Math CentralQuandaries & Queries


For numbers in bases larger than ten, what are the digits by mathematicians' conventions?

I assume the first 10 are always 0-9. I know two computer science conventions (for hexadecimal (A-F case-insensitively) and for base 85 (case-sensitivity plus 23 other characters)) and as a school student I learned that base 12 uses either T and E (which confused me when I thought they stood for Eleven and Twelve) or A and B, the latter as capitals only. Clearly, these conventions are in conflict.

I saw your reply in as accessed a few minutes ago but hope that in the 10 years since then mathematicians have arrived at a convention. Have they?

Thank you.



There is a base 60 system that we use every day, though the fact is not often noticed: We count in hours of 60 minutes of 60 seconds, which is the Babylonian base 60 version of the metric system with kilometers of 1000 meters of 1000 milimeters.

At the time of the French revolution, there was an attempt to introduce "hours'' of 100 "minutes'' of 100 "seconds'', but people never got used to it, so the system was abandoned. However when we divided the second in smaller parts, we didn't stick to the base 60 system, but used our familiar base ten: Instead of sixtieths of seconds, you'll see tenths, hundredths of seconds in the Olympics, and nanoseconds in physics. Just like the Babylonian kept the old system of dividing the night and day in parts of 12 hours each when they introduces their minutes and seconds.

How does that answer your question?

  1. Even with a pervasive use of base 60 in minutes and seconds, we never found it necessary or convenient to devise special words and characters to represent the digits in that system. The base 10 digits are simple and convenient, so the convention is not likely to change. This being the most common use of a base other than 10 in everyday lives, I think it unlikely that any convention for any base would gain a wide acceptance outside a circle of specialists.

  2. To warm up a soup for a minute and a half, you can enter either 90 or 1:30 on the microwave timer. That is, the words or symbols (like 90) of another base are not incompatible with a given base. Therefore, the "practical'' words or symbol from other numbering systems can always enter our vocabulary. In a sense, the "four scores and seven years ago'' of the Gettysburg address is an example. However, this is quite rare. I don't think anybody will start singing an hexadecimal "The B days of Christmas'' anytime soon.



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