6 ms·

Fun fact: you can count to hexadecimal on the fingers of one hand. The way it works is that each of your four fingers has four well-defined places on them: the

Fun fact: you can count to hexadecimal on the fingers of one hand. The way it works is that each of your four fingers has four well-defined places on them: the three joints, plus the tip. You use your thumb to point at one of these. The first finger represents 1-4, the second 5-8, the third 9-12, the fourth 13-16. You count up and down by moving your thumb. Also, because you use the thumb of the same hand for pointing, you can count two different hexadecimal numbers if you use both hands --- which means you can count to 256. (I don't know anybody limber enough to do this with their toes.) As a special bonus: if you point to the fleshy pads of the fingers instead of the joints, you can use the same system for base 12.

daeken 10y ago
Counting in binary gets you even further. If each finger is one bit, you can easily count to 2^10-1 == 1023. I frequently use this to keep track of running totals over 10.
- kmill 10y ago
  In my high school orchestra, I'd count the rest measures on my cello's fingerboard using binary. Usually I'd only need four fingers, but occasionally I'd need my thumb as well if the composer mistook the cello for a bass.
- khedoros1 10y ago
  There was a teacher in high school that taught his classes to count in binary on their fingers. It became common for students to flip the double bird at each other in the halls yelling "132!"
- lern_too_spel 10y ago
  This method is impractical to use due to the way the human hand works. Representing numbers where the middle and ring fingers are in different positions is tricky, and using this method for counting quickly is nearly impossible because it requires the simultaneous movement of multiple fingers up and down. The method of counting on finger creases, on the other hand, is entirely practical, and I've known people who have used it for more than 50 years.
  - yorwba 10y ago
    There is quite some variation in the way the human hand works. I can move my middle and ring fingers completely independently, but when I stretch my left middle finger, my left pinky tries to extend, too. I can prevent this by pressing its tip against my palm. Except for this difficulty, simultaneously moving multiple fingers is no problem for me. For binary counting you just need to flip a group of fingers inward and extend the next one. There are instruments that require more difficult movement patterns to play. I just timed myself repeatedly counting to 32 as fast as possible on my left hand, and I averaged at 13 seconds. That seems entirely practical to me.
    - lern_too_spel 10y ago
      Now try counting to 32 using the creases method. You'll beat that time without any practice. I can count on my creases faster than I can count in my head, and it requires little mental effort, which is why you would use an external counter in the first place.
  - thaumasiotes 10y ago
    > using this method for counting quickly is nearly impossible because it requires the simultaneous movement of multiple fingers up and down Simple solution: use the Gray code. :D
ptaipale 10y ago
Or if you point just the joints, not the tips, you get base 12. If you count multiples of 12 with the five fingers in the other hand, you get to 60. And this is the reason we have the 12-hour, 60-minute system, and other sexagesimal systems, ever since ancient Babylonia. https://en.wikipedia.org/wiki/Sexagesimal https://en.wikipedia.org/wiki/Sexagesimal
- hackuser 10y ago
  Coincidentally, I am reading a leading history of mathematics, and just read the section on Mesopotamian math (the source of our base 60 customs, such as time, degrees in circle, etc.). The Sumerians (of Mesopotamia, as was Babylon) not only used base 60, but also were the first known to use position or place value to indicate power of 60 the same way we do in base 10, a significant invention. For contrast, the Romans did not use position: IV is 4, not 15. Thus 11 (imagine Sumerian characters; sorry I don't know my Unicode well enough) in Sumerian is 61 in our contemporary system, and 563 in Sumerian would be 18,363 today. I'm impressed: Could people work with these numbers in their heads? Did someone doing Sumerian addition and multiplication have to memorize 60 x 60 tables, the way we memorized 10 x 10 tables in grade school? (The historian is silent on this matter.) > if you point just the joints, not the tips, you get base 12. / If you count multiples of 12 with the five fingers in the other hand, you get to 60. / And this is the reason we have the 12-hour ... The historian doesn't mention this (and how would anyone know today how the Mesopotamians counted on their hands?), but does say, "There is reason to believe that this choice of 60 rather than 10 as a unit occurred in an attempt to unify systems of measure, although the fact that 60 has many divisors may also have played a role." Other societies commonly have used base 20 (my speculation: maybe natural if you are barefoot or wear sandals) and base 5. I highly recommend the book, the leader in its time (1987) and maybe still today: A Concise History of Mathematics, 4th revised Ed., by Dirk Struik
  - thaumasiotes 10y ago
    > Other societies commonly have used base 20 I've seen this claim several times about French. Apparently it's a standard description of the French number system, but it is totally and obviously wrong on the facts. Here are the characteristics you'd see in a base-20 number system: special words for 0/1/2/3/4/5/6/.../19 (the digits); special words for 20/400/8000/maybe 160000/... (the place values). No special word for 100, which is the not-obviously-significant value "50". No special word for 1000, which is the even-less-significant value "2A0". Here are the characteristics you'd see in a base-10 number system: special words for 0/1/2/3/.../9; special words for 10/100/1000/maybe 10000/... . Here's what French has: special words for 0/1/2/.../16; special words for 10/20/30/40/50/60/80/100/1000/... . It's quite plain that French uses a base-10 number system with some irregular number names. Is there a society that actually used base 20 numbers? I also note, in https://en.wikipedia.org/wiki/Sexagesimal#Usage https://en.wikipedia.org/wiki/Sexagesimal#Usage , that the Sumerians didn't have symbols for 0..59 . They represent large numbers as a series of coefficients of powers of 60, but their "digits" have a base-10 internal structure, precisely as we represent minutes taking one of 60 values today ("6:32"). That suggests to me that the language, as opposed to the number writing system, was probably base 10.

daeken 10y ago

Counting in binary gets you even further. If each finger is one bit, you can easily count to 2^10-1 == 1023. I frequently use this to keep track of running totals over 10.

kmill 10y ago
In my high school orchestra, I'd count the rest measures on my cello's fingerboard using binary. Usually I'd only need four fingers, but occasionally I'd need my thumb as well if the composer mistook the cello for a bass.
khedoros1 10y ago
There was a teacher in high school that taught his classes to count in binary on their fingers. It became common for students to flip the double bird at each other in the halls yelling "132!"
lern_too_spel 10y ago
This method is impractical to use due to the way the human hand works. Representing numbers where the middle and ring fingers are in different positions is tricky, and using this method for counting quickly is nearly impossible because it requires the simultaneous movement of multiple fingers up and down. The method of counting on finger creases, on the other hand, is entirely practical, and I've known people who have used it for more than 50 years.
- yorwba 10y ago
  There is quite some variation in the way the human hand works. I can move my middle and ring fingers completely independently, but when I stretch my left middle finger, my left pinky tries to extend, too. I can prevent this by pressing its tip against my palm. Except for this difficulty, simultaneously moving multiple fingers is no problem for me. For binary counting you just need to flip a group of fingers inward and extend the next one. There are instruments that require more difficult movement patterns to play. I just timed myself repeatedly counting to 32 as fast as possible on my left hand, and I averaged at 13 seconds. That seems entirely practical to me.
  - lern_too_spel 10y ago
    Now try counting to 32 using the creases method. You'll beat that time without any practice. I can count on my creases faster than I can count in my head, and it requires little mental effort, which is why you would use an external counter in the first place.
- thaumasiotes 10y ago
  > using this method for counting quickly is nearly impossible because it requires the simultaneous movement of multiple fingers up and down Simple solution: use the Gray code. :D

ptaipale 10y ago

Or if you point just the joints, not the tips, you get base 12. If you count multiples of 12 with the five fingers in the other hand, you get to 60. And this is the reason we have the 12-hour, 60-minute system, and other sexagesimal systems, ever since ancient Babylonia. https://en.wikipedia.org/wiki/Sexagesimal https://en.wikipedia.org/wiki/Sexagesimal

hackuser 10y ago
Coincidentally, I am reading a leading history of mathematics, and just read the section on Mesopotamian math (the source of our base 60 customs, such as time, degrees in circle, etc.). The Sumerians (of Mesopotamia, as was Babylon) not only used base 60, but also were the first known to use position or place value to indicate power of 60 the same way we do in base 10, a significant invention. For contrast, the Romans did not use position: IV is 4, not 15. Thus 11 (imagine Sumerian characters; sorry I don't know my Unicode well enough) in Sumerian is 61 in our contemporary system, and 563 in Sumerian would be 18,363 today. I'm impressed: Could people work with these numbers in their heads? Did someone doing Sumerian addition and multiplication have to memorize 60 x 60 tables, the way we memorized 10 x 10 tables in grade school? (The historian is silent on this matter.) > if you point just the joints, not the tips, you get base 12. / If you count multiples of 12 with the five fingers in the other hand, you get to 60. / And this is the reason we have the 12-hour ... The historian doesn't mention this (and how would anyone know today how the Mesopotamians counted on their hands?), but does say, "There is reason to believe that this choice of 60 rather than 10 as a unit occurred in an attempt to unify systems of measure, although the fact that 60 has many divisors may also have played a role." Other societies commonly have used base 20 (my speculation: maybe natural if you are barefoot or wear sandals) and base 5. I highly recommend the book, the leader in its time (1987) and maybe still today: A Concise History of Mathematics, 4th revised Ed., by Dirk Struik
- thaumasiotes 10y ago
  > Other societies commonly have used base 20 I've seen this claim several times about French. Apparently it's a standard description of the French number system, but it is totally and obviously wrong on the facts. Here are the characteristics you'd see in a base-20 number system: special words for 0/1/2/3/4/5/6/.../19 (the digits); special words for 20/400/8000/maybe 160000/... (the place values). No special word for 100, which is the not-obviously-significant value "50". No special word for 1000, which is the even-less-significant value "2A0". Here are the characteristics you'd see in a base-10 number system: special words for 0/1/2/3/.../9; special words for 10/100/1000/maybe 10000/... . Here's what French has: special words for 0/1/2/.../16; special words for 10/20/30/40/50/60/80/100/1000/... . It's quite plain that French uses a base-10 number system with some irregular number names. Is there a society that actually used base 20 numbers? I also note, in https://en.wikipedia.org/wiki/Sexagesimal#Usage https://en.wikipedia.org/wiki/Sexagesimal#Usage , that the Sumerians didn't have symbols for 0..59 . They represent large numbers as a series of coefficients of powers of 60, but their "digits" have a base-10 internal structure, precisely as we represent minutes taking one of 60 values today ("6:32"). That suggests to me that the language, as opposed to the number writing system, was probably base 10.