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(Motivation: I am going to be working with a high school student next week on long division, which is a subject I strongly dislike.)

Consider: $\frac{1110}{56}=19\frac{46}{56}$.

This is really a super easy problem, since once you realize $56*20=1120$ its trivial to write out $1110=56*19+46$.

You can work out the long division for yourself if you want; needless to say it makes an otherwise trivial problem into a tedious, multi-step process.

Long division is an "effective procedure", in the sense that a Turing machine could do any division problem once it's given the instructions for the long division procedure. To put it another way, an effective procedure is one for which given any problem of a specific type, I can apply this procedure systematically to this type of problem, and always arrive at a correct solution.

Here are my questions:

1) Are there other distinct effective procedures for doing division problems besides long division?

2) Is there a way to measure how efficient a given effective procedure is for doing division problems?

3) Does there exist an optimal effective procedure for division problems, in the sense that this procedure is the most efficient?

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    I would certainly try to figure out the greatest common factor of the numerator and denominator first...2010-12-30
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    @Matt: I would suggest breaking him out of the habit of writing $19\frac{46}{56}$; it is just too easy to misinterpret it as a product. Either $19+\frac{46}{56}$, or stick with improper fractions.2010-12-30
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    @Arturo: Oh, you hate mixed numbers too? I had thought I was championing a lost cause... :D2010-12-30
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    @J.M.: I keep telling my calculus students to break out of the habit. Not a semester goes by that at least three of them make the mistake somewhere along the lines of multiplying the integer by the fraction.2010-12-30
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    @Arturo: That won't beat a (former) student of mine who once asked if you can do something like mixed numbers for the quotient of two *polynomials*... \*shudders\*2010-12-30
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    @J.M. Certainly, you win.2010-12-30
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    @J.M.: Writing a quotient of polynomials (rational expression) in the form polynomial + remainder/divisor, which is sort of the mixed number analog in polynomials, is useful for pulling apart the end behavior versus the strange (discontinuous) behavior in the middle of the graph.2010-12-30
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    @Isaac: Certainly, but that student was asking if the "+" was absolutely needed since one could just jut together the polynomial and rational function *a la* mixed numbers... e.g. expressing $\frac{x^3+2x^2+3x}{x+1}$ as $(x^2+x+1)\left(\frac{x-1}{x+1}\right)$, to which I replied "how sure are you that you won't be confusing that with multiplication when you're in the middle of a long derivation?"2010-12-30
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    Just so you know I wasn't looking for help with my student! I hope at this point you guys realize I can handle something as trivial as teaching division =) I actually have worked out several different procedures, including one I believe to be optimal. I could have kept working things out on my own, but I thought it was a nice problem that people might find interesting.2010-12-30
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    @Matt: no worries, this is a very good question. When more people wake up and check the site, you'll probably get more upvotes.2010-12-30

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