Talk:Origin

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With regards to your comment "My mind wandered to base6 and base12 because sometimes it's nice to divide things into thirds... but again, they don't give compact representations of binary numbers.", I'd like to reply that dyadic fractions in dozenal (base twelve), even though of course not as compact as in a dyadic base such as hexadecimal, are still much more compact than in decimal. Compare:

  • 1/2: dec 0.5, doz 0.6, hex 0.8
  • 1/4: dec 0.25, doz 0.3, hex 0.4
  • 1/8: dec 0.125, doz 0.16, hex 0.2
  • 1/16: dec 0.0625, doz 0.09, hex 0.1
  • 1/32: dec 0.03125, doz 0.046, hex 0.08
  • 1/64: dec 0.015625, doz 0.023, hex 0.04
  • 1/128: dec 0.0078125, doz 0.0116, hex 0.02
  • 1/256: dec 0.00390625, doz 0.0069, hex 0.01
  • 1/512: dec 0.001953125, doz 0.00346, hex 0.008
  • 1/1024: dec 0.0009765625, doz 0.00183, hex 0.004
  • 1/2048: dec 0.00048828125, doz 0.000A16, hex 0.002
  • 1/4096: dec 0.000244140625, doz 0.000509, hex 0.001

As you can see, decimal adds one digit for every iteration, quickly leading to cumbersome lengthy strings of digits. Hexadecimal does it for every four iterations, greatly reducing the fuss indeed. But dozenal, which adds one digit for every two iterations, doesn't perform anywhere nearly as bad as decimal. Also, the number of significant digits grows like crazy in decimal (9 significant digits —244140625— needed for the fraction 1/4096), whereas in dozenal the number of significant digits required is much more reduced (3 significant digits —509— for the same fraction), although of course hexadecimal beats both in this respect with only 1 significant digit needed for any dyadic fraction. Overall, it is true that dyadic fractions are not as compact in dozenal as in hexadecimal, but they are still decently compact, being nowhere near as annoying as in decimal, plus as you mentioned dozenal supports many more useful everyday fractions which hexadecimal can't finitely cope with (thirds, sixths, ninths, twelfths...). 213.37.6.23 13:05, 9 October 2007 (UTC)