Thread: Question about square roots!

Is it mathematically correct to say √16=±4, or would it be correct to say that √16=4, unless specified ±√16? This isn't for any specific problem or anything, but I'm just curious if is correct to have ±, because I know every number has two square roots.

the point is that you cannot tell wether squareroot16 is composed of +4 or -4 therefore you must assume both however if you know from previous calculations that +4 was used I dont see a problem not providing a -4. Other reasons too,assuming you have a length l the length must be a positive value because it makes no sense having a table -4m

The question is whether or not it's mathematically correct to do the + and - on the root, and yes, it is.

Excuse me ... .

As a real function on non-negative real line, the square root function is a non-negative single-valued-function ... .

As a complex function on complex plane, the square root function is a multi-valued-function ... .

Originally Posted by Muon321

Is it mathematically correct to say √16=±4, or would it be correct to say that √16=4, unless specified ±√16? This isn't for any specific problem or anything, but I'm just curious if is correct to have ±, because I know every number has two square roots.

Yes, every number has "two square roots" but when talking about the square root function as applied to real numbers, "the" square root of a is "the positive number, x, such that ."

So it is correct to say that and then .

In fact when we say that "the solutions to the equation are " the reason we need the "" symbol is that , alone, does NOT give both.

Yes, agree. Square root function of 16 is defined in real field as the positive solution of equation x^2 = 16.

Originally Posted by fred91

Yes, agree. Square root function of 16 is defined in real field as the positive solution of equation x^2 = 16.

That is true in the theory of real-valued functions. However, after switching to complex numbers it always gives headaches to students, because there the root is actually multi-valued and its value has to be chosen carefully depending on the values of other roots in the expression. In many ways the complex definition of the root is much more natural.
So, in the end the exact understanding of the root depends on the application, but to write sqrt(16)=4 and think that the sign gets "included" in the root sign is just wrong.

I have been giving a hand to some middle-school guys and I found out, with great sadness, that there are books in which there is written sqrt(16) = +- 4.
Then teachers get puzzled when the guys show no interest in Mathematics! Not being rigorous on definitions is the first step to transform Math in a series of indemonstrable theorems..

Originally Posted by fred91

I have been giving a hand to some middle-school guys and I found out, with great sadness, that there are books in which there is written sqrt(16) = +- 4.
Then teachers get puzzled when the guys show no interest in Mathematics! Not being rigorous on definitions is the first step to transform Math in a series of indemonstrable theorems..

Yeah, that's bad without a proper definition. It would be ridiculous to see for example the formula for quadratic equation written as (-b+sqrt(b^2-4ac)).
Anyway, the question that was asked is still not clear to me, because in real-valued science sqrt(16)=4 and in the complex world it's +-4.

Originally Posted by level1807

Originally Posted by fred91

I have been giving a hand to some middle-school guys and I found out, with great sadness, that there are books in which there is written sqrt(16) = +- 4.
Then teachers get puzzled when the guys show no interest in Mathematics! Not being rigorous on definitions is the first step to transform Math in a series of indemonstrable theorems..

Yeah, that's bad without a proper definition. It would be ridiculous to see for example the formula for quadratic equation written as (-b+sqrt(b^2-4ac)).
Anyway, the question that was asked is still not clear to me, because in real-valued science sqrt(16)=4 and in the complex world it's +-4.

This wasn't a high level question. What is the square root of 16? Both -4 and 4, when squared, yield 16. That's it. Nothing else to it.

when you are resolving square root of 16 which is positive always the value will be positive. so, √16= 4 always and not to mention -4 with that.
The correct way is √16=4 and +-√16 = +-4