Question about square roots!

[John]

That's perfectly true, but [math]\sqrt{x}[/math] normally refers to the positive root specifically. There is no conflict here.

[studiot]

 

That's perfectly true

 

No special weight or treatment is afforded to either the positive or the negative in the definition. No preference is expressed.

 

This is mathematics, where we follow definitions.

 

The issue is easy to resolve, as I keep saying and you keep avoiding.

 

 

Looking back I first made the point in my post#2 and have been consistent ever since.

 

The requirement for positivity is a direct consequence of the definition of a function.

It is not a requirement for an expression containing an equals sign, that is not a function.

Edited December 18, 2013 by studiot

[John]

I am following the (usual) definition.

[math]x^2 = 16[/math] has the solutions [math]x = \pm \sqrt{16} = \pm 4[/math].

 

[math]x = \sqrt{16}[/math] has the solution [math]x = 4[/math].

 

When Amaton says, in his post, that [math]\sqrt{16} = 4[/math], he is correct. There is no issue here.

I think we may be arguing about different things.

Edited December 18, 2013 by John

[studiot]

But it is not usual, and hasn't been since at least 1880. I have no data prior to that.

 

Maybe current vogue in some places is to call it something different.

 

IMHO that is just counterproductive. There is so much genuinely new to uncover, it is a waste to simple rename the old.

Edited December 18, 2013 by studiot

[John]

The passage you quoted doesn't seem to mention the radical sign at all, but simply explains that even roots may be positive or negative, which is not in dispute. I'm talking about the definition of the symbol itself, which is the principal square root.

As an example of ubiquitous use, check out the quadratic formula. Further examples are given by the Wikipedia articles on the square root and nth root, and the results of plugging something like x = sqrt(16) into Wolfram Alpha (which also has a page of information about the square root here). While these may not be century-old books, they are the results of the collaborative efforts of hundreds or thousands of individuals, which at least suggests the definition is common. These sources are also more up-to-date than a book written in 1880.

In any case, I think all points that can be made about this have been made. I appreciate you forcing me to reexamine my thoughts and statements on this, even if I have ended up reaching the same conclusions I had before.

[studiot]

 

The passage you quoted doesn't seem to mention the radical sign at all, but simply explains that even roots may be positive or negative, which is not in dispute. I'm talking about the definition of the symbol itself, which is the principal square root.

 

Let us examine the logic of this.

 

If the roots may be positive or negative then why does the radical also need a sign?

 

Surely if you negate the radical of a negative root you obtain a positive value?

 

 

 

What I don't understand is why so are so wedded to the idea that [math]\sqrt {16} [/math]may not equal -4 ?

 

You have not presented a single reason beyond 'others also do it' for this view.

 

I can readily understand the mathematical benefits that flow from choosing only positive values for the square root function. It guarantees an inverse function and fits in with a heap of other mathematical structure about sets and functions etc.

 

 

But 16 is just a number, it is not a function.

and there are many ways to get to 16.

 

16 = 15+1

16=17-1

16 = 8*2

18= (-8)*(-2)

 

and so on

 

Do you also reject these?

Edited December 18, 2013 by studiot

[John]

Let us examine the logic of this.

 

If the roots may be positive or negative then why does the radical also need a sign?

 

Surely if you negate the radical of a negative root you obtain a positive value?

 

 

 

What I don't understand is why so are so wedded to the idea that [math]\sqrt {16} [/math]may not equal -4 ?

I've explained and shown in several sources that the radical sign is used to denote specifically the principal square root. I've also explained that notation isn't sacred, and one can easily use the radical to denote both roots, but this isn't the common usage.

 

You have not presented a single reason beyond 'others also do it' for this view.

This is sort of the definition of "convention" in this context.

 

I can readily understand the mathematical benefits that flow from choosing only positive values for the square root function. It guarantees an inverse function and fits in with a heap of other mathematical structure about sets and functions etc.

 

 

But 16 is just a number, it is not a function.

and there are many ways to get to 16.

In the context of the square root function, 16 would be the input and the radical sign would be the function. In general, [math]\sqrt{16}[/math] represents the principal square root, as I've stated many times.

 

16 = 15+1

16=17-1

16 = 8*2

18= (-8)*(-2)

 

and so on

 

Do you also reject these?

I reject the last one.

 

Edit: Also, why are we even still discussing this? We agree on the concepts involved. The only disagreement is on the default meaning of a single symbol. It's silly to get this caught up in notation.

Edited December 18, 2013 by John

[studiot]

 

I reject the last one.

 

 

Fair comment, my finger slipped as I was getting tired of typing.

 

 

 

I've explained and shown in several sources that the radical sign is used to denote specifically the principal square root. I've also explained that notation isn't sacred, and one can easily use the radical to denote both roots, but this isn't the common usage.

 

 

How can a negative radical sign applied to the principle square root denote a positive value?

You either have the positive and/or negative signs before the radical, which is then defined as always having a positive value, or you allow the value itself to be positive or negative, but not both. If the latter were to be the case the sign before the radical is worse than redundant since it can oppose the sing of the value.

 

The radical sign by itself denotes not both roots, but either root. It is up to the employer of that radical to determine which particular one.

 

 

In the context of the square root function, 16 would be the input and the radical sign would be the function. In general, represents the principal square root, as I've stated many times.

 

 

Yes the function has an inverse. But the number 16 does not in the snese that it does not have an operation that can send it back to whence it came. Of course it has a reciprocal but that is a different thing.

 

And no, you have not stated many times. You have argued, many times that you are not discussing functions, that they are irrelevant etc.

[John]

How can a negative radical sign applied to the principle square root denote a positive value?

You either have the positive and/or negative signs before the radical, which is then defined as always having a positive value, or you allow the value itself to be positive or negative, but not both. If the latter were to be the case the sign before the radical is worse than redundant since it can oppose the sing of the value.

 

In the case of 16, [math]-\sqrt{16} = -4[/math] and [math]\sqrt{16} = 4[/math]. The negative radical sign doesn't denote a positive value unless we take the radical sign to specifically refer to the negative root, which is something we probably agree would be unusual. As for the rest, I agree with you. If we take the radical to denote either root (thanks for the correction of my sloppy wording), then the [math]\pm[/math] is redundant. But usually we do prepend [math]\pm[/math] in contexts where either root is a solution, which lends further credence to my claim that the radical itself is usually taken to be the principal square root.

 

The radical sign by itself denotes not both roots, but either root. It is up to the employer of that radical to determine which particular one.

I absolutely agree. My point is that, without an explicit declaration by the mathematician in question, the radical sign is normally taken to denote the positive root.

 

 

And no, you have not stated many times. You have argued, many times that you are not discussing functions, that they are irrelevant etc.

I have stated that [math]\sqrt{x}[/math] denotes the principal square root of x in every post I've made in this thread (though a couple of times it was somewhat indirect). I've irritated even myself with the repetition, so surely you've noticed.

Edited December 18, 2013 by John

[studiot]

 

I have stated that denotes the principal square root of x in every post I've made in this thread (though a couple of times it was somewhat indirect). I've even irritated myself with the repetition, so surely you've noticed.

 

 

Yes, you have indeed.

 

But I think that this is the first time you have stated it (added the condition)

I was observing that all other times you have eschewed the function.

 

 

In the context of the square root function,

 

 

then the is redundant. But usually we do prepend in contexts where either root is a solution, which lends further credence to my claim that the radical itself is usually taken to be the principal square root.

 

 

I don't view the [math] \mp [/math] sign as problematic. Again this is because it means either... or not both as some read it. Something cannot be both minus and plus at the same time.

 

So saying minus or plus something implies that there are (at least) two possibilities.

 

I consider it unwise to put in front of the radical because it implies information we may not have.

 

So we have three situations

 

1 We know the answer is positive so we use + (or nothing if we like)

 

2 We know the answer is negative so we use -

 

3 We don't know which sign the answer has so we use [math] \mp [/math]

 

Note I have use the inverted form for variety.

 

Edit a final thought, how do you interpret the expression

[math] \pm \sqrt z [/math]

 

 

where z is complex since complex numbers do not associate with a sign?

 

Obviously it the [math] \pm [/math] has to be associated with the operations of subtraction or addition but then you are assigning a different meaning to the symbol.

 

Edited December 19, 2013 by studiot