Patrick (patrickwonders) wrote,
Patrick
patrickwonders

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Over-specified number bases...

Isaac and I play a ton of Lego Star Wars on the PS2. I got sick of constantly answerring, Do you have more money than me? So, I sat down with Isaac and taught him how to tell if one (non-negative) integer is bigger than another. All of this went fairly well. There were still some kinks after the first fifteen minute session (sometimes something like: 333,333,343 vs. 333,333,335 would catch him), but he got it nailed down during the second session.

Anyhow, this got me to thinking about number bases. One of the features of decimal, hexidecimal, octal, ternary, binary, balanced ternary, etc. is that each integer can be written in only one way (as an integer... cuz 3 == 2.99999999...).

Not all numeric systems have this property. Unless you're very strict, IIII and IV both represent four in Roman numerals. If one took some liberties with Binary Coded Decimal (BCD), one could specify the number ten as either the bit string 0001 0000 or the bit string 0000 1010. There are three ways to represent the number fifteen on a standard abacus:

  • [[xx | xxxxx] [ xx|xxxxx ]]
  • [[xx |x xxxx] [x x| xxxxx]]
  • [[xx |x xxxx] [xx |xxxxx ]]

Note: each column of the abacus can take on values from zero through fifteen making the abacus really hexadecimal coded decimal.

For the abacus, the goal is to save you from having to carry while you're in the middle of doing everything else. You can go through your basic addition and then sort out the carries in a second pass. For Roman numerals, there was not really any sort of goal as near as I can tell. And, BCD would try to deny that there really is any extra room in there.

All of that said, we do mix bases all of the time with dates and times and coins and paper money and English measurements and so on. We say 75 minutes or an hour and fifteen minutes. We say three feet five inches or 41 inches.

This got me wondering... if the Gettysburg address had been ten years later would Lincoln have said, Four score and seventeen? or does the score remainder have to stay small? Would he have switched to eight dozen and one?

Anyhow.. maybe we could benefit from adding a few more digits into decimal as the abacus did.

Tags: math, numbers
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