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.
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.
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!"
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.
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.
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.
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.
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
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 , 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.
> I've seen this claim several times about French.
Who said it was about modern French?
French has been heavily influenced by different cultures. Some apparently used full or partial vigesimal systems such as in Basque, Breton, Normans, maybe Gaulish too. And some others like the Roman, Saxon were mostly decimal systems.
There are examples in French literature of the Middle Ages of numerals based on twenty (30 was (20+10) "vingt et dix", 40 was (2×20) "deux vingt", 50 was ((2×20)+10) "deux vingt et dix", 60 was (3×20) "trois vingt", etc).
One of the most famous vestigial example is the "Hospice des Quinze-Vingts" (15×20) founded in 1260 in Paris and was intended to house 300 patients.
That system was progressively replaced by decimal system at the end of Middle Ages, and most of the etymology can be now related to latin and a bit of greek.
Still currently in France numbers from 80 to 99 seems to follow the old system : 80 is (4×20) "quatre-vingt", and 90 is (4×20)+10 "quatre-vingt dix".
Note that I didn't say anything about France. You seem very doubtful that any society would use something besides base 10; why?
> Is there a society that actually used base 20 numbers?
From Struik, the book mentioned above, here is what I could find quickly:
Of 307 number systems of primitive American peoples investigated by W.C. Eels, 146 were decimal, 106 quinary [base 5] and quinary decimal, vigesimal [base 20] and quinary vigesimal. The vigesimal system in its most characteristic form occurred among the Mayans of Mexico and the Celts in Europe.
Give the book a try if you are interested in these things; he writes clearly and engagingly, with a personality and sophistication; the short factual passage above isn't representative. With due respect, I'm going to rely on the highly-regarded writing of arguably the leading historian in the field over Wikipedia. None of these number systems map directly to ours; I've omitted plenty of complexity (and Struik probably omitted more).
> Note that I didn't say anything about France. You seem very doubtful that any society would use something besides base 10; why?
That's all you. I'm not doubtful that a society would use something besides base 10, but when you make the same claim that a lot of other people make, you risk being perceived as similar to those other people. The form of my comment is "I hear this same claim a lot from people who clearly don't know what they're talking about. Do you have something better than they have?"
> The vigesimal system usually occurs in connection with some other system.
> Quinary-vigesimal. This is most frequent. The Greenland Eskimo says "other hand two" for 7, "first foot two" for 12, "other foot two" for 17 and similar combinations to 20, "man ended." The Unalit is also quinary to twenty which is "man completed." But 40 is "two sets of animals' paws," 60 "three sets of animals' paws" and so on regularly to 400 where there is an interesting change in the formation of this primary base (20 X 20) from animals back to man, for 400 is "20 sets of man's paws."
A list of nahuatl numbers can be found here ( http://www.omniglot.com/language/numbers/nahuatl.htm ), and it is quinary-vigesimal. Maya numerals, in their symbolic written form, are also quinary-vigesimal (in the same sense that written Sumerian numerals are decimal-sexagesimal -- that is to say, they are purely quinary, but the forms 20 to 24 aren't used), and Celtic numerals (the words) seem to be a weird quinary-decimal-vigesimal hybrid. This doesn't fill me with confidence in Struik. But Celtic aside, I am satisfied that several American number systems were vigesimal and not decimal.
>> I also note, in 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
> With due respect, I'm going to rely on the highly-regarded writing of arguably the leading historian in the field over Wikipedia
If you looked at the link, you might notice that all I was relying on was pictures of the cuneiform glyphs. But hey, why read when you can just generally slam wikipedia?
I'm not sure what inspired these somewhat angry responses. I'm just discussing ancient mathematics, which is not a hot button issue. Sorry if I offended you somehow.
> It's quite plain that French uses a base-10 number system with some irregular number names
Well, it has two interesting irregularities on two of the multiples of 10 that are the origin of the base-20 theory: 80 and 90. They are "quatre vingt" and "quatre vingt dix", literally "four twenty" and "four twenty ten". 70 is also "soixante dix", literally "sixty ten".
Your phrasing is quite a bit stronger than your source's:
> It is possible for people to count on their fingers to 12 using one hand only, with the thumb pointing to each finger bone on the four fingers in turn. A traditional counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60
It was a long time ago; we don't actually know why the Mesopotamians liked cycles of 60, and it's unlikely that we ever will. But the argument "it's easy to divide 60 by 2, 3, 4, 5, and 6" is a lot more parsimonious. We do know that the Romans, despite using a base-10 number system, had money which divided into 12, not 10, subunits, precisely to make money easier to divide. 12 and 60 are technologically superior to 10, and this can explain their use even in societies which we know used base-10 numbers. (Such as... the Mesopotamians, who, as shown in your link, represented the value 47 as 4*10+7.)
The bibi-binary notation is somewhat similar (16 === 2^2^2). It was patented in 1968 by Bobby Lapointe, a French variety singer, but didn't get much traction AFAICT. Bibi-binary also has a custom pronunciation for hex digits.
The article is in French[0], but the main illustration that describes the system is self-explanatory[1].
The shapes are all able to be written in a way that visits points in the corners of the square in the same order (UL, LL, UR, LR). So 0111 and 1110 can be written with the skewed V shape because the 1's are in sequence, while 1011 and 1101 are written with rotated L shapes because they skip one of the middle bits. (Compare also 1111, a backwards N shape).
It could also interestingly be mixed with reflected binary code [1] with only one segment changing between two successive values.
> Theoretically, the Unicode Consortium could decide to add the Birkana symbols to the Unicode specification and some enterprising font designer could come up with a set for general use.
There are already some runes encoded in unicode, but most of the symbols of the birkana system does not match any existing historical runes though.
Yet, there is something in unicode that is somehow visually close [2]: braille patterns (2 columns of 4 rows of dots).
But sadly the braille dots are not ordered in the same way because it was extended from 6 bits (from top to bottom : left column 123, right column 456) to 8 bits (from top to bottom : left column 1237, right column 4568).
What I find the neatest is that it's not just showing hexadecimals, but the binary representation at the same time. In the Birkana symbol for 9 it's very easy to recognize the binary 1001: it has the top and bottom 'bits' set.
It also made me aware that I never thought about the symbols we use for numbers. Our digits 0 through 9 suddenly seem very arbitrary, looking at Birkana. Though ten digits are more difficult than sixteen to represent this way, they could certainly have had logic in them.
As for the logic behind our decimal digits, there was an explanation going around for it, but at best it's probably reaching very hard. I'll let you decide: http://message.snopes.com/showthread.php?t=49183
Once upon a time, the digit for 0 was a dot, the digit for 1 was (close to) ι, the digit for 2 was (close to) μ, the digit for 3 was close to w with a leading descender like μ has. The digit for 4 might have been closer to a + sign that eventually morphed to look more like a backwards 2. It was the number of vertical lines which encoded 1, 2, or 3; with 4 and 5 and 6 there was more need to come up with more-abstract symbols.
In Arabic these first 4 digits have actually survived relatively unmolested, ٠ - ١ - ٢ - ٣, while the backwards 2 for 4 started hooking down to become a ٤ and some other stuff happened with the next 5 digits.
In English the digits for 2, 3, and 4 all rotated about 90 degrees counterclockwise while the tail disappeared for the 3 and the 4 didn't close up on the top until relatively late.
The symbols themselves neatly match the binary representation of 4 bits, so you can look at the numbers and see them as binary numbers in groups of four if you wish.
These would be great hexadecimal symbols for programmers.
I get what the parent is saying, though. It's just binary digits on a weird looking vertical number line. Each part of the "digit" may as well be a zero or a 1, because that's all it represents. Position tells us what the value is, 2^0, 2^1, 2^2, or 2^3, no need for different symbols. And while we're at it, let's write it left-to-right MSB first, because that's how other numbers work, and it fits on a line nicely, like so: 0101.
For fun, I have designed a font like that once. Each hex digit was 4x4 grid, and there were (visually distinct and symmetric if possible) patterns of 0-15 filled dots.
I also thought about an extension for having a symbol for each byte value in 16x16 grid. The lower half byte would determine a 4x4 pattern, which would be repeated over the 16x16 grid according to the pattern corresponding to the higher half byte.
The point is that as you get used to reading them, you will recognise the complete glyph as a hex digit, without puzzling over the binary details. Just like reading any other script.
This is a good point, but you get the same from standard Hex notation.
But standard Hex notation has the advantage that I can say FF to someone, but how do I pronounce these things? I guess this could have merit, but I just don't have much of an issue using standard Hex notation.
Easier for us because it's the conventional way hexadecimal was represented in computer history, and how we learned it.
That rune-like system is more "logical" in the way you can visually read the value by summing the segments/bits value composing the berkanam rune from least significant at top to the most significant at bottom.
> AMONG number systems, the hexadecimal system of counting (or 'radix') has a special place in the hearts of programmers, being closely related to binary, the fundamental number system used by all modern computers.
Could somebody explain how hexadecimal is more closely related to binary than any other base is? My only conclusion is that one hexadecimal digit is equivalent four binary digits.
I would say your conclusion is right. Some bases don't really make sense to use in conjunction with binary (5, 7, 10, etc.) but any power of two "works". I prefer hex to octal because a byte splits nicely into two hex digits.
Octal is 3 bits per digit, hexadecimal is 4. Either was / is widely used wherever you need to interact with bits (eg embedded development). But 10-based cannot be easily mapped to binary, so does not work as well.
I think there is a problem with this notation - it is not self aligning. Meaning that if you hand wrote these you could then easily mis-read the values. It would probably be okay in printed form or if you were using lined paper, using the ruled line as a centre point.
I think its quite easy to judge whether a downwards accent starts at the top of the line or at the middle, similarly for the upwards accent if it starts at the middle or at the bottom.
I may have made a mistake in assuming that the vertical line, because it was drawn in a different weight, was not part of the symbol. Without the line 1 & 4, 2 & 8 and 3 & 12 would rely on the vertical relationship with other symbols.
How would you misread the values? I suppose the symbols for 0x1/0x4 or 0x2/0x8 could be written sloppily enough to be confused, but it's no worse than trying to distinguish sloppily-written arabic numeral symbols. In fact, I suspect that these would be significantly easier to distinguish.
I've also seen various notation for writing numbers in hex used by some reversers/crackers/demosceners mostly from Eastern European countries, who probably used hex enough to develop something similar to a stylised form of 0-F but distinctly different from those letters/numbers for easier reading and writing. (This was when home computers were still relatively new, and it was usual to make great amounts of handwritten material when working with them.)
I had reinvented these symbols, without knowing Birkana, for one of my hobby alphabets at high school. I had given a lot of thought with my high-schooler leisure time and haven't been able find more simple system with enough data capacity. I think readability suffers from similarity though.
it would be nice, just generally, not to introduce another character which is just a plain vertical line, even if 99.9% of the time it is identifiable by context
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.