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How do we define division?

  1. Sep 28, 2003 #1
    Multiplication is defined as repeated addition.

    3x5 = 5+5+5

    How do we define 10/2?
     
  2. jcsd
  3. Sep 28, 2003 #2
    Repeated subtraction?
    10-2-2-2-2-2=0 5 equal parts.

    10/3
    10-3-3-3=1 3.3333333333333 parts. 3 and 1/3
     
  4. Sep 28, 2003 #3

    HallsofIvy

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    You also posted this under "general mathematics" where Hurkyl pointed out, correctly, that multiplication is NOT defined as "repeated addition".
     
  5. Oct 8, 2003 #4
    Reminds me of a truly awful joke . . .

    Q: How many times can you subtract 5 from 21
    and what do you have left?

    A: I can subtract 5 from 21 as many times as I like,
    and I always have 16 left.
     
  6. Oct 10, 2003 #5
    If A/B = C, then I would define C as the the number of times that you have to subtract B from a quantity that starts out at A until you get to 0. Hence, 10/5 = 2, because you have to subtract 5 twice from a quantity that starts out at 10 in order to get 0, and 0/0 is undefined, because you always have 0, no matter how many times you subtract, and 10/0 is undefined because there is no answer(There is no amount of times that you can subtract in order to arrive at zero--"no amount of times" is NOT the same as "zero times", because zero is an amount of times; you have an empty set, as opposed to a set with an element 0).
     
  7. Oct 10, 2003 #6

    hypnagogue

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    What about something like 1/3? How many times do you have to subtract 3 from 1 to get 0? The answer is "1/3 of 3" times, but the answer here using the above formulation doesn't get us any closer to a meaningful answer than the initial question. It's circular.
     
  8. Oct 15, 2003 #7
    we define division of two integer numbers a and b as follows a/b is
    a=b(q)+r
    for some integers q and 0<= r< b. Here q and r are uniquely deteremine. 28/5 is the same thing as 28=5(5)+3 If you want to get fancy schmancy. Take the integers Z and since Z is an integral domain define the field of rationals Q as all the numbers(quotients) that satisfy the following equation for x

    xm=n

    all solution are n/m where m and n are integers, here you have the field of quotients or Q
     
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