Roman Finger Counting | Hacker News Reader

Roman Finger Counting

tookmund.com

180 pointsJNRowe86 comments

Cool! I've noticed a similar way of counting that's still quite common today in China. They point with their thumb to each segment of their four fingers to count up to 12 on one hand. The other hand tracks the number of 12 counts, which lets you keep track all the way up to 144 with both hands.

Not quite on the level of the Romans, but still a solid improvement over how I've done it my whole life!

And the Babylonians, whose base-60 system's legacy lives on in our timekeeping, used the finger segments on one hand and entire fingers on the 2nd hand to count to 60.

Yeah, I use that one when I need to hand-count somewhat large quantities. The Roman system is better (larger numbers), but I can't independently bend my pinky, ring or middle fingers, so it's unusable sadly.

forgive my question, I am probably missing context and I am curious: why are you hand counting large quantities?

I mean, why do you keep it on your fingers rather than just counting out loudly, or possibly just keeping on the fingers numbers up to ten repeatedly for "double checking"?

I know there are clicker tally counters which can be useful for e.g. counting cattle or people on a plane, but counting up to 60 seems feasible in your mind.

Not parent, but back when I was young and couldn’t afford a sports watch, I kept track of distance while running by counting steps. I would count up a hundred step pairs in my head, and increment a counter on my fingers on each hundred. (I also used a simple system of my own devising to allow counting up to 99 on my fingers.)

I one little sentence, you answered a lifelong question. Why 60?

Thank you!

60 has nice properties, in that it is evenly divisible by 2,3,4,5,6,10,12,15,20,30. Helpful as a base since you can divide it into smaller units easily (monetary, measurement, etc).

Funny i just started reading(half way through, its a hard book to read) Micheal Hudson book "...And forgive them their debts" talks about bronze age economics etc and debt forgiveness and how their number base was usually taken for interests calculations.

The Babylonian Base-60 also obviously survives in a lot of circle measurements: degrees, minutes:seconds. While the metric system has mostly moved on to radians, metric proponents (and esp. the French Revolution) failed to find a base-10 time system that people could agree on that matches the convenience of Base-60 minutes:seconds.

Which arguably suggests we didn't settle on base 10 because we have 10 finger as seems to be often told. We settled on base 10 likely because of politics (in the broader sense of the word)

I imagine there's a natural gravitational pull toward base 10 from having fingers, and throughout prehistory and early civilization occasionally systems deviated to suit certain purposes (like sibling comment about even divisibility of 60) but usually came back to using 10. We've always needed to count, been smart enough to count, and had 10 fingers readily accessible to count, so I wouldn't count that theory out :)

I think the biggest strength of base 10 is not hand-counting (the OP and the Babylonian/Chinese base-12 method are both superior in that regard), but ease of performing pen-and-paper operations. You can literally teach a 6-year-old to multiply huge numbers effortlessly.

Couldnt you do the same using two additional digits? Say binary and hexedicmal arithmetic are as easy as decimal if you substract the bias of being used to decimal.

If you use any other base for writing numbers down, it's just as easy to perform pen-and-paper operations. The only problem with larger bases is that the multiplication tables increase quadratically. Whereas a base-10 multiplication table has 100 entries, a base-16 table already has 256 entries.

>a base-10 multiplication table has 100 entries

Not quite! You can safely ignore identities (0, 1, and 10 itself) so you only have 8 numbers in your table. And multiplication is commutative so you only need 8+7+6... (= (8+1)(8/2) as per Gauss) = 36 entries.

Base 16 would have (14+1)(14/2) = 105 entries. So proportional to base-10, actually slightly harder than you said.

The classic - detailed correction followed by "but yea you're totally right anyway" lmao

A fine demonstration how "basically correct in two sentences" is better than "technically correct in five sentences" in most situations.

This video convinced me that base 6 would be even better for simple pen-and-paper math, as well as just about everything else: https://youtu.be/qID2B4MK7Y0

It was a long time ago, so not likely anybody wrote it down. But wasn't it from India? Counting-sticks in boxes, when you got to nine (maybe all that would fit in the box?) you put one stick in the next box and 'cleared' the lower-significant box. Apparently zero is a drawing of an empty box...

This reminded me of the system of musical notation using finger segments. http://www.openculture.com/2014/03/watch-the-guidonian-hand-...

A bit complicated. I am learning Tabla, and my teacher counts to 16 on one hand; 3 finger segments plus the bit below the finger, for a total of 4 per finger. Starts from top of index finger and ends at bottom of pinky finger.

In this way, each bar is one finger.

Thinking about my fingers while hitting something with my fingers might be a bit much for my brain! But actually it seems quite intuitive to layout a beat pattern that way. Thanks for prompting me to search what Tabla is, there is some great artists out there.

Actually, you don't do those things at the same time. The finger thing is mostly for clarifying the beat patterns before you start playing. On the Tabla, or actually in Northern Indian music, you do things like play a 27 beat bar and a 16 beat bar at the same time, and pseudo merging back on the 80th beat (16x4 and 27x3-1).

For a 5th position per finger you can add your fingernail. Count to 400 with both hands. Also has the nice property of 100 per finger on the second hand.

I grew up counting the segments on each hand, twice. But the 2 segments of the thumb are also included (you use the index to point at the thumb).

Total comes up to a much-shorter 14x2 = 28 though.

Sum types versus product types.

You’re doing

     data Number = Left Segment | Right Segment

but

     data Number = BothHands Segment Segment

is possible.

I just love how this illustrates the names sum and product types.

    permutations(sum of Segment) == permutations(Segment) + permutations(Segment)

vs.

    permutations(product of Segment) == permutations(Segment) * permutations(Segment)

(Edit: permutations is the wrong term, it should be cardinality or a non-mathematical concept like some colloquial concept of “variations” )

I don't think "permutations" is the right word. You probably want "cardinality".

You are right, I was thinking more from a programmers practical perspective and started with the word «varations» and changed it to «permutations» without thinking it through.

Thanks for the correction!

And if you add the under part of each finger you get 16 x 16, which is pretty handy for software engineers because it overlaps the increments used in the binary system (16, 32, 64, 128, 256)

This is the same way I learned growing up in India as well.

I've read that societies that used such a system for counting used base 12 (instead of base 10), and had names for one digit representations of numbers 10 and 11.

Such as system allows for quarters and thirds to be whole numbers, and is much more powerful once you get used to it.

Many of our numbers come from merges between the two base systems (12 hour days, 12 inch foot, etc.)

To clarify, I was taught that the reason we use base 10 is that we have ten fingers. In societies that used a twelve jont counting from a young age, they use base 12. I suspect that the reason that most people find subtraction harder than addition is that they have been doing addition from a younger age.

One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve... they're unique names. Then comes thirten, fourten.

Ten, eleven and twelve are unique names and probably have their entymology derived from a base-12 numbering system once primarily used in Anglo-speaking countries.

“Twelve” is suspiciously similar to “two” however, which makes me wonder about its origin.

It's simple enough to Google "etymology eleven" to see that it is a base 10 word.

Modern proponents of a dozenal system seem to prefer the words "el" and "doz" for eleven and twelve. I would totally be in for changing, but realistically it'll never happen.

Numberphile on the dozenal system: https://youtu.be/U6xJfP7-HCc

I and some friends in college sometimes used binary finger counting: it's a simple 10-bit finger up/down that counts up to 1023 on two hands.

It's also kind of obvious communication style if you are doing a lot of assembly programming or logic gates programming, so we certainly didn't invent the idea.

It got to the point that "132 to you" was a verbal shortcut/joke for a particular rude gesture that naturally results from counting to 132 this way.

I've used this before for counting. The way I've done it is based on what fingers are touching the desk, or my leg. Reason being, it's kind of awkward holding up/down some finger combinations in the air enough where it's distinct (eg only ring finger up).

So for me it doesn't work too well for communicating numbers, but counting works fine.

Yes, some finger combinations are interesting dexterity challenges (varying among individuals, too), which is why the hysteresis/expectation setting of what's up or down can be important when using it for communication (outside of 132 of course, which is usually pretty obvious). I vaguely recall having at least one drunken conversation with an electrical engineer about what the "voltage equivalent" was for various finger positions might be and what your finger logic gate would need to accept to properly determine finger binary state.

I'm surprised no one has mentioned counting in binary on their hands yet. You can get to 31 with just one hand.

This just blew my mind. Also almost broke my fingers.

Some of the finger strain can be alleviated by using a Gray code.

Also by switching from emacs to vim.

...oh sorry, wrong thread. :)

Yes. Though gray code is a bit harder to work with directly for anything but counting up.

Rude.

132

hey.. 5 man. be cool.

13 (considering the thumb as the most significant digit, for obvious evolutionary reasons)

Thumb should be least significant, because the least significant digit changes most often when counting, and therefore you want the most dextrous digit doing that.

You can go a lot higher on two hands, though it gets annoying to hold your left hand in one awkward position for long periods.

Still, a useful little thing I picked up in my teens. And taught at a conference one year when I was doing a presentation that was glorified babysitting for smart kids so their parents could attend other stuff at the conference.

Just be careful about 27.

Well there are specific combinations that are pretty hard for some people.

That's also true of the linked article. I can't bend my pinky finger to make the symbol for 1 in the diagram without my ring finger also being significantly bent.

My favorite is Chisenbop[0], the Korean finger-counting system, that lets you count/add/subtract easily to 100. It isn't as intuitive, but it is quite fun.

[0]: https://en.wikipedia.org/wiki/Chisanbop

Some indigenous Americans had a wonderful octal counting system based on the spaces _between_ fingers which becomes exponentially more awesome when you hold small twigs, string, grass whatever between your fingers as placeholders in your manual abacus.

If we could have seen one more planet in the sky maybe we would be counting in 8s, much easier for switching to binary then

I remember when I used to teach at a kids programming camp part of my curriculum included teaching them to finger count in binary. It's fairly straightforward to understand and some kids would write the base-10 equivalent (1, 2, 4, 8, ...) on the back of their fingers to practice.

Occasionally a camper would ask why don't computers use base-3 instead of binary. In fairness, with 10 trits you could count to almost 60k, way better than silly binary's 1024. I imagine some of those campers went on to become excellent surgeons.

Somebody published a paper (back in the 80's?) that posited, digital logic should use base-3. They solved some equation and base-e was the optimum (2.718...) and that is closer to 3 than 2.

But so much easier to do circuits 0-1 I guess. Or the science was wrong. Anyway, just be glad we have the sensible binary system we have. Used to use bi-quinary, ones-complement and other strange schemes. Took a surprisingly long time to settle on twos-complement integers.

Some of the finger dexterity here I'm pretty sure I won't be able to do without assistance or faking or similar; for example, my physically getting the difference between "1" and "9" boarders on Vulcan-level fingering for me. I am impressed.

My mother grew up using a similar system using knuckles and positional counting. I remember she taught it to me as a kid but I've forgotten. She's Indian.

Counting knuckles and gaps across both hands from left to right also works for tracking which months have 31 days (with February as the exception). Knuckle = 31 days, gap = 30 days.

It's actually quite interesting thinking about all the number systems you could accommodate by counting on your fingers, and some of the interesting difficulties (try holding your fingers out, then folding down only your index and ring fingers... Not comfortable).

Using base 10 is probably one of the worst ways you could use your hands for counting, funnily enough. Base 6 would be pretty convenient, quick and comfortable (counting up to 35) - each finger on the right hand is the "units" column and each finger on the left hand is the "sixes" column.

There's a compromise between complexity, comfort and capacity.

Also re counting: we also have Bede to thank that we call this year 2020, and not e.g. the 4th year of Trump :

"At the time Bede wrote the Historia Ecclesiastica, there were two common ways of referring to dates. One was to use indictions, which were 15-year cycles, counting from 312 AD. There were three different varieties of indiction, each starting on a different day of the year. The other approach was to use regnal years—the reigning Roman emperor, for example, or the ruler of whichever kingdom was under discussion. This meant that in discussing conflicts between kingdoms, the date would have to be given in the regnal years of all the kings involved. Bede used both these approaches on occasion but adopted a third method as his main approach to dating: the Anno Domini method invented by Dionysius Exiguus. Although Bede did not invent this method, his adoption of it and his promulgation of it in De Temporum Ratione, his work on chronology, is the main reason it is now so widely used."

https://en.wikipedia.org/wiki/Bede#Use_of_Anno_Domini

I was just reading in his History the other day, such a fascinating book. It's a history of England (focusing on Christianity) from Julius Caesar to the 600s. His monastery was a renowned centre of learning, with 200 books. Just imagine trying to do historical research in England in 700AD..

You can count to a 100 on a single hand easily. Imagine each finger has four facets, and combining segments and joints you can count to 5 on each facet of each finger, gives you the ability to count to 20 on a single finger, so 100 for the hand. I haven't tried it but combining both hands would then allow you to go up to 10,000

Tuna auction is also using hand numbers like this https://youtu.be/H-eoIxfycf8?t=316

They also sometimes do the gestures with the hands under a sheet to do a silent auction. Pretty impressive.

The most interesting use of my fingers for numbers and computation were for playing perfect strategy in the game of Nim (back in high school). I just represented numbers between 0 and 15 in binary on one hand by slightly bending the fingers corresponding to bit positions for binary digits.

Nim is a simple game played with coins or markers distributed into a number of piles. Players alternately remove one or more markers from a pile of their choosing. The player removing the last marker loses. The game starts with any number of piles, typically 3 or 4, and any number of markers in each of the piles.

Using one hand (and knowing binary) it’s easy to mystify opponents.

I found something very interesting about roman numerals. If you had to set up a physical score table for a game (say basketball), in base 10 you would need 10*3 digits (10 digits for each position in a 3 digit number).

If you try other bases, you'll find 3 is the cheapest (you'll only need 19 digits to show all numbers from 0 to 999).

But with roman numerals, you only need 16 (15 plus one for zero).

It is the most economical base by far. You would need I (3), V (1), X (4), L (1), C (4), D (1), M (1), and N for nihil.

I find it interesting that no culture used the binary system. It allows to count to 1024 on 10 fingers.

I just tried - comfortable practice requires more supple dexterity than I can muster.

Maybe you are doing something wrong - all agility the binary counting requires is bending one finger per bit, which is the same as counting from 1 to 10.

There are 1024 configurations for counting in binary, and only 11 for counting "normally". Some of those other ~1000 configurations are harder to do (like having every other finger up). Mostly stuff with your ring finger.

Apart from my index finger and thumb, the tips of my fingers all kind of curl together to some degree, so my roman one and two would look much the same, and I have to strain to get a rough 5. I often wonder if I could retrain my brain about this with enough time.

I'm also surprised how hard it was for me. All it took was counting up to 8 a couple o times, trying to do it somewhat fast and visually-clear and I feel the strain, particularly on my ring finger.

I could use it as a mnemonic device to help me "store" a number, perhaps, but likely not to visually communicate numbers to others in an effective way

American Sign Language can represent numbers up to 999 with one hand, and numbers past that with two hands (formally, though you could still be understood if you were holding a beer in the other hand.) A useful skill to have!

I find it interesting that, according to the chart, the Roman's were not only comfortable flexing their pinkie finger without their ring finger, but so comfortable they did it rather than flexing their index finger for 1.

I habitually count from the little finger out to the thumb.

The 'one' sign, pinky out, is accomplished by locking the other three fingers with the thumb.

Then I sort of flick the other three fingers out, and extend the thumb for five. It's quite comfortable, and easier than going in the other direction.

It's interesting to see that they had a notation for zero in that system.

Interesting, if you ask a computer scientist what is the max number that can be counted with two hands, he will say 2^10 = 1024. 9,999 is far from 10-bit.

It is not binary because 1 through 3 and 7 through 9 bend the same finger but at the different joints. As a result it is much more awkward to bend than binary counting.

That assumes two finger states - extended or bent.

Introduce half-extended as a state and you have 3^10 = 59049

It’s powerful. I recently contemplated that a hand has 6bit/sec or so bandwidth.

5 fingers + two sides of the arm. Cool stuff isn’t it?

I can't move my pinky independently of my ring finger, so I can't do a unique 7 and 9.

That must be a thing. I have a sister and a son, same way.

Glad I'm not the only one. I was picturing myself being laughed out of the Roman Forum.

I wonder if there's a scientific notation

I just grow more fingers. Problem solved.

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