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Spivak's Calculus Prologue Problem 3 (v)

  1. Nov 2, 2013 #1
    Here's my attempt at this proof. Is this correct?


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

    Prove [itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = [itex]\frac{ad}{bc}[/itex]

    2. Relevant equations

    P 1-12

    3. The attempt at a solution

    [itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = [itex]\frac{ad}{bc}[/itex]

    [itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = (ad)(bc)[itex]^{-1}[/itex]

    [itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = (ad)(b[itex]^{-1}[/itex]c[itex]^{-1}[/itex])

    [itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = (ab[itex]^{-1}[/itex])(dc[itex]^{-1}[/itex])

    [itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = (ab[itex]^{-1}[/itex])(d[itex]^{-1}[/itex]c)[itex]^{-1}[/itex]

    [itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex] = [itex]\frac{a}{b}[/itex][itex]/[/itex][itex]\frac{c}{d}[/itex]



    Also, do proofs have to be in if, then, hence form like they are when Spivak is presenting the basic properties of numbers?
     
  2. jcsd
  3. Nov 3, 2013 #2
    The proof can be in any format as long as your steps are clear.

    Your steps in this proof are clear, and you are correct. However, there are two things for you to consider.

    1. You started with the answer and worked backwards to a statement that had to be true. This is okay as long as all your implications (steps) go in both directions, so that you could have started at the bottom and worked your way up to the top correctly. In fact, the best presentation of this proof is to do that. Start with a known fact or facts and work your way to the conclusion you want.

    2. You took a long way around to do this. Do you remember in 5th grade arithmetic when they taught you things like 3/4 divided by 4/5 is 3/4 x 5/4? Is that not true for every number? So can't you just say ##\frac{a/b}{c/d}## = a/b x d/c = ad/bc (as per your studies in the 5th grade). Or did they explicitly want you to use negative exponents?
     
  4. Nov 3, 2013 #3
    Ok thank you that cleared up a few questions I had concerning proofs.

    Yes, I had thought about doing that but I didn't know if that would be considered "rigorous" enough for a proof. Spivak's text is my first encounter with any rigorous math and I assumed my proof could only be based on the properties presented earlier in the chapter/prologue - Spivak presents division as being defined in terms of multiplication of a negative exponent after listing the property for a multiplicative inverse. That's why I stuck directly to the negative exponents
     
  5. Nov 3, 2013 #4
    Yes, for a course you should always do things the way they tell you.

    I imagine Spivak said said ##x\cdot x^{-1}## (x inverse) has to be 1, so ##x^{-1}## has to be 1/x. He is probably trying to start in on the general idea of inverses.

    You can always ask if you think your book or professor has done something wrong or created a harder way than necessary. It's a good way to learn more.

    That all said, it's not how I would have done it. But then, I never wrote that kind of book.
     
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