# Examples of (use of) position-systems

As you now, the most used number system is the position system with base 10, where for instance $101$ means $1\cdot 10^2+0\cdot 10^1 + 1 \cdot 10^0$. Likewise we can define binary number system with base 2, ternary number system with base 3, and so on.

But we can also define non-standard number systems with a position-dependent base (mixed radix), as an example a 2,3,2-system, where we have \begin{alignat} 0 &= 0_2 0_30_2 &= 0\cdot 2^0 \\ 1 &= 0_2 0_3 1_2 &= 1\cdot 2^0 \\ 2 &= 0_2 1_3 0_2 &= 0\cdot 2^0 + 1 \cdot 2 \\ 3 &= 0_2 1_3 1_2 &= 1\cdot 2^0 + 1\cdot 2 \\ 4 &= 0_2 2_3 0_2 &= 0\cdot 2^0 + 2\cdot 2 \\ 5 &= 0_2 2_3 1_2 &= 1\cdot 2^0 + 2\cdot 2 \\ 6 &= 1_2 0_3 0_2 &= 0\cdot 2^0 + 0\cdot 2 + 1\cdot 3\cdot 2 \\ 7 &= 1_2 0_3 1_2 &= 1\cdot 2^0 + 0\cdot 2 + 1 \cdot 3 \cdot 2 \\ 8 &= 1_2 1_3 0_2 &= 0\cdot 2^0 + 1\cdot 2 + 1 \cdot 3 \cdot 2 \\ 9 &= 1_2 1_3 1_2 &= 1\cdot 2^0 + 1\cdot 2 + 1\cdot 3 \cdot 2 \\ 10 &= 1_2 2_3 0_2 &= 0\cdot 2^0 + 2 \cdot 2 + 1\cdot 3 \cdot 6 \end{alignat} and so on. Such mixed-radix systems seems unnatural at first sight, but are not: whenever, in a computer program we use a large array with multiple dimensions, such as an array with dimensions [12,8,5,7] we are implicitely using a mixed-radix number. Other examples are calendar and timekeeping.

So then to the question: I want a (large) list of uses of such mixed-radix numbers, and better yet, list of uses of even other, stranger, non-standard systems for writing numbers.

• I'm not sure if the tag is correct!! anybody can do better? also, should be community-wiki – kjetil b halvorsen Jul 1 '14 at 17:56

## 1 Answer

Once upon a time, some people counted money in pounds, shillings, and pence. I have forgotten what the ratios were, something like 12 and 20.