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Topic: 2 simple? questions and proposed answers
Replies: 4   Last Post: Jan 29, 2012 7:50 AM

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Kent Holing

Posts: 342
Registered: 12/3/04
Re: 2 simple? questions and proposed answers
Posted: Jan 29, 2012 7:50 AM
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Please, see a good and extensive discussion on the (-a)(-b) = ab issue in Mazur?s delightful book Imagining Numbers (a to-the-point summary of the discussion is given on page 102). Also, Honsberger?s Mathematical Delights gives a discussion on a nice geometric representation of multiplication where (-a)(-b) = ab pops up as a special case. (See Section 5, pp. 53.)

Finally, let me tell you a story which migth convince non-mathematical people that (-1)(-5) = 5:
I have some friends which each has lent me 5 dollars, so for me this is totally Nx(-5) dollars. It is fair to say that if I had no friends (bad for me), then this would be 0x(-5) = 0 dollars.
If I pay back one of my friends, for me this means (N-1)x(-5) dollars, which must be Nx(-5) + 5 dollars. If this is also true in the case N = 0, we have (0-1)x(-5) = 0x(-5) + 5. Now, the LHS here is equal to (-1)(-5) and the RHS is equal to 0 + 5 = 5; so LHS = RHS gives (-1)(-5) = 5.
Convinced? (Of course, the distributive law is lurking behind the scene.)

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