Dumb Square Root Ques.

[SkepticalScience]

Why doesn't my calculator, computer, even google say that the square root of 4 is 2 AND -2?

Aren't both of them the square root of 4?

(Now, on to moving the hair from my forehead, so it doesn't get in the way of my hand!)

 

[wollery]

Why doesn't my calculator, computer, even google say that the square root of 4 is 2 AND -2?

Aren't both of them the square root of 4?

(Now, on to moving the hair from my forehead, so it doesn't get in the way of my hand!)

Because it isn't worth the effort.

All numbers which have real roots have both a positive and a negative root, but unless you know that fact already it's extremely unlikely that you'd need to know it, and if you do already know it then you'd know that the negative root is the same as the positive root, except that it's negative, so you only actually need to be told what the positive root is and then you'd know what the negative root is.

 

[Matabiri]

Re: Re: Dumb Square Root Ques.

Because it isn't worth the effort.

Also, the sqrt() function is defined as returning only the positive root, and this is what is being implemented.

This is because returning both is problematic and unnecessary in computing.

 

[rppa]

Why doesn't my calculator, computer, even google say that the square root of 4 is 2 AND -2?

Two responses. First of all, how do you suggest it return two solutions?

Second response:
Aren't both of them the square root of 4?

Actually, no. In mathematics, the term "square root" of a positive real number A is reserved for the positive solution of x^2 = A. It's the "principal" solution to this equation, the principal root of x^2 - A.

The situation is similar to "the" arctan of 1 being 45 degrees, despite the fact that there are infinitely many angles whose tangent is 1.

 

[SkepticalScience]

Thanks Guys.

I can accept that, simply because of the way sqrt was defined, the number should be positive.

But why did they decide to define it that way?

I like to think of square roots as "What number when multiplied by itself will yield this other number?"

In the case of square roots, there are always two numbers. I assume that in the cause of cube roots, 4 roots, 100 roots, etc, there are that many solutions as well.

I'd think that people would want to know ALL those solutions, not just a "principal" one.

Ha. . . This is math, not english!

 

[drkitten]

I can accept that, simply because of the way sqrt was defined, the number should be positive.

But why did they decide to define it that way?

Because in order to talk about the square root function, by definition, it needs to have a unique value (for each input). And since the positive square root is usually more useful than the negative square root, it makes sense to define the function to take the positive value. If anyone really needs the negative square root (which doesn't often happen), it's intellectually trivial to note that -sqrt(x) is also a valid root.

 

[rppa]

I'd think that people would want to know ALL those solutions, not just a "principal" one.

Declaring one branch (that's the technical term) a "principal" one doesn't somehow mean you've forgotten the others. It's just that if you're going to number things, *something* has to be called number 1.

And as has already been pointed out, the positive square root is more often useful than the negative one. Nobody is forgetting that the two solutions to x^2 = A are +-sqrt(A), but adopting a particular meaning for "sqrt(A)" as the positive value lets you understand that "+-sqrt(A)" refers to two values, one positive (sqrt(A)) and one negative (-sqrt(A)).

If you said "sqrt(A)" could mean either root, then you wouldn't be able to write a notation like "+-sqrt(A)" to mean the two roots. And then whenever a square root appeared in an equation, you'd have to ask, "does this mean the positive one or the negative one"?

For instance, is 2 + sqrt(3) a number which is more than 2, or less than 2? Since we've adopted a meaning for sqrt(3), we know that refers to a number more than 2, whereas 2 - sqrt(3) is a number less than 2. Make "sqrt(3)" ambiguous and you have to come up with another way of indicating what the value of 2 + sqrt(3) is.

 

[SkepticalScience]

Hmmm. . . I don't like these answers at all. I accept them, but it's not obvious why the rule was defined this way.

new drkitten says:
"And since the positive square root is usually more useful than the negative square root"

But who gets to decide what is useful? And since when does usefullness trump complete solutions?


New drkitten also says:
"Because in order to talk about the square root function, by definition, it needs to have a unique value"

Why would the function *need* to befined that way?? I can write a computer function, that returns both the positive and negative root of a number. . . why couldn't the math expression "sqrt()" do the same?

New drkitten:
"it's intellectually trivial to note that -sqrt(x)"

Er...... what does trivialness have to do with anything??

So, if I ask you for the 10th root of a number, would you only give me one of the 10?

 

[wollery]

Hmmm. . . I don't like these answers at all. I accept them, but it's not obvious why the rule was defined this way.

new drkitten says:
"And since the positive square root is usually more useful than the negative square root"

But who gets to decide what is useful? And since when does usefullness trump complete solutions?

Fine, how would you like to give me the complete set of solutions for arcsin0.5?

The whole infinite set please!

New drkitten also says:
"Because in order to talk about the square root function, by definition, it needs to have a unique value"

Why would the function *need* to befined that way?? I can write a computer function, that returns both the positive and negative root of a number. . . why couldn't the math expression "sqrt()" do the same?

It could, but as I explained earlier, anyone needing to know the negative root of a number would certainly already know that it was just the negative of the positive root, so why overcomplicate matters?

New drkitten:
"it's intellectually trivial to note that -sqrt(x)"

Er...... what does trivialness have to do with anything??

So, if I ask you for the 10th root of a number, would you only give me one of the 10?

Errrrrr, what gives you the idea that the 10th root of a number has ten solutions? A cube root has only one solution, a 4th root has two solutions, a 5th root has only one, a 6th root has 2, and so on. You seem to be confusing simple numbers with polynomial functions.

 

[Terry]

Hmmm. . . I don't like these answers at all. I accept them, but it's not obvious why the rule was defined this way.
[...]
New drkitten also says:
"Because in order to talk about the square root function, by definition, it needs to have a unique value"

Why would the function *need* to befined that way?? I can write a computer function, that returns both the positive and negative root of a number. . . why couldn't the math expression "sqrt()" do the same?

Because math != computer science. If I recall correctly, a mathematical function is defined as a mapping from a domain to a range, where each element of the domain is mapped to exactly one element of the range. Both the domain and the range are (possibly infinite) sets. Note that the mapping need not be 1:1 - in other words, more than one element of the domain can map to the same element of the range.

--Terry

 

[rppa]

New drkitten also says:
"Because in order to talk about the square root function, by definition, it needs to have a unique value"

Why would the function *need* to befined that way?? I can write a computer function, that returns both the positive and negative root of a number. . . why couldn't the math expression "sqrt()" do the same?

Because a mathematical "function" is not the same as a computer "function". A function by definition is single-valued. If it's not single-valued, it's a different kind of thing (a one-to-many mapping). Both are useful. The function form is the mapping where we make a choice as to which is the main output. Again, nobody is pretending the other roots don't exist.

And though you could write a computer procedure to return two values in some sort of array or data structure, you couldn't use that in an expression to evaluate, for instance 3 + sqrt(5) - sqrt(3).

 

[SkepticalScience]

Hmm. I didn't realize this:

Wollery Said:
" Errrrrr, what gives you the idea that the 10th root of a number has ten solutions? A cube root has only one solution, a 4th root has two solutions, a 5th root has only one, a 6th root has 2, and so on. You seem to be confusing simple numbers with polynomial functions."


I thought that simple numbers and polynomials were the same in this regard.

I could be wrong though. . . .

I think I rememberd learning somewhere that , there imaginary numbers fill in all the solutions for simple numbers.

So, that the cube root of 1 for instance, has one simple number solution, and two imaginary number solutions.

I can agree that a math function and a computer function are different.


Woolery Said:
"how would you like to give me the complete set of solutions for arcsin0.5?"

Not sure. . . what is the answer to that?

So why, when we deal with polynomials, do we expect multiple answers but not simple numbers??

 

[Brian the Snail]

Errrrrr, what gives you the idea that the 10th root of a number has ten solutions? A cube root has only one solution, a 4th root has two solutions, a 5th root has only one, a 6th root has 2, and so on. You seem to be confusing simple numbers with polynomial functions.

Be careful...you're forgetting that real numbers can have complex roots too...

 

[drkitten]

Errrrrr, what gives you the idea that the 10th root of a number has ten solutions?

I don't know what gives him the idea that it does, but what gives me the idea that it does is, well, the fact that there are ten solutions to the equation x^10 = A for constant A.

Of course, eight of the ten solutions for a positive real A are complex numbers.

 

[phildonnia]

Errrrrr, what gives you the idea that the 10th root of a number has ten solutions? A cube root has only one solution, a 4th root has two solutions, a 5th root has only one, a 6th root has 2, and so on.

Square roots of 1:
1,
-1.

Cube roots of 1:
1,
0.5+0.87i,
0.5-0.87i

Fourth roots of 1:
1,
i,
-1,
-i

Fifth roots of 1:
1,
0.31 + 0.95i,
-0.81 + 0.59i,
-0.81 - 0.59i,
0.31 - 0.95i,

etc.

 

[Art Vandelay]

So what's the positive square root of -1?

[nitpick edit]I don't know what gives him the idea that it does, but what gives me the idea that it does is, well, the fact that there are ten solutions to the equation x^10 = A for nonzero constant A.

Of course, eight of the ten solutions for a positive real A are non-real complex numbers.[/nitpick edit]

 

[phildonnia]

So what's the positive square root of -1?

The "principal" square root is i; -i is also a square root.

 

[pgwenthold]

Errrrrr, what gives you the idea that the 10th root of a number has ten solutions?

Isn't this the Fundamental Theorem of Algebra.

For any polynomial of order n, there are n solutions.

A cube root has only one solution, a 4th root has two solutions, a 5th root has only one, a 6th root has 2, and so on. You seem to be confusing simple numbers with polynomial functions.

Nah, the poster has just taken a complex analysis course.

There are indeed three cube roots. For example, the cube roots of 1 include

1
.5 +/- i*sqrt(3))/2

(by using De Moivre's Formula)

 

[Art Vandelay]

The "principal" square root is i; -i is also a square root.

What's i?

pgwenthold

Isn't this the Fundamental Theorem of Algebra.

I think the FTA just says that one exists. The fact that n exist can be derived from the FTA, but I don't think it's part of it.

 

[drkitten]

What's i?

Um, it's the principal square root of negative one.

If you know enough to make nit-picky corrections to my earlier post, you already know the answer to your question.


And my algebra textbook does phrase the Fundamental Theorem of Algebra as implying n complex roots for an n-th degree polynomial. Of course, complex numbers were well-established by the time Gauss proved the FTA, and the specific question of how many roots a given polynomial possessed had been asked -- and mis-answered-- many times before. (The claim of n solutions dates back to the 1600s, I think.)