Division algorithm for R

  • #1

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
mathman
Science Advisor
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Assume that it is not unique, subtract one representation from the other. The resultant equation is obviously false.
 
  • #3
Yeah, but how can I show existence?
 
  • #4
mathman
Science Advisor
7,743
406
Yeah, but how can I show existence?
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
 
  • #5
355
3
or even more straightforward, let k = floor(x/a)
 

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