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Division Algorithm proof explanation

  1. Jan 7, 2013 #1
    This is taken from "Passage to Abstract Mathematics," Watkins and Meyer.

    "Theorem 2.3.9 (Division Algorithm) If [itex]a \in\mathbb{Z}[/itex] and [itex]b\in\mathbb{N}[/itex], then there exist [itex]q,r \in\mathbb{Z}[/itex] such that [itex]a = qb+r[/itex] and [itex]0 \leq r < b[/itex].
    Furthermore, for each [itex]b\in\mathbb{N}[/itex], this representation of [itex]a[/itex] is unique.


    Proof.
    Let [itex]a\in\mathbb{Z}[/itex] and [itex]b\in\mathbb{N}[/itex] be given, and let [itex]S = \{ a + xb : x\in\mathbb{Z} \wedge a+xb\geq 0 \}[/itex].

    Note that there is an element in [itex]S[/itex]. (If [itex]a\geq 0[/itex], pick [itex]x=0[/itex], so that [itex]a\in S[/itex]; if [itex]a<0[/itex], pick [itex]x=-a[/itex], so that [itex]a-ab = a(1-b) \in S[/itex].) Thus, by the Well Ordering Principle, [itex]S[/itex] has a least element, which we denote by [itex]r[/itex]. Since [itex]r \in S, r=a-qb[/itex] for some [itex]q\in\mathbb{Z}[/itex].
    By definition, [itex]r \geq 0[/itex].

    Suppose that [itex]r \geq b[/itex]. Then,
    [itex]a - (q+1)b = a - qb - b = r - b \geq 0[/itex],
    and [itex]a-(q+1)b = r-b<r[/itex], since [itex]b>0[/itex]. This gives an element [itex]r-b\in S[/itex] which is strictly less than [itex]r[/itex], contradicting the minimality of [itex]r[/itex]. Thus [itex]r<b[/itex]. This yields the representation [itex]a=qb+r[/itex] with [itex]0\leq r < b[/itex]..."​

    (The rest of the proof concerns the uniqueness of the representation, which I understand somewhat.)

    I was hoping someone could help explain the red text. My main questions are
    • What is the advantage to using the set [itex]S[/itex]? Or, in other words, why is this set used?
    • What is the "Note that ... (If [itex]a\leq 0[/itex]...)" saying? I don't really see what's going on here.
    • How do we know that [itex]r=a-qb[/itex]?

    I think once these questions are addressed I'll be better equipped in understanding the proof overall. Thanks!
     
  2. jcsd
  3. Jan 7, 2013 #2
    The bit in the parentheses is establishing that the the set S is nonempty. This needs to be known so that the well ordering principle can give us an element r to work with.

    We know that r = a-qb since r is an element of S. You may be thinking "doesn't S have elements that look like a+xb?" It does, but we can put x = -q and have our desired representation.

    I think that proof goes through every detail, and if I were you, I'd stare at it until it makes sense.

    If you're not convinced that you need to use a set like S, try to write the proof without it. (I'm not suggesting you won't be able to)
     
  4. Jan 10, 2013 #3
    So I've been looking over this proof and throwing bits of it around in my head for a bit. I've got two more questions:

    • Since [itex]a + xb \geq 0[/itex], don't we already know that [itex]0\in S[/itex], or is that only if we're specifically told that [itex]a=0[/itex]?
    • Where does the [itex]a - (q+1)b[/itex] come from? I don't understand where [itex]q+1[/itex] comes from, specifically.
     
  5. Jan 10, 2013 #4
    1: Let's look at the case when a=7 and b=2. Then a+xb = 7+2x. Keeping in mind that x is an integer, we see that there's no x that makes 7+2x = 0. So, in this case, 0 is not in S. Ask yourself when 0 will be in S.

    ----------

    2: Still with a=7 and b=2, the division algorithm gives us 7= (3)2 + (1).

    Let's say that you're fiddling around with this (without knowing the division algorithm). Maybe you stumble upon the expression 7 = (2)2 + (3) [note that here q=2 and r=3]. Now, 3 is not less than 2, so this seems to go against the division algorithm. Is this division algorithm wrong or can we fix it?

    Well, we can just bump up q to 3 (=2+1). This is all that part of the proof is doing. It's establishing that we can find an r between 0 and b.
     
  6. Jan 10, 2013 #5

    haruspex

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    [itex]0\in S[/itex] if and only if [itex]\exists[/itex] x s.t. a+xb=0. Clearly that will only be the case if b divides a.
    r is defined as the least element in S. q is then defined in terms of r, q+1 in terms of q, and finally a - (q+1)b in terms of that. The result is to show that if r >= b then we can construct a smaller member than r.
     
  7. Jan 11, 2013 #6
    Okay, I see that now. I also reread the beginning and noticed that the proof starts off with letting a being given, so I think that answers my question.

    Ah, alright. I see what's going on now, but I still need some time to digest all this information and put it all together. Thanks to both of you for the help!
     
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