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chocolate-chip chooky

Maths whizzes, please help :)

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Can the resident maths whizzes please help?

 

Forever and a day I've always thought that the square root of a number had two possible solutions - the positive or the negative - and then you have to consider the context and disregard one if it isn't applicable eg a negative length would be tossed out.

 

But then we came across the section in AoPS prealgebra where he says the definition of a square root is the non-negative answer only.

 

 

The reactions in our family:

 

20 yr old university maths student:  Huh?

 

Engineer dad: What?

 

11yr old: That's negativist!!

 

Me: Oh my golly goodness, I'm ruining her education with my ignorance and ineptitude. 

 

23 yr old criminologist:  - - -      *crickets chirping*

 

 

 

So, we looked up Wolfram, and it seems there's a *principal* square root ie the positive answer.

But, why haven't any of us ever come across this before? Does it matter? Is it a definition to file away for a later time? 

Edited by chocolate-chip chooky

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Given a number a, in a lot of cases there are 2 solutions for x in the equation

x^2=a

So sometimes you would like to say that all the solutions are square roots of a.

But sometimes you want to canonically* choose one particular solution from among the others, e.g. within real numbers

x^2=4

has two solutions +2 and -2, but in this case it is easy to choose a convention that we'll choose the solution +2 to be the square root of 4. In this case there's an easy rule for choosing. If a is a positive real number, then there are exactly two solutions to

x^2=a

with exactly one positive and one negative, so we set up a convention where the positive solution is the one that is chosen to be called the square root of a.

 

For other types of numbers, it's not so easy. For example with complex numbers, the equation
x^2=-1

has exactly two solutions +i and -i. This time, there's no clear way to choose one of them to be designated to be called the square root of -1. One option is to just let them both be called square roots. Another is to make a choice of which one, but to also make such a choice for picking one of the two solutions for all equations

x^2=a

for complex a (except there's just one solution when a=0). There is unavoidably some arbitrariness in the choice, and no matter how you do it, you can't get the square root function to be continuous, which you may have wanted (or you can have a continuous multi-function if you keep both solutions).

 

Note that sometimes there can be lots of solutions to

x^2=a

For example within the quaternions, every negative real number has infinitely many square roots.

 

 

*

https://en.wikipedia.org/wiki/Canonical_form

http://mathworld.wolfram.com/Canonical.html

 

Edited by epi
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This came up as my DW taught my DD algebra last week. My understanding, agreed to by my AoPS star student DS:

When asked in WORDS, the answer of the "square root" is the positive and negative.

When asked by the funky square root symbol, it is the positive answer only, unless the radical symbol is preceded by a little +/- notation.

Why? No clue. The technical term is that the positive root is the "principal root" and is understood to be the desired answer when only the square root symbol is used. If you're being asked "x^2=144", what is x?", then 12 and -12 are correct answers. But for square root questions using the symbol for square root, it is the positive answer only unless otherwise specified. One reason might be that in calculating areas and sides of a square, it makes no sense to answer with a negative answer for the length of a side.

Edited by tj_610
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Yes, I agree with what the others have posted.

 

Another way of thinking about it would be this: If the question is regards to the square root *function*, then the answer is always positive. But it's because the graph of x = y^2 looks like a parabola on its side, and functions by definition can only have one output per input. in other ways, a U shape on it's side can't represent a function, because each positive input (x) corresponds to two outputs (y): a positive and a negative. So in order to turn a square root map into a function, mathematicians just define it in such a way as to erase the bottom half of the sideways U, thereby giving you only positive solutions and voila! The map that wasn't a function is now a function! :) But, when you're just solving an equation like x = y^2, then your solution set will still have both a positive and a negative solution.

 

This becomes more clearly defined when you consider more complex numbers: What are the solutions to x^8 = 1, for instance? In the complex plane, there are exactly eight solutions, but we define a certain one to be the principal one (and it will generate all of the others, in fact).

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(Forgive me, as it's been a very long time since I've tried to explain this and I haven't slept in about 9 years...)

 

The number âˆš4 is a real number, and can be plotted on a number line, at +2. It cannot be plotted at +2 AND -2, as it is only one real number.

 

The solution to the equation x^2=4 can be found by using the inverse of the square function, x=√4; it is here that x=+/-2. It is ONLY in the solving of an equation that the solution set now has 2 options, +/-.

 

 

Now, watch me as I hide because I'm positive Dr. Panasuk is going to sense a disturbance in the force and come kick my behind for forgetting how to explain this... *weeps for my old brain*

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It is simply how "square root of x" is defined. The definition is that  sqrt(x) is the positive number whose square is x.

 

Now, of course the equation z^2=x has two solutions for z: z can be plus or minus sqrt(x). But the sqrt itself is always positive.

Any prealgebra text should be clear about that.

 

Edited by regentrude
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But, why haven't any of us ever come across this before? Does it matter? Is it a definition to file away for a later time? 

It does matter. You should try and sort it out now.

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It does matter. You should try and sort it out now.

 

The explanations all make sense. I can understand and accept it. Everyone else here too.

 

What I still can't get my head around is the fact that all of us here have never heard of this before. Ever. And I feel like we've all been reasonably well educated.

 

 

We're in Australia. I wonder if our curriculum here is different? Is it possible that things are defined differently? (Like trapezium vs trapezoid, for example).

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Mathematical thunder from down under LOL.

 

Simple and clear. But it doesn't really explain WHY that is so, and that falls short of the standard that our house (OK, mainly my DS16) has for math explanations. :-)

 

I think epi's and others' explanations spell it out. I suspect it comes down to the mathematical desire for square root to be considered a "function"; every value of x must have only one value of y. As indicated above, to show this, if one graphs y=sq rt x, you get a parabola that fell down to the right. Every value of x > 0 has two solutions. So in order to keep square root to meet the conventional definition of "function", let's say "it only means the positive answer".

 

Mathematicians do things like this, to keep the whole math shebang consistent and working. :-)

 

PS I'm not a professional mathematician, so take someone else's word for it if they correct me.

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Mathematical thunder from down under LOL.

 

Simple and clear. But it doesn't really explain WHY that is so, and that falls short of the standard that our house (OK, mainly my DS16) has for math explanations. :-)

 

I think epi's and others' explanations spell it out. I suspect it comes down to the mathematical desire for square root to be considered a "function"; every value of x must have only one value of y. As indicated above, to show this, if one graphs y=sq rt x, you get a parabola that fell down to the right. Every value of x > 0 has two solutions. So in order to keep square root to meet the conventional definition of "function", let's say "it only means the positive answer".

 

Mathematicians do things like this, to keep the whole math shebang consistent and working. :-)

 

PS I'm not a professional mathematician, so take someone else's word for it if they correct me.

 

:001_smile:

 

Sounds similar to my 11 year old. She really doesn't like having to just accept things.

We have been reading Isaac Newton biographies recently and we came across the motto for the Royal Society: Nullius in Verba (Take no one's word for it).

 

She wants this on a shirt  :)

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Hilarious! My DS is 16; maybe we can arrange a marriage in 10 years. They would get along swimmingly.

 

I can't remember the context, but I remember my professor in a course on Aristotle telling us that the only necessary rule of any system of mathematics is that it be internally consistent. I don't know why, but it's the only thing I recall from the whole class LOL. Many mathematicians would probably revolt at such a relativistic view of "the purest science". :-)

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I can't remember the context, but I remember my professor in a course on Aristotle telling us that the only necessary rule of any system of mathematics is that it be internally consistent. I don't know why, but it's the only thing I recall from the whole class LOL. Many mathematicians would probably revolt at such a relativistic view of "the purest science". :-)

 

Actually, since Gödel in the 1930s, mathematicians have come to terms with the fact that you can have different versions of mathematics based on different axioms. For example you could include the the Axiom of Choice

https://en.wikipedia.org/wiki/Axiom_of_choice

or you could include its negation.

 

Earlier I mentioned

Inverse function

https://en.wikipedia.org/wiki/Inverse_function

Example: squaring and square root functions

https://en.wikipedia.org/wiki/Inverse_function#Example:_squaring_and_square_root_functions

 

Indeed a square root function is a right inverse or section

https://en.wikipedia.org/wiki/Section_(category_theory)

of a squaring function. In general, the existence of a right inverses or sections is equivalent to the axiom of choice (which, recall, is an optional axiom).

 

For real numbers, it's easy to define a square root function (just choose the positive square root). But what about in general. In general, for the squaring function in a field or ring or group or semigroup, can we be sure that a square root function (i.e. a right inverse or section of the squaring function) even exists. Or does it depend on which version of based on which axioms. Is the statement that "a square root function (i.e. a right inverse or section of the squaring function) always exists", equivalent to the axiom of choice? Something to think about.

 

 

 

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Hilarious! My DS is 16; maybe we can arrange a marriage in 10 years. They would get along swimmingly.

 

I can't remember the context, but I remember my professor in a course on Aristotle telling us that the only necessary rule of any system of mathematics is that it be internally consistent. I don't know why, but it's the only thing I recall from the whole class LOL. Many mathematicians would probably revolt at such a relativistic view of "the purest science". :-)

 

:laugh:

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Actually, since Gödel in the 1930s, mathematicians have come to terms with the fact that you can have different versions of mathematics based on different axioms. For example you could include the the Axiom of Choice

https://en.wikipedia.org/wiki/Axiom_of_choice

or you could include its negation.

 

Earlier I mentioned

Inverse function

https://en.wikipedia.org/wiki/Inverse_function

Example: squaring and square root functions

https://en.wikipedia.org/wiki/Inverse_function#Example:_squaring_and_square_root_functions

 

Indeed a square root function is a right inverse or section

https://en.wikipedia.org/wiki/Section_(category_theory)

of a squaring function. In general, the existence of a right inverses or sections is equivalent to the axiom of choice (which, recall, is an optional axiom).

 

For real numbers, it's easy to define a square root function (just choose the positive square root). But what about in general. In general, for the squaring function in a field or ring or group or semigroup, can we be sure that a square root function (i.e. a right inverse or section of the squaring function) even exists. Or does it depend on which version of based on which axioms. Is the statement that "a square root function (i.e. a right inverse or section of the squaring function) always exists", equivalent to the axiom of choice? Something to think about.

 

 

 

 

Epi, you're amazing, but you're making my head all  :willy_nilly:  :willy_nilly:  :willy_nilly:

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Epi, you're amazing, but you're making my head all  :willy_nilly:  :willy_nilly:  :willy_nilly:

 

Sorry, I wasn't doing it on purpose. :rolleyes: Nah, just kidding. I really was doing it on purpose. :) 

 

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