# How to check that my number representation is consistent?

I'm experimenting with a positional numeral system in which I do not use Zero in representing numbers. I have a separate symbol for Zero and I introduced a new symbol for the base of the system.

For an example I take the current base 10 representation and modify it as so:

1. The symbol for $\ 0_{10}$ remains $\ 0_Y$
2. I introduce new symbol for the number $\ 10_{10}$: $\ P_{Y}$
3. The digits $\ [1_{10}; 9_{10}]$ remain with the same meaning and symbols
4. The symbols that I use to represent any number in a unique way are: 1, 2, 3, 4, 5, 6, 7, 8, 9, P

With that in mind I will list a few examples below. Every line contains one expression in the current numeral system, followed by a semicolon and the same expression in my system:

1. $\ 10_{10} + 1_{10} = 11_{10}; P_{Y} + 1_{Y} = 11_{Y}$
2. $\ 19_{10} + 1_{10} = 20_{10}; 19_{Y} + 1_{Y} = 1P_{Y}$
3. $\ 20_{10} + 1_{10} = 21_{10}; 1P_{Y} + 1_{Y} = 21_{Y}$
4. $\ 99_{10} + 1_{10} = 100_{10}; 99_{Y} + 1_{Y} = 9P_{Y}$
5. $\ 100_{10} + 1_{10} = 101_{10}; 9P_{Y} + 1_{Y} = P1_{Y}$
6. $\ 109_{10} + 1_{10} = 110_{10}; P9_{Y} + 1_{Y} = PP_{Y}$
7. $\ 110_{10} + 1_{10} = 111_{10}; PP_{Y} + 1_{Y} = 111_{Y}$

I want to know if there are any logical errors in my representation.

I tried a few examples and they all work fine. Is there anything else that I should do before I can declare that my system is consistent?

Thank you for your time and effort

• How should I represent 101 in your numeral system? – didgogns Feb 19 '17 at 6:01
• It's very hard to deduce what your numbering system is, especially since part of the question is whether you made any logical errors. You could start by giving a complete list of all single-digit values, then explain how we can tell the value of a $1$ if it is $n$ digits to the left of the rightmost digit, then describe in every difference between your number system and an ordinary base-$b$ system by mathematical rules that can be applied to every situation, not just by showing a handful of examples. Use Mathjax to write subscripts on the numbers so we can know which are base ten. – David K Feb 19 '17 at 6:18
• As a guess, it appears to me that $100_{10} = 9P_{\mathrm{Yordan}},$ not $PP,$ but it's hard to tell since you did not explain in enough detail how your system is supposed to work. – David K Feb 19 '17 at 6:19
• You are right - Im sorry. Ill edit the question with more details. @David K is right - I made a mistake. – Yordan Feb 19 '17 at 6:48