# In arithmetic, what is the generic name for a number written in an arbitrary base?

A number can be written in various bases. We commonly use base 10 and the resulting number is called decimal. If it is written in base 2 then it is called binary. A list of over 100 such names, as in binary, ternary, octal, decimal, hexadecimal, etc., are given in Wikipedia. I am looking for their generic name. In math books they sometimes use "n-ary", but I was wondering if there is another name.

Examples:

1- An example of technical use of "n-ary" in Wikipedia. Here it is recorded as n-nary.

2- Binary numbers are long strings of zeros and ones, and as the base increases the resulting "n-ary" number becomes somewhat of a shorter string but then the digits themselves become long cumbersome numbers or symbols. So, for example, when 267,401 is written as binary it becomes 19 digits long, i.e. 1000001010010001001, then in base 10 it is 6 digits, and that is a significant decrease in length, but there is not much gain in succinctness as we increase the base, for example in base 70 that number becomes 54:40:1.

(This example is rather artificial, but nevertheless, the emphasis is on the changing the base of the number. So the adjective "n-ary" is significant.)

Perhaps a candidate is arity.

• Can you give an example sentence? What's wrong with number? Are you specifically referring to numbers in positional notation (excluding Roman numerals, etc)? The word numeral is often used for a representation of a number, but some people use it specifically for a single digit. Commented Jun 27, 2022 at 16:08
• @StuartF An example is now included. The word "n-ary" will be a significant adjective in "n-ary number" and using plain "number" will not be clear. Yes, I am only interested in positional notation. Commented Jun 27, 2022 at 16:40
• total tangent, but the loss of "succinctness" comes from using decimal digits to represent higher-base numbers, which means you need mulitple digits to represent one value. If you were writing base-10 by separating place values by `:` and writing individual numbers in binary, you would end up in a much less "succinct" place. Commented Jun 27, 2022 at 18:15
• @Esther Point well taken. As stated, that paragraph is rather artificial. Folks use a-z or A-Z to extend digits from 10 to 35 and do away with : for a while. But then you need to remember that for example m stands for 22 etc. I'll try to find a better example. Commented Jun 27, 2022 at 20:08