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Division algorithm for R

  1. Apr 17, 2008 #1
    How might I prove a Division Algorithm for the Real numbers?

    That is to say, if [tex]x, \alpha \in \mathbb{R}[/tex], then [tex]x=k \alpha + \delta[/tex] for some [tex]k \in \mathbb{Z}, [/tex] [tex]\delta \in \mathbb{R}[/tex] with [tex]0 \leq \delta < \alpha[/tex] where [tex]k, \delta[/tex] are unique.
     
  2. jcsd
  3. Apr 17, 2008 #2

    mathman

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    Assume that it is not unique, subtract one representation from the other. The resultant equation is obviously false.
     
  4. Apr 18, 2008 #3
    Yeah, but how can I show existence?
     
  5. Apr 18, 2008 #4

    mathman

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    The sequence na for n=1,2,... is unbounded. Therefore for some n, na>x. Find lowest bound, subtract 1 and you will have k. ka<=x, (k+1)a>x, so x-ka(remainder)<a
     
  6. Apr 18, 2008 #5
    or even more straightforward, let k = floor(x/a)
     
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