sqrt(16) - how many answers?
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.
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Casey said,
September 24, 2005 at 6:03 am
http://mathforum.org/library/drmath/view/52613.html
zac said,
September 24, 2005 at 10:19 am
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?
w said,
September 27, 2005 at 10:10 pm
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
Casey said,
October 5, 2005 at 4:06 am
(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
When you can root both sides, you always include the +- … otherwise they wouldnt be equal.
zac said,
October 5, 2005 at 1:02 pm
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
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.
Alan said,
October 9, 2005 at 11:21 am
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.)
squareCircleZ » Happy anniversary said,
November 2, 2006 at 12:40 pm
[...] The posts that got the most reaction were on √16 - how many answers? (with a follow-up here), my critical post on Heymath! and my Review on Game of School. [...]
Paul said,
November 8, 2007 at 12:27 am
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.
zac said,
November 8, 2007 at 7:36 am
Hi Paul. Yes, your summary is quite correct. Good luck with your studies!