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#26 2013-08-06 01:47:17

Au101
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Re: Change your subject.

Sorry to but in, but I was just browsing and couldn't resist the chance to do some algebra and show how bob bundy's and bobbym's answers are the same:

Last edited by Au101 (2013-08-06 01:47:43)

#27 2013-08-06 01:48:25

Au101
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Re: Change your subject.

Sorry, posted that before I saw bob bundy had done the same thing smile

#28 2013-08-06 02:06:07

bob bundy
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Re: Change your subject.

That's OK.  Great minds think alike and all that.  smile

I now have a diagram for my statement about the position of mathematica in the hierarchy of intelligence, etc.

Bob


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View Image: mathematica.gif      


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#29 2013-08-06 02:07:20

bobbym
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Re: Change your subject.

I will have to concede that is probably true.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#30 2013-08-06 02:24:55

bob bundy
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Re: Change your subject.

OK.  Don't get me wrong.  It's a useful tool but that's all.  I'm a better driver than my car.  Last time it tried driving on its own there was a right crunch.

Bob  smile


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#31 2013-08-06 02:30:32

EbenezerSon
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Re: Change your subject.

Au101 wrote:

Where from the negative or positive sign in front of the root sign?

#32 2013-08-06 02:31:40

bobbym
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Re: Change your subject.

Look at post #23, he is continuing from there.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#33 2013-08-06 03:03:36

Au101
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Re: Change your subject.

Yep, because a negative number times a negative number is a positive number, so:



Try it:



Therefore, when we 'undo' the squaring, by square rooting, the answer could be positive, or negative. The square root of 4 is either 2, or -2. We can't tell.

Again:



Edit: this is why bobbym's original solution had two answers



And:



Which is the same as:

Last edited by Au101 (2013-08-06 03:06:24)

#34 2013-08-06 03:11:45

EbenezerSon
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Re: Change your subject.

bob bundy wrote:



Still I cant grasp why there is positive and negative sign before the root.

I have other questions but want this cleared.

Amen.

#35 2013-08-06 03:20:53

bobbym
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Re: Change your subject.

When you take a square root of a number there are two possible roots. (-3)(-3) = 9 and (3)(3)=9


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#36 2013-08-06 03:26:38

bob bundy
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Re: Change your subject.

eg.  √9 = 3

But this is not the only square root of 9.

-3 x -3 = 9 as well, so √9 = -3 is also correct.

to show you have got all possible answers you write






Let's look at an actual question.

Consider the graph y = x^2

Find x when y = 9

see graph below.

If you said x = 3, you would loose some marks because you hadn't given all the possible values.

Bob


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You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#37 2013-08-06 03:39:02

anonimnystefy
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Re: Change your subject.

Actually,

only, but both 3 and -3 satisfy the equation
.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#38 2013-08-06 03:53:19

Au101
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Re: Change your subject.

anonimnystefy wrote:

Actually,

only, but both 3 and -3 satisfy the equation
.

Excellent technical observation, we should, really, say that:



However, it is true to say that the square root of 9 is plus or minus 3. The problem we have is that the sign √ refers to the principal square root only.

This does make the thing a little harder to understand, though tongue

Suffice it to say that when we square root both sides of an equation, we must include the ± sign, as an equation of the form:



Has two solutions.

Last edited by Au101 (2013-08-06 03:54:17)

#39 2013-08-06 04:11:57

bobbym
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Re: Change your subject.

Hi;

I knew that about the principal square root and should have phrased post #35 better. Sorry for the confusion.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#40 2013-08-06 04:15:40

bob bundy
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Re: Change your subject.

Interesting.  smile 

When I use this: √, I have always meant 'the square root of' and never worried about the term 'principle square root'.  Somehow I've managed.  This probably means that the same error has occurred before in one of my posts, but no one noticed.  I'll try to stick to the convention in future, but I cannot promise I'll succeed.

If you read the Wiki article on square roots

http://en.wikipedia.org/wiki/Square_root

you'll see it all starts nicely, with the principle root defined and then it gets in a muddle when complex roots are introduced.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#41 2013-08-06 04:19:39

Au101
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Re: Change your subject.

I agree, it was sloppy phrasing on my part as well. Although, bobbym, I think your post #35 was correct. 9 does have two square roots (the principal root being 3, the other being -3) the problem is that the notation √9 gives us the principal square root. This is why we have to write the ± sign before the radical.

Thus ± √9 = ±3 and √9 = 3.

But "the square root of nine is plus, or minus, three" is correct (or, perhaps I should say "the square roots of 9 are..."). It is a problem of formal notation that we have, I believe - if I have understood everything correctly.

Last edited by Au101 (2013-08-06 04:27:30)

#42 2013-08-06 04:30:23

bob bundy
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Re: Change your subject.

hi

It matters when you are labelling the button on a calculator.  Some of us are old enough to remember when there was no such thing (as a calculator).  Has this convention come about because calculators and computers have to generate single values?  It would be interesting to know if the convention existed pre-1960.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#43 2013-08-06 04:37:24

Au101
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Re: Change your subject.

I would imagine it dates back to early geometry. If, say, you're trying to calculate the length of a hypotenuse, you're not interested in the negative values. I imagine this general precedence of the principal square root was incorporated into the notation when it was defined. Because, of course, we define our notation to be useful to us and easy to work with. But, without realising it, you've always been using the convention whenever you've gone:



If it weren't for the fact that the √ sign only referred to the principal value, you wouldn't need the ±, that would be implied by the √. Then you could just write:



It's just the way we learn to think about it conceptually smile

Last edited by Au101 (2013-08-06 04:38:06)

#44 2013-08-06 04:40:07

Au101
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Re: Change your subject.

Another way to think about it is if √25 = ± 5, then the ±'s would cancel each other out. You would have:

#45 2013-08-06 06:12:10

zetafunc.
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Re: Change your subject.

Au101 wrote:

Another way to think about it is if √25 = ± 5, then the ±'s would cancel each other out. You would have:

Surely

?

#46 2013-08-06 06:24:11

anonimnystefy
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Re: Change your subject.

Again, the square root returns only the principal value by convention, not because it would cause contradiction otherwise.

Last edited by anonimnystefy (2013-08-06 06:25:14)


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#47 2013-08-06 06:24:13

Au101
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Re: Change your subject.

Hmmmm....on second thought, maybe? I'm not sure. My original thinking was that when the first ± is +, so is the second one and when the first ± is -, so is the second one. So, for example:



When the first ± is +, so is the second one and when the first ± is -, so is the second one. So we have:



Hence the need for a ∓ sign. In this case we would have +(+5) or -(-5) and only these options. But it seems reasonable to be able to say:



So, yes, I think you're right. Ignore my second post. The first one still stands, though, I think smile

But yes, anonimnystefy is right. Sorry for confusing the matter further, the important thing to note is √x is always the positive, principal root, hence the need for the ± sign before the radical sign.

Last edited by Au101 (2013-08-06 06:26:57)

#48 2013-08-06 06:32:36

zetafunc.
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Re: Change your subject.

bob bundy wrote:

Interesting.  smile 

When I use this: √, I have always meant 'the square root of' and never worried about the term 'principle square root'.  Somehow I've managed.  This probably means that the same error has occurred before in one of my posts, but no one noticed.  I'll try to stick to the convention in future, but I cannot promise I'll succeed.

If you read the Wiki article on square roots

[link]

you'll see it all starts nicely, with the principle root defined and then it gets in a muddle when complex roots are introduced.

Bob

It can become quite a complex issue (pun not intended). For example, I did this problem via contour integration:



Naturally, the first step is to compute the sum of the residues of that function. Tell Mathematica to do that, and it won't give you the correct answer, thanks to the square root.

#49 2013-08-06 17:35:56

bob bundy
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Re: Change your subject.

So does power one half mean take the principle value as well?

ie is the following true



Generally what about other fractional powers?

eg.



Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#50 2013-08-06 17:43:50

anonimnystefy
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Re: Change your subject.

I'd say so.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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