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Below is a slightly-modified comment I wrote 59 days ago on benefits of Duodecimal (Base-12) over Decimal and Hexadecimal. Sexagesimal takes the benefits of Du

Below is a slightly-modified comment I wrote 59 days ago on benefits of Duodecimal (Base-12) over Decimal and Hexadecimal. Sexagesimal takes the benefits of Duodecimal further by introducing a nice factor of five into the mix, though at the expense of 48 extra digits. Base-12 is a nice sweet spot for having high number of divisors vs the number of digits in the base. ----- I was about to post that in my opinion base-12 is superior to base-10. But someone beat me to it. In a sci-fi novel I'm writing, an advanced alien civilisation uses base-12. As to your question specifically regarding base-16 instead of base-12, it depends. Decimal itself is just a bizarre choice, most likely due to humans having literally ten digits. In decimal we can represent exact fractions of 1/2, 1/5, and 1/10 (without repeated decimals like 0.33333 for 1/3). Counting by fives (and twos) is very easy. But choosing prime factors of 2 and 5 is a strange choice in itself. Why skip 3? Why is it more useful to easily represent fraction 1/5th as 0.2 instead of 1/3rd? How often do we use fifths? Hexadecimal in one sense is easier, all prime factors are two. So we can represent 1/2, 1/4, 1/8, and 1/16 exactly. Duodecimal (Base 12) is very convenient for having a high proportion of exact fractions. Eg - 1/12, 1/6, 1/4, 1/3, and 1/2 can all be represented exactly. I'd argue in everyday use we're more likely to consider 1/3rd of something than 1/5th. Counting by twos, threes, fours, and sixes is easy. Watch, let's count to 20 (24 in decimal) by different amounts. By 2's : 2, 4, 6, 8, A, 10, 12, 14, 16, 18, 1A, 20. By 3's : 3, 6, 9, 10, 13, 16, 19, 20. By 4's : 4, 8, 10, 14, 18, 20. By 6's : 6, 10, 16, 20. And conversely counting to 1 exactly by different fractions. By 1/6th : 0.2, 0.4, 0.6, 0.8, 0.A, 1.0 By 1/4th : 0.3, 0.6, 0.9, 1.0 By 1/3rd : 0.4, 0.8, 1.0 By 1/2th : 0.6, 1.0 Base-12 offers four handy subdivisions (excluding 1) instead of two for decimal or three for hexadecimal. That beats hexadecimal using fewer unique digits. It beats decimal by two using only two extra unique digits. And I think it's these reasons it was chosen for various historical subdivisional units (inches per foot, pence per shilling). The other item to consider is the relative number of unique values per digit. I'm not sure of the utility of having 10, 12, or 16 here. At one extreme, while binary is useful for discretising signals in digital logic, using only zeroes and ones becomes cumbersome for daily use at higher numbers. Once we're at base 10 and higher, I'm not sure how much here extra digits help or hurt.

wahern 9y ago
Decimal itself is just a bizarre choice, most likely due to humans having literally ten digits. I think that's probably a myth and there's little reason to think it's related to the number of fingers on our hands. People who count using body parts to keep track of multiples don't just use their fingers, but their knuckles, too, and sometimes other body parts. Lest you think that would naturally give rise to a power of 10, some extant systems only use the 12 knuckles of your four fingers--index, middle, ring, and pinky. The 10 fingers explanation seems obvious only because of confirmation bias, representativeness bias, etc. Even a cursory survey of appendage-based counting systems makes the number 10 much less likely than we naturally think. Include all historic numeral systems and base 10 disappears into the mix altogether. We get base-10 from the Hindus. Perhaps some ancient Hindu philosopher actually choose 10 because of our 10 fingers. But there's nothing obvious about it. As you amply point out, there's plenty of reasons to choose some other base, and plenty of civilizations did just that.
- jacobolus 9y ago
  We get the strictly positional numeral system from the Indians. But base 10 was also used by people in various other places and some earlier (Greeks, Minoans, Egyptians, Romans, Semites, Chinese, ...) The source for most (all?) base ten systems was almost certainly finger counting, though I'm not sure how useful archaeological / linguistic evidence is about this in many cases.
- jandrese 9y ago
  Isn't knuckle counting and the like just a hack on top of finger counting, and finger counting would naturally lend itself to base 10 in humans. It would take some pretty extraordinary evidence to convince me that we use base 10 for a reason other than starting off with finger counting.
kutkloon7 9y ago
The Babylonians already used a composite-ish system for numbers < 60: They had a different symbol for 10. However, smaller numbers were represented by repeating the unit symbol (which is not very clever). I guess that the least confusing transcript to modern notation would be something like 7, 24 to represent 7 * 60 + 24 = 444

wahern 9y ago

Decimal itself is just a bizarre choice, most likely due to humans having literally ten digits. I think that's probably a myth and there's little reason to think it's related to the number of fingers on our hands. People who count using body parts to keep track of multiples don't just use their fingers, but their knuckles, too, and sometimes other body parts. Lest you think that would naturally give rise to a power of 10, some extant systems only use the 12 knuckles of your four fingers--index, middle, ring, and pinky. The 10 fingers explanation seems obvious only because of confirmation bias, representativeness bias, etc. Even a cursory survey of appendage-based counting systems makes the number 10 much less likely than we naturally think. Include all historic numeral systems and base 10 disappears into the mix altogether. We get base-10 from the Hindus. Perhaps some ancient Hindu philosopher actually choose 10 because of our 10 fingers. But there's nothing obvious about it. As you amply point out, there's plenty of reasons to choose some other base, and plenty of civilizations did just that.

jacobolus 9y ago
We get the strictly positional numeral system from the Indians. But base 10 was also used by people in various other places and some earlier (Greeks, Minoans, Egyptians, Romans, Semites, Chinese, ...) The source for most (all?) base ten systems was almost certainly finger counting, though I'm not sure how useful archaeological / linguistic evidence is about this in many cases.
jandrese 9y ago
Isn't knuckle counting and the like just a hack on top of finger counting, and finger counting would naturally lend itself to base 10 in humans. It would take some pretty extraordinary evidence to convince me that we use base 10 for a reason other than starting off with finger counting.

kutkloon7 9y ago

The Babylonians already used a composite-ish system for numbers < 60: They had a different symbol for 10. However, smaller numbers were represented by repeating the unit symbol (which is not very clever). I guess that the least confusing transcript to modern notation would be something like 7, 24 to represent 7 * 60 + 24 = 444