May 5th, 2014, 11:54 PM  #11 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
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? 
May 6th, 2014, 04:51 AM  #12 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond  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.

May 6th, 2014, 07:43 AM  #13  
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Quote:
 
May 6th, 2014, 12:50 PM  #14 
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717  
May 6th, 2014, 12:52 PM  #15 
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717  
May 6th, 2014, 03:02 PM  #16  
Member Joined: Apr 2014 From: Florida Posts: 69 Thanks: 4  Quote:
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.  
May 7th, 2014, 04:17 AM  #17 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  
May 7th, 2014, 02:10 PM  #18 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  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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