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Every rational number can be written in one way

  1. Feb 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that every positive rational number x can be written in ONE way in form
    x=a1+ a2/2! + a3/3! + ..... + ak/k!
    where a1,a2,...,ak are integers and 0<=a1, 0<=a2<2,.... ,0<=ak<k
    I wrote my solution below. Please check if it is correct and rewrite it for me in a neater way. Thank you!


    2. Relevant equations
    None


    3. The attempt at a solution


    I proved by induction a1 > a2/2! > a3/3! > a4/4!>.... ak>k! when taking a1,a2,a3,..., ak not= 0 ( I started with 2 as base case since a1 has different interval from them). Can I prove it without induction since I don't use the term before?

    Thus I notice that every fraction has a unique interval for a1,a2,a3,...,ak not = 0
    and when it is 0, the fraction is 0 and doesnt add up. Thus it is unique

    I also need to prove that every rational number can be written in this form. I take x=p/q and I add up the fractions to get that q=k! and p=a1 k!+ a2 k!/2! + .... + ak .
     
  2. jcsd
  3. Feb 18, 2012 #2

    HallsofIvy

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    Looks to me like you can just construct it. For any rational number, x, let [itex]a_1[/itex] be the largest integer less than or equal to x. Then [itex]x- a_1< 1[/itex] so that [itex]2!(x- a_1)= 2(x- a_1< 2[/itex]. Let [itex]a_2[/itex] be the largest integer less than or equal to [itex]x- a_1[/itex], etc.
     
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