## Wheres the flaw in this...?

Post any maths and/or physics related questions here.

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Alvaro wrote:It is true though that for positive reals, the notation sqrt(x) usually means just the positive square root of x.
I've never seen it used any other way — and, besides, it's this definition that's being used in the original post.

Beer Hunter

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Beer Hunter wrote:
Alvaro wrote:It is true though that for positive reals, the notation sqrt(x) usually means just the positive square root of x.
I've never seen it used any other way — and, besides, it's this definition that's being used in the original post.

I'm not sure that is - the original post contained a sqrt(-1), for which choosing the "positive" root is not defined.

Also, i was always taught (in a physics sense more than a maths sense) to consider both roots. There's the famous example where Dirac predicted the existence of the positron by theory alone, but then discarded it because it was a negative root to his equation...

GeekDog

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GeekDog wrote:I'm not sure that is - the original post contained a sqrt(-1), for which choosing the "positive" root is not defined.
It's easily extended to the complex plane by choosing the root with the smallest complex argument. This is the common usage, which is assumed unless specified otherwise. I could go off and define sin(x) to be a vector of the smallest pair of twin primes such that the largest one is greater than or equal to x, unless the day exactly x days after the release of the SpongeBob SquarePants movie happens to be a Friday-the-thirteenth (in which case the function returns a vector of the number of intervening Friday-the-thirteenths and the number of prime numbers less than x). That doesn't mean that this is a sensible definition, and it definitely doesn't mean that I can use it in a proof without mentioning it.
GeekDog wrote:Also, i was always taught (in a physics sense more than a maths sense) to consider both roots.
Nobody disagrees. Nobody is saying that you can go from r² = s to r = |sqrt(s)|. We have always gone from r² = s to r = ±sqrt(s), and we will continue to do so, because sqrt() does not have the ± implicitly. That's my whole point.

Beer Hunter

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so houw would you like to see for what x E R, it is possible that :
x^2 = 4?
x = sqrt(4)
so only 2? ( x= |sqrt(4)| ??)
frea

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frea wrote:x^2 = 4?
x = sqrt(4)
The second line does not follow from the first, since it discards the possibility that x = −2. The second line should read, x = ±sqrt(4).

Beer Hunter

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What is the diffrence between ±sqrt(4) and ±2? They mean the same thing.

It looks like we have 'argued' only about the place of signs(before and after 'sqrting'), which even doesn't matter.
frea

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Joined: Wed Nov 24, 2004 9:21 am

frea wrote:What is the diffrence between ±sqrt(4) and ±2? They mean the same thing.
That's right, they're the same. The things that aren't the same are sqrt(4) and ±sqrt(4). The first one is 2 and the second one is ±2.

Beer Hunter

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Beer Hunter wrote:
frea wrote:What is the diffrence between ±sqrt(4) and ±2? They mean the same thing.
That's right, they're the same. The things that aren't the same are sqrt(4) and ±sqrt(4). The first one is 2 and the second one is ±2.

have i said that they'r the same? I've just siad that it doesn't matter when you change the sign.
Mabye it is so, if you threat sqrt as a function, then you should write +- sqrt(x), but it is easier, causes less problems if you say that sqrt(4) = +-2, and sqrt isn't a ordinal function? It is easier for some calculations to know what exacly sign has sqrt.
frea

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frea wrote:but it is easier, causes less problems if you say that sqrt(4) = +-2, and sqrt isn't a ordinal function?
It doesn't cause fewer problems, because there were no problems to begin with. However, it does add a problem: it's not the convention, and you have to juggle your modified definition with the normal definition used in any mathematical laws that you apply to your work. (You also have to declare that you're using a modified definition. The original post in this topic did not declare it as such, and is therefore not using it.)
frea wrote:It is easier for some calculations to know what exacly sign has sqrt.
Yes; I know exactly what sign sqrt() has. What're you saying?

Beer Hunter

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for example if you have :
x=±sqrt(2)±sqrt(3)
so x = sqrt(2)+sqrt(3) or x = -sqrt(2)-sqrt(3).
Only these two, because if we r reading ±, and want to take the one with +, the second we must need with + too.
(as the ± sign says, if we read the upper one which is by sqrt(2), we have to read the upper one by sqrt(3), the same is with the lower one.)
but if we write :
x=sqrt(2) + sqrt(3)
and we know that sqrt(y)=±x, we have got all 4 x's.

btw if you trust wikipedia http://en.wikipedia.org/wiki/Square_root Properties(8'th point).
frea

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frea wrote:for example if you have :
x=±sqrt(2)±sqrt(3)
so x = sqrt(2)+sqrt(3) or x = -sqrt(2)-sqrt(3).

Only these two, because if we r reading ±, and want to take the one with +, the second we must need with + too.
According to Mathworld, the first line refers to four values. (To get the situation on your second line, you'd need to write x=±(sqrt(2)+sqrt(3)).)
frea wrote:btw if you trust wikipedia :) http://en.wikipedia.org/wiki/Square_root Properties(8'th point).
This one?
Suppose that x and a are reals, and that x² = a, and we want to find x. A common mistake is to "take the square root" and deduce that x = √a. This is incorrect, because the principal square root of x² is not x, but the absolute value |x|, one of our above rules. Thus, all we can conclude is that |x| = √a, or equivalently x = ±√a.
But this agrees with me! I'm not saying that x² = 4 ⇒ x = 2, because that's obviously wrong. I'm just saying that the sqrt() (or √) notation refers to the positive value, and that you need a ± sign when you want to refer to both square roots. My point is echoed all the way through that Wikipedia article, but especially by these points:
• The principal square root function √x is a function which maps the non-negative real domain ... into the non-negative real codomain ... .
• The principal square root function √x always returns a single unique value.

Beer Hunter

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I never said that i think here of sqrt as a function, if i would i wouln't discuss with you. (or at least i dont' remember such thing)
Beer Hunter wrote:and that you need a ± sign when you want to refer to both square roots

sure i want !

Thanks for that mathworld link , i've known a little bit other explanation of these signs.
frea

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frea wrote:so houw would you like to see for what x E R, it is possible that :
x^2 = 4?
x = sqrt(4)
so only 2? ( x= |sqrt(4)| ??)
What do the | | mean?

Syn

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Location: Birmingham, UK

Modulus, i.e. take only the positive value (that's not quite right for complex numbers, but so long as you're talking reals it's OK as an explanation). It's a similar idea to the C function abs().

GeekDog

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Syn wrote:
frea wrote:so houw would you like to see for what x E R, it is possible that :
x^2 = 4?
x = sqrt(4)
so only 2? ( x= |sqrt(4)| ??)
What do the | | mean?

absolute value, if it's negative, make it positive, if positive, leave it positive.

Wizard