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May 5th, 2014, 11:54 PM   #11
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If a statement is really truly a definition in the best mathematical terms then it should be IMPOSSIBLE to prove it using other statements because if it were possible to prove it using other statements then there would be no need to call it a definition. It would follow from other statements , right?

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May 6th, 2014, 04:51 AM   #12
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Originally Posted by kevintampa5 View Post
Humor me. Whether or not it need be done, I asked for it to be proved so I understand why the definition is so. I do appreciate your input, however.
Definitions don't need to be proved but there must be some consistency with the body of mathematics in which they are used. This consistency has been shown numerous times in this thread.
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May 6th, 2014, 07:43 AM   #13
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Originally Posted by mathman
√n is BY DEFINITION the number which when squared = n.
That could be numbers which when squared = n. For example, 1 has two square roots; -1 and 1.
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May 6th, 2014, 12:50 PM   #14
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Originally Posted by Hoempa View Post
That could be numbers which when squared = n. For example, 1 has two square roots; -1 and 1.
True - all numbers (except 0) have two square roots.
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May 6th, 2014, 12:52 PM   #15
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Quote:
Originally Posted by kevintampa5 View Post
Humor me. Whether or not it need be done, I asked for it to be proved so I understand why the definition is so. I do appreciate your input, however.
Theorem: (√x)² = x
Proof: Definition of square root.
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May 6th, 2014, 03:02 PM   #16
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Quote:
Originally Posted by greg1313 View Post
Definitions don't need to be proved but there must be some consistency with the body of mathematics in which they are used. This consistency has been shown numerous times in this thread.
I wasnt asking if a definition need be proved or not. I was asking for it to be shown why and how it is proven so I know for myself how it is done. The first time it was ever done it wasnt a definition. The actions that were taken resulted in a definition through proof the definition will always be this.
I just requested it be shown how it is done so I can understand it. This lead to requesting confirmation of the way I perceived. These questions were answered in this thread.


Example: As a fisherman, I can tell you "Bait a hook and you can catch fish". If you have never fished before and know nothing about the subject, me telling you to bait a hook to catch fish reveals nothing to you other than it is what you have to do to catch fish but it doesnt teach you how to bait the hook and how to use that to catch the fish. You would need to be taught. This comes from many other peoples experience that teaches others to teach you how to bait a hook, and how to use that to catch the fish.
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May 7th, 2014, 04:17 AM   #17
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Quote:
Originally Posted by mathman View Post
Theorem: (√x)² = x
Proof: Definition of square root.
$\displaystyle x = -1 \ne 1 = (\sqrt{x})^2$
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May 7th, 2014, 02:10 PM   #18
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Quote:
Originally Posted by Hoempa View Post
$\displaystyle x = -1 \ne 1 = (\sqrt{x})^2$
If you work from the inside out , you don't run into that difficulty.

Let x = -1

$$\sqrt{-1} = \pm i$$

Using

$$(\sqrt{x})^2 = x$$

And being careful to work from the inside out , we get ,

$$ (\sqrt{-1})^2 = ( \pm i )^2 = -1$$

Let x = 1

$$ \sqrt{1} = \pm 1$$

Again as above ,

$$( \sqrt{1})^2 = ( \pm 1)^2 = 1$$

So we never bump into the annoying $1 = -1$ or $-1 = 1$

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