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Spivak's Calculus: Chap 1: Problem 3i - Simple Proof

  1. Sep 2, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that

    [tex]\frac ab = \frac{ac}{bc}[/tex]

    2. Relevant equations

    Basic properties of numbers



    3. The attempt at a solution

    I really don't understand what is left to show here? By definition c/c means c*c-1 = 1. So is that is? That is:

    [tex]
    \begin{array}{l}
    \frac ab &= \frac ab \cdot c\cdot c^{-1} \\
    &= \frac ab \frac cc \\
    &= \frac {ac}{bc}
    \end{array}
    [/tex]


    I dunno. Is that it?
     
  2. jcsd
  3. Sep 2, 2011 #2

    HallsofIvy

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    Please state the entire problem. Are a, b, and c numbers or from some abstract algebraic structure? Is it assumed that c is not 0?
     
  4. Sep 2, 2011 #3
    The whole problem is:
    Prove the following:

    [itex]\frac{a}{b}=\frac{ac}{bc}[/itex] if b and c do not equal 0.


    What spivak wants you to do is flip back to about page 9 where he lists his P1-P12 properties and use those to document each step.
     
  5. Sep 2, 2011 #4
    Hello :smile:

    As QC has noted, yes c and b not 0.


    Hi QC So that's what I thought. Does my proof lack anything? I am not sure what is left to document, if anything. I guess to be complete I should show that [itex]\frac ab \frac cc = \frac{ab}{cc}[/itex] by writing them as [itex](ab^{-1})(cc^{-1})[/itex] and using the associate property to rearrange them.

    Just trying to get a feel for how these proofs should be written (in a formal sense).

    Thank you. :smile:
     
  6. Sep 2, 2011 #5
    The first chapter of Spivak's book really emphasizes rigor. If you really wanted to formalize it, you'd have to add more statements justifying your proof. For instance, there's no property that says you can multiply any number by [itex]c \cdot c^{-1}[/itex]; there is one (P6) that says you can multiply any number by the identity, symbolized by [itex]1[/itex], and there's another one (P7) which says that [itex]c \cdot c^{-1} = 1[/itex].

    For these beginning exercises, it's probably worth it to go a little overboard with the rigor. Line by line proof, with properties or theorems to the right justifying each step. I think Spivak's intention is to really build a solid base, and to do that you have to really forget everything you know, which is near impossible. That being said, you clearly have the right idea.
     
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