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Spivak Calculus Chapter 1 Problem 3.i

  1. Oct 23, 2014 #1
    1. The problem statement, all variables and given/known data
    Prove the following: ##a / b = ac / bc##, if ##b, c \neq 0##.

    2. Relevant equations

    P1-12

    3. The attempt at a solution
    ##a/b = a*b^{-1}##
    ##1 = c*c^{-1}##
    ##a/b*1 = (a*b^{-1})(c*c^{-1})##
    ##a/b = (a*c)(b^{-1}*c^{-1})##

    Now, if ##b^{-1}c^{-1} = (bc)^{-1}##, then the problem is easy to solve. However, you don't prove this until problem 3.iii and I'm assuming I can only use P1-12.

    Some hints about how to proceed would be very helpful.
     
  2. jcsd
  3. Oct 23, 2014 #2

    Ray Vickson

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    What is the meaning of ##a/b##; that is, if we say ##a/b = r##, what are we actually saying? In other words, how could you re-write the relation between ##a,b,r## in another way?
     
  4. Oct 23, 2014 #3

    PeroK

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    What about looking at ##ac(bc)^{-1}bc##?
     
  5. Oct 23, 2014 #4
    How did I not see that, I feel so stupid.

    ##1 = b* b^{-1}##
    ##a = a *(b* b^{-1})##
    ##a = (a*b^{-1})*b##
    ##ac = ((a*b^{-1})*b)*c##
    ##ac = (a*b^{-1})*(bc)##
    ##(ac)*(bc)^{-1} = (a*b^{-1})*((bc)*(bc)^{-1})##
    ##(ac)/(bc) = a/b##

    I think this is it.
     
  6. Oct 23, 2014 #5
    For someone unfamiliar with Spivak, what are the variables? Numbers?
     
  7. Oct 23, 2014 #6
    They are real numbers.
     
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