Inca Knot Numbers - Numberphile
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- čas přidán 9. 06. 2024
- Alex Bellos discusses how the Incans used knots in string (Quipu) to record numbers.
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Ok
I want to gift this to my dad (a math teacher)
When will the book be available in the US (not from Amazon)? I've been waiting what seems like months, and I can't purchase it.
Perhaps the leader of the tribe gave each person a bunch of strings based on there bank accounts!
Lol
@@pukku1 Why can't you by it thru Amazon?
One of my professors is the guy who did a lot of the work on deciphering the more linguistic khipus. The reason why we know they can't all be numerical is because the knots store more information than just number of twists in the knot. Some are left handed, some are right, some are dyed different colors, some use different materials, some use other knots than are needed for the numerical representations. The khipu as a writing system is poorly understood, but it's certain it was used as more than maths! It's fascinating.
Am I wrong or a Spanish missionary wrote a book whith a rough translation of some strings of quipus, mainly names?
Yes, there's more to it than just adding!
I always thought there must be more than numbers! It makes sense, they had developed this elaborate system for registering numbers, there had to be a moment when they said :" hang on, i need to write down also what are these numbers about..cause i have this quipu about llamas i sold and this about soldiers that need to be paid. Here, let me put this knot here to distinguish them..."and boom, written language baby
It seems easy to show that it is not all numbers. Just check the math. If there are samples that are completely wrong, then you know it's not all numbers. Of course this doesn't tell you what it does mean.
@@Eisenwulf666 If i had nothing but just numbers at my dispossal, I would assign a number to each type of thing I am counting. Like an SKU number. If that was the case with those strings, we should see a lot of repeating numbers, like Animals = 100, Llama = 101, Chicken = 102, and so on.
"these are not numbers" "well actually, these are knot numbers"
:D
Get out. :D
hmm... Yes. And the floor is made of *f l o o r*
Wtf you're here too
Carlos!...
I'm Peruvian and I didn't know how Quipus worked. Great video.
im from Bolivia and i didn't know either
so incas invented CSV; storing integers as strings.
Except they're space-separated instead of comma
@@andrewlalis Take my upvotes and get outta here.
@@andrewlalis, tab delimited
And the checksum to catch errors.
CPU!!!
When he said "No number goes beyond six" I thought a base-6 system, so I tried to sum the numbers in base 6 but the answer wasn't correct, so I started wondering...
...It was base-10, simply.
I summed in base 10, but then started thinking "It shouldn't have been this easy, he said there are something to do with base 6"
Maybe there are other quipos which go up to larger sequences of knots or maybe quechua uses a base 10 number system?
Wouldn't that be base-7?
@@williammiles9926 Yeah, but I was SO distracted that I didn't even notice there were two 6s, lol.
Base 6 is my favorite base!
A quipu with 100 strings is like an excel sheet.
Yay. I'm glad he pointed out that the reason they didn't use wheels was probably because of mountains and terrain, and not llamas.
Llamas can be made to pull a cart, and the Inca actually did know of wheels, but they were confined to very specific uses or to childrens toys.
So it's basically ancient excel?
Based on the historic and modern evidence I'm convinced some quipu contain written language. Probably in the form of a syllabary, instead of an alphabet.
From what I learned, the Tawantinsuyu had about 19 million inhabitants at the time of invasion. Only a few decades later it was reduced to about 5 millions. This included most intellectual and scientific L337 and those required for the infrastructure of the empire, among them the khipu scolars. Practically another Alexandrian library burnt to the grounds, only 400 years ago.
It's fortunate these knot numbers didn't turn out to be INCAlculable.
I'll show myself out.
2020 Dad Joke contender right here folks
Well that's CUS COunting turns out to be quite universal!
Is that a Les Luthiers reference?
@@erumabo I actually paused this video shortly after starting it to go listen to that Les Luthiers sketch
??
The Inca's did have the wheel. This seems to be persistent myth that confused the fact that they didn't really use wheels in the same way we Europeans didn't because well they don't work well on steep mountains and in dense jungle but they did have them. We have multiple Inca toys that had wheels on them.
In the southwest Okinawan islands there as a very similar system of knotted ropes called barazan. They used knots to keep track of who had paid what taxes (to the Japanese overlords) and who owned what. There were all kinds of innovations like using bigger knots for 5s, and using a separate string for each unit of volume, with the biggest volumes being made from the thickest ropes. Some people even tried to fashion ropes into the shapes of the things they were counting, but there aren't too many examples of that. It is fascinating to see people separated by so much space come up with the same solutions to the same problems!
I wonder if their system included a nought.
Gottem
Get out
@2C (02) Chan Kwan Yu Why knot?
Yes, nought is not a knot.
I’m guessing it’s just not having a string.
The Inca were very weird in having developed a system for recording numbers without apprently having the ability to record language. Which, I suppose, is why people are thinking that maybe qipu's are also a writing system.
In the drawing of the knots/table the green string only has 7 knots on the ‘8’ side.
oops
The error has been glaring at me the whole time. I'm glad that someone else noticed it.
such a cool system of numbers! the strings look so pretty!
If they ever make a Numberphile film, I nominate Michael Sheen to play this dude. It's uncanny!
Is he not Michael Sheen ??
And a key plot point in one of the Dirk Pitt novels.
Which one is that?
It was used in a Chad Flenderman novel too
@@Seffero ha I read your profile picture but didn't read your channel banner you lose I win
Fun fact (from a Peruvian) : Quechua is more than a single language, several dialects of Quechua are only spoken and understood in their respective areas but yes, they , for the most part, had a "standardized" language
So, is the green string an historical example of checksum?
Haha, ya, that's what it is.
No, I think it checked all. :p
I imagine that if you had a few of these counting the same thing you could take the sum lines from each and put those on a new one more easily.
I was thinking the same. Checksums and SKU numbers.
No, it's a normal sum.
Checksum would be, let's say, 40 (sum of all digits).
What a great video. I'm from Peru (the capital of the Inca Empire was located in Cusco, Peru of course) and this is the first time someone shows me how to properly read a Quipu!.
I've done the first three chapters of the book and loved it. There's a great balance of difficulty in each chapter, and really interesting history between problems.
6:00 fun fact: the incas didn't use "money" or currency. They still kept track of their goods thoug.
The Inka were such an amazing civilization, leading a whole continent. They mastered roads, mummification and astronomy. A few quechua words climbed up the language chain to become worldwide recognizable, such as jerky and coke
And llama
The Incas had the concept of zero.
They didn't have a symbol for zero (according to the video) which is different than the concept of zero.
There's no knot for zero - they just left a big gap. So they have a positional/place-value notation but they don't have zero as both a number and a place-holder in the way we think of it these days in the Hindu-Arabic number system. It's similar to the old Chinese rod numerals and also the Babylonian number system which was also positional/place-value but without a symbol for zero.
@@stephenbutler3929 I also don't see how this couldn't indicate the concept of zero. Especially since the Mayans also had it. Absence as in 0-digit speaks in favor of it. More evidence in terms of possible negative numbers (someone above mentioned LHS, RHS- knots) or multiplication rules including zero might cement it
I agree with the OP. There's no symbol for zero, nor for any other digit, but there is the CONCEPT of indicating zero, of having no knots in a specific position. So in their minds a position can have "zero" knots and still count as a position. The ancient Greeks e.g., couldn't imagine "zero" as a quantity.
I remember learning about these in sixth grade, and I always wondered how these worked! Gotta love ancient number systems!
Loving this exploration of number systems from other cultures and stuff!
Ah, you brought Michael Sheen back. Excellent.
I always thought a quipu would make an interesting spellbook for an Incan-culture-informed wizard character in a D&D campaign, perhaps with tiny bits of spell components tied in along each spell's thread
Consider it stolen, but for a whole setting
I find it surprising the system is decimal
Decimal counting systems appearing in so many unrelated civilizations probably has to do with the fact that people use their fingers to count
Most counting systems were either decimal or vigesimal (base 20). If they just counted on their fingers, they'd use base-10. If they counted on fingers and toes, they'd use base-20. This suggests that the Incas wore shoes.
@@cabra500 yes, probably. The thing is there is more than one way to count with your fingers. For example, you could use the thumb tho count each of the 3 parts of the other 4 fingers. This would give you a 12 number system in a single hand
@@arnaldo8681 yes, I remember my parents teaching me the months like that, each part of the finger representing one month, it’s an interesting idea indeed
@@cabra500 So based on what you're telling me, it's undeniably alien intervention, what else could explain that.
I find it absolutely AMAZING how they were able to build an empire and manage an entire bureaucratic system with more than 20 million people without written laws or records.
I guess when most of your civilization is illiterate (as was the case with all old empires) the oral law is way stronger
I mean, they have been living there for hundreds of years, solving all kinds of disputes. They probably had all kinds of traditions about how to handle those disputes, and all this was public knowledge, or at least you knew who was the closest 'smart guy' that knows it
This sort of stuff is already there when the conquerors come. And can be a big source of conflict when the conqueror's and the conquered's code is too different. But if the differences arent too big in the important parts, as long as you can keep the army fed (and thats why the kipus were important) the oral law should be enough to keep the empire together
And well, they did have lots of revolts. And lasted less than 100 years
Could this be an example of a society based on anarchy? Maybe there was no bureaucracy, just people with a common language.
@@samstewart4444 no, probably there was a lot of coercion and forced work. It was just not written
They obviously had non-numeric usage of quipu, just it hasn't been deciphered yet.
@@samstewart4444 There were lots of languages spoken in the Inca empire: Wikipedia lists Quechua, Aymara, Puquina, Mohica, etc, etc, although Quechua was used as a lingua franca (and had been before the Inca took over). The Inca had a centrally planned economy (I've heard them described as "agrarian socialists" before) and didn't have a market economy, tax being payed via labor services known as the mit'a and mink'a (which apparently is still in use today). That is to say, I don't think they could really be classified as a 'society based on anarchy.'
Thanks ! Happy new year !
I think the most frustrating thing about this video is that there's no acknowledgement that most of the stored khipu the Inca had were burned by the Spanish. Nor is the preservation of khipu by Indigenous people acknowledged - some of what researchers have learned have been because some Indigenous person comes forth and says "oh, this was how we used them, and we still have that knowledge."
Yes, and it's worsened by they not acknowledging that Quipu is more than counting, it's considered a writing system (and he say's there was none). And then in the ending they ask "it's so simple [sic], why it was so hard to decipher??" That's the answer: Spanish brutal colonialism.
Yes! And Indigenous ontology of numbers were not just for capitalistic concerns!!The ontology is tied to the Indigenous culture. Oh, you want a book actually with Quecha perspective? Sure, The Social Life of Numbers: A Quecha Ontology of Numbers and Philosophy of Numbers.
@@jk-lu3zb that book costs 1/2 a minimum (monthly) wage here on brazil. 🥺
This was the best birthday present I could have wished for. I have long waited for a mention of kipus. Greetings from Brazil.
“Developed a decimal system without knowing it”... um.... excuse me? How do you mean they didn’t know. They obviously understood base 10 (aka decimal) because of how they carried in the addition when hitting 10. Of course they knew.
It also looks like they independently developed 0 (in eastern hemisphere/old world math, 0 came from India)
I would think these other knots represented objects. 27 sheep. 1208 units of grain, 310 units of gold, etc. These symbols may have differed slightly between users.
that's not a string, that's an int array
A very interesting and worthwhile video. A must see video for everyone.
thank you,for touching on the cultire most acosiated with Peru!
Its amazing that they also used base 10? Maybe because of our fingers
Pretty sure most systems are base 5 or 10 and then there's special ones, like dozens were used to count eggs because buyer and seller would count them together, three per hand with four hands
More likely because of _their_ fingers. ;)
But yes.
Base 20 shows up fairly often and often the remnants of base 20 are still visible even if a society later switched to base 10. .ex: French, Welsh, Mayan.
Some tribes in papua new guienea ise base 27 number system !
@@hatebreeder999 Any idea why 27?
Having studied the anthropology of the region, my professor stated nowhere in Peru would you find a statue of a conqueror of the empire, rather Pizarro founded Lima. Pizarro at no time had the military might to defeat the Inca as they out numbered him in size many times. He did copy Cortez. The history and anthropology of the quite unique for many reasons.
Since the knowledge of these was ‘lost’, I’m still wondering if there’s any other possible explanation 🤔
There was another theory that said it represented family trees back in the 1990;s.
The specifics were lost, but knowledge of what in general they were used for (including how the khipukamayuqs would have an accompanying abacus - the yupana - to help with quick calculations) was very much attested to.
We've learnt about the Inca's numbering system and arithmetic in Professor Richard Feynman's book. But here we had an illustration. Thanks 👍
Quipu were used in Death Stranding. (Took me a while to remember where I had come across them before).
Very intersting video
Wish there were more like this one .
I had a dream about these knot numbers, and then woke up to this in my feed.
On the shelves behind him: a menorah, a Klein bottle, a bottle of Jack Daniel's, and a Rubik"s cube
Which of those do you prefer?
@@blizzy78 a menora puzzle with self intersection filled with Jack daniels
hanukkiah, as it has 8+1 candle holders :)
New, to me, information. Very interesting. Thank you.
Wow, this is fascinating and cool! 👍🏻😀 Well done!
Thank youuu for this videooooo! Greetings from Perú.
I think I speak for all numberphile fans when I demand a quipo(??) Of Grahams number 🤭
I knew QUIPU from constant Scrabble play! It's useful for dumping the Q, but I had no idea what it meant until this video.
This is some beautiful madness, have you guys covered quaternions yet?
I've saved this until I could stop everything and pay attention. I'm glad I did.
Can you image a Type I civilization that does its calculation with this number? It will be very funny
You should collab with a linguist and talk about the Quipu as a writing medium.
I'd love for Tom Scott to do another video with them
Thanks !
Use of Base 10 is interesting on its own.
@@williamcurtis2145 "because we have 10 fingers" is an assumption ... just sayin'
I once saw a documentary about Petroglyph National Monument in which the white American narrator talks about how mysterious the symbols are ... and then they interview a Pueblo guy and he immediately starts explaining them. I've also read articles about European explorers finding "lost" cities in the Central American rainforest, only for locals to say "um, we all knew that was there the whole time." Makes me wonder if any of these mathematicians bothered to ask a Quechua person about how khipu work.
I can tell just from looking at those that they were used as maps. You would just count certain rocks in the terrain, and you would use your fingers to "feel" each bump in the string, as if you are feeling bumps in the road.
Whats so special about it, is that it is very easy to change the map orientation, by grabing the other end of the strings.
I mentioned "counting rocks", but it makes more sense if they counted mountains, considering that the Inca region is filled with lots of mountains. Those strings COULD be correlated with the current mountains and their paths in that region.
@@jco997 Can they? Do you have a source on that?
@@lunkel8108 my english is rusty. I meant to say that I think (guess) that the ropes could had been used to count mountains and other objects along a path. They could serve as maps without having any need to orient yourself, since I believe (guess) they didnt had any compass to do it so.
There's a bit about quipus in The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph. I've only started on the book but I've found what I've read so far fascinating.
I sometimes wonder if you and Matt Parker plan your video releases together. He posted a video yesterday about an ancient number system as well.
I'd imagine that in order to identify a link between the quipu and language, it would be easiest to find speakers of the original language or records of diaolgue and then look for "numeric slang" where the sound for a number had been morphed to indicate something else, or visa versa. It makes sense to me, at least, that the counting system would merge with the spoken system if they were recorded in the same way, a modern example being text speach such as "h3110"
omg! i'm from Perú >w
¡Compatriota! 🇵🇪 🇵🇪
Yesterday Matt Parker made a video about the first math error. Today we have this.
How do you get math special effects?
By using a green string.
I thought the thumbnail was jellyfish counting beans
IIRC the reason we couldn't just read the khipu were because the Spanish had all the khipukamayuqs - the people who could read them - killed, along with destroying all the khipu they could find because the histories they told about the interactions between the Spanish and Inca disagreed with the Spanish versions of those histories.
It's kinda ironic that we are using a writing system to understand their knots which was used because they didn't have a writing system to begin with!
But we both had a number system.
We aren't using a writing system. This is a video system 😂
Imagine an alternate universe where the inca took over and they'd use knots to explain our paper-written numbers
That's part of the problem of inventing a writing system from new. If _you_ were going to invent a writing system, you could make two columns, put the symbols (new letters) in one column and their pronunciation in the second column. But if you're making a writing system from new (if you don't already have a writing system), you can't make that second column. So it takes memorization
As an exercise, invent a new writing system for English without using the old system as an intermediary.
not really. I mean, we dont háve to use a writing system to explain it. It's just so much clearer than using only spoken word.
I would have been ironic if our writing system was used sólely to explain the lack of one in the Incan society. But we use the same writing system to explain almost éverything in the universe as well.
I would say "irony is not coincidence" but it's not even a coincidence. It's a self conformation bias.
a brilliant instrument for registering events, the Qipu \m/
I've been wondering about this ever since I found out about those strange things.
Very intriguing. Imagine this still being implemented, and receiving your gas bill as a bunch of knots on a string.
Fascinating
Maybe the total string was some kind of checksum providing gubernamental seal or a similar kind of certification. Thank you for the video!
Quipus are *far* more complex than just spreadsheets. They didn't just encode a sequence of digits, but also encoded information in the color of the strings, and how the strings were tied together. In many cases, strings were tied to other strings in nested structures that could get at least 6 levels deep. There's also evidence of encodings for locations via a zip-code-like system, as well as a tagging system that served a similar purpose to pointers in computers. As much as 20% of quipus are believed to have also stored linguistic information, though this is currently undeciphered. It's not even just one language either - the oldest quipus date back to the Norte Chico civilization almost 5000 years ago!
There are also some quipus that broke the standard format, doing things like weaving strings together into a patterned fabric before diverging into separate strings. Quipu bitmaps?
Quipus weren't just spreadsheets; they were arbitrary data structures encoded in string, shockingly similarly in format and complexity to the data structures used by modern computers. Calling quipus "just a counting system" is like calling a modern computer "just an adding machine".
They organized society in base 10, as well. A family (understood as a couple and their children) was the nucleous of society, then a group of 10 families had a chief or lider, a group of 10 of the previous (i.e. 100 families) had another chief, and so on until groups of 10000 families, I think
You can read more about this in The Information by James Gleik!
Y'all should do a series where you guys start from grade one math to graduate level and start explaining it all.
Nah, there are tons of courses that cover that. Brady and his army of scientists and mathematicians are better off doing these special cases and interesting phenomena, which then _inspire_ people to pick up one of aforementioned courses. Or that is at least my humble opinion.
there is enough material about school math out there
Its like in scoobydoo when you figure out the answer before its revealed, I'm proud of myself 😎
Reminds me of the AppleTV+ series ‘SEE’ starring Jason Momoa (AquaMan, Game Of Thrones).
In ‘SEE’ nearly everyone in society is blind at birth, so they use knots along a cord/string to record messages.
Can you please make a video on -- How can we get into research in mathematics.
New numberphile yasssssss
how to know if you have to read it from left to right or right to left?
A knot writing form that would be cool.
Knowledge of how these worked wasn't lost - it was intentionally and systematically destroyed. There's a difference.
So were their virgin-women sacrifices.
@@mikedoe1737 is that your only quarrel? As the other person said sit down and put your head down if you just want to be racist.
A string went into a bar and ordered a beer. "Get out of here, we don't serve strings here", he heard this over and again. Then a light went on. He tied himself into a shoe string bow, walked into a bar and ordered a beer. "get out of here, you're a string, we don't serve strings." Said he "No, I'm knot, .........."
I'm surprised the Incans used Base 10 numbers, since they presumably developed their number system completely separately to the rest of the world. Maybe the fact that humans have 10 fingers and toes just makes it a favourite to use everywhere.
The largest strand represents a commodity and the other strands represent the number sold or traded
I had a cat named Quipu. I lived next to a dive bar called Knotheads and I think it gave me the idea.
Could you guys possibly do A video on Friedman’s SSGC Function?
Looks like a great way to record binary numbers. It would have been interesting to see a civilization of that era use it in everyday life.
Yes
Just an observation: Khipus have not been invented by the Inca, but have been used by different Andean cultures and civilizations stretching back for at least four thousand years before the Inca. If they truly are writing - which I personally doubt, but I seriously hope I'm wrong on this one - than they're a serious contestant for the oldest writing system in the world.
There's a really interesting trick that can be used to calculate the powers of 2. if you divide 1 by a lot of nines followed by an 8, you obtain a series of powers of 2. For example, 1/998=0.001002004008016032064128...
Didn't they also use counting boxes or did Carmen Sandiego lie to me?
Came here to say this! I loved that game.
My birthday is 22772 and I've had a strange life. I'm very in tune with the earth and emotions.
Can you please do our own numbering ststem, the stock; from which we get the stock exchange.
It's weird how even the Incas used decimal system, and not some other base, like said 6 in the video.