Some theorems regarding decimal representations

[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.

 

[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.

 

[M98Ranger]

To prove these theorems, we will use the definition of a decimal representation of a number. A decimal representation of a number is a way of writing a number using the digits 0-9, with a decimal point separating the whole number part from the decimal part. For example, the number 3.14159 can be represented as 3.14159.

1) If m < m', then we can write m' as m+1. This is because m' is a larger number than m, and the difference between them is 1. Therefore, the decimal representation of m will have one less digit than the representation of m'. This means that all the digits after the decimal point in the representation of m will be 0, as they do not exist in the representation of m. Similarly, in the representation of m', all the digits after the decimal point will be 0, as they do not exist in the representation of m+1. Therefore, every di'=0 and di=0.

2) If m = m' and there is an i where di does not equal di', then we can write m as m'+0.000...0, where there are n zeros after the decimal point. This is because m=m' and there is at least one digit where di does not equal di'. This means that the representation of m will have at least one digit that is different from the representation of m'. Let N be the least element of {i/di does not equal di'}. This means that all the digits after the decimal point in the representation of m will be equal to the digits after the decimal point in the representation of m' up to the Nth digit. After the Nth digit, the digits will be different. For example, if m=3.14159 and m'=3.14169, then N=5 and d5=5 and d5'=9.

If dN<dN', then we can write dN' as dN+1. This is because dN is smaller than dN', and the only way to make them equal is by adding 1 to dN. Similarly, all the digits after dN' will be 0, as they do not exist in the representation of m'+0.000...0. Therefore, di'=0 for all i>N.

On the other hand, di=9 for all i>N. This is because we are adding