Subject: Base Representations

Hi, My Name is Garret Magin I attend a high school in New Berlin, Wisconsin, US. We are doing a lesson on numbers of other bases than 10. We are working with binary, octal, and Hexadecimal. I was wondering what is used to represent number of different bases other then 16? Does it just continue on with the alphabet and if so what happens when you get to Z. It would be a help if you could answer this because it is really bugging me. And none of the math teachers at my school could let me know.

Garret Magin

Hi Garret,

The choice for symbols is actually completely arbitrary. You could, if you wanted, define your own numbering system based on the symbols 0,1,2,3,4,5,6,7,8,9,T,E,W, etc. for (T)en, (E)leven, T(W)elve, etc. The logic and mathematics behind these numeration systems does not change if you pick different letters to represent your digits.
   However, some numeration systems (such as Hexadecimal) are widely used in a number of applications, such as Computer Science. Hence it is useful to use a standardized system such as 0,1,2,...9,A,B,C,D,E,F so that people can communicate their results without having to explain which particular system they are using.
   As a result, if a base-20 system was used, it would very likely use the symbols 0 - J, but there is no "rule" that tells you which symbols must be used. If you got beyond Z you'd probably have to start using different symbols altogether, such as perhaps the Greek or Hebrew alphabet.
   It may seem impractical to use a number system with a base as large as 20 but that is precisely what was done by the Maya. Their sysbols and faces can be found at The Babylonian went even further, they used a sexagecimal notation, that is base 60, way past Z,

Claude and Patrick


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