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Question about unique real number

  1. Dec 10, 2013 #1
    hello guys im new in this forum
    i start to self learn calculus tom apostol (in my college we got another book)
    i start from the very begining of the chapter. so this not really a homework. if i post thread in wrong sub-forum please remind me :)

    1. The problem statement, all variables and given/known data

    prove : If a+b = a+c, then b = c.

    2. Relevant equations

    this also prove zero number is unique.

    3. The attempt at a solution

    the prove was easy

    using AXIOM EXISTENCE OF IDENTITY ELEMENTS & ASSOCIATIVE

    there is a number y such that y + a = O
    then
    y+(a+b) = y+(a+c)

    (y+a)+b=(y+a)+c

    0+b=0+c

    b=c

    but why in the book say this also prove number 0 is unique? in other words only one real number have the property zero?!

    thank u :)
     
  2. jcsd
  3. Dec 10, 2013 #2
    How did you define the zero number?
     
  4. Dec 10, 2013 #3
    axiom says : there exist real number denote by 0, such that for every real x, we have x+0=x

    see the underlined, it doesnt say : there exist ONE real number.
    so we have to prove only exactly one real number denote by 0. but i still dont get it how to prove this.. :)
     
  5. Dec 10, 2013 #4

    PeroK

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    Science Advisor
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    Gold Member

    Can you see how to use the result you have just proved:

    a + b = a + c => b = c

    To prove that there is only one 0?
     
  6. Dec 10, 2013 #5
    Assume there are two real numbers 0 and 0'. Both satisfy x+0 = x and x+0' = 0' for all x. Can you deduce 0 = 0'?
     
  7. Dec 10, 2013 #6
    ok guys thanks for the help..
    i dont have problem anymore.. its clear now
     
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