbages) would be meaningless, because there is no negative number of things in the world. I realize that debt is in a sense negative equity, when you owe money, they don't issue you negative coinage or currency. Hence, practically speaking, the square root of 16 objects of anything would be four objects, but only in a theoretical sense is there a negative square root of 16. I'm not arguing that it is unimportant in mathematics, only that it is confined to the world of the abstract (with the possible exception of the negative spin and charge of atomic particles, quantum states, etc.)

34. Asey says:

the sqrt of 16 is 4, what is the term for that?

35. Randy says:

Look at the graph of the equation y=x^2. For all real values of x. It results in a hyperbola centered around the y-axis. It contains the points (0,0), (1,1), (2,4), (3,9) and so on. It also includes the points (-1,1), (-2,4), (-3,9) and so on.
Then consider the graph of the equation y=16. That would be a straight line parallel with the x-axis, 16 units above it. This line would intersect the hyperbola in two places. At (4,16) and (-4,16). Clearly, when you take the square root of a number, there are always two answers; one positive and one negative. Unless you indicate somehow that you only want positive answers.
You could do that by stating so in the problem or by placing the square root symbol within absolute value brackets.
In the quadratic equation used to solve quadratic equations, there are both a plus sign and a negative sign simply to remind people to use both roots. The quadratic equation would yield the same answers with or with the double signs.
Without a doubt, ✓16 equals both +4 and -4.

36. zac says:

@Randy: Thanks for your perspective.

Firstly, a quadratic equation is a parabola, not a hyperbola.

I feel you are perhaps missing the fundamental difference between a function and a relation. A function can only have one y-value for each x-value. Rotate your parabola 90° and you will have a relation, and when x = 16, we will only get one value of y.

It's actually a convention that there is one square root. It's a bit like the BODMAS convention (do inner brackets, then divisions and multiplications, then additions and subtractions, in that order). Without such a convention, we'll all end up with different answer for an expression like

3 + 2 × 5 − 2

37. Sanchit says:

Yuh so basically....the principal solution of √x² is the non-negative value of x .
But otherwise there are 2 solutions for √x² ,i.e, +x and -x

38. Chandan says:

Okay. Let's say √16 equals both + and - 4. Then 4 = √16 = -4 would end up -4 = 4. Which mathematics won't permit.
Tho elaborate it mathematically...how do we get to the principal value 4.
√16 = √(4×4) = √4 × √4 = (√4)^2 = 4
But when we do the same for -4...
√16 = √(-4×-4) = √-4 × √-4 = (√-4)^2= -4
Here we got the following step wrong...i.e.
√(a×b) = √a × √b
This step is valid only if a,b > or = 0.
Hence -4 is not the solution.

If we were able to do the above step then...
-1 = i×i = √-1 × √-1 = √(-1×-1) = √1 = 1
Wait what ?
- 1 = 1...Hence...√16 = 4.

39. wutek says:

If √16 is equal to 4 and -4 then you can't write √16 = 4 nor √16 = -4. (otherwise you would get a contradiction: 4 = √16 = -4, as shown in the previous answer). The only way you could write it down would be like this:
√16 = {-4, 4}, where {-4, 4} is a set of two numbers.
Doing any calculations involving √ would be more complicated (not impossible). You would have to always remember that the result of √ is not a number (like for every other function you know from school) but a set consisting of two numbers. In fact √ wouldn't be a function, because functions by definition always return exactly one value.

To sum it up, the fact that √16 = 4 is just a convention, but it's a very good convention.

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