# Some theorems regarding decimal representations

1. Apr 27, 2004

### Ed Quanta

I have to prove the following, and while I understand why the following is true, and I am not sure how to begin writing it out

Let m.d1d2d3... and m'.d1'd2'd3' represent the same non-negative real number

1)If m<m', then I have to prove m'=m+1 and every di'=0 and di=0

2)If m=m' and there is such i that di does not equal di' then we let N=least element of {i/di does not equal di'}. If dN<dN' then dN'=dN + 1, di'=0 for all i>N, and di=9 for all i>N.

Once again, I understand why this is true simply because of the nature of 1=.9999... and therefore there being at most two decimal representations of any number, yet I am not sure how to go about proving such a statement.

2. Apr 27, 2004

### NateTG

First off, you should be able to get both of them in a single proof -- they're essentially the same.

Consider that the difference of the numbers is zero.