Ethiopian Binary Math
uh.edu
uh.edu
https://en.wikipedia.org/wiki/Ancient_Egyptian_multiplicatio...
http://mitpress.mit.edu/sicp/full-text/book/book-Z-H-11.html...
First of all it isn't an article it's a transcription. If you can't picture simple math operations with stones and holes then something might be wrong with you.
Maybe try to keep personal invective out of the discussion, it's not nice.
Also ''No. 504''. Sounds like a series of something? by John H. Lienhard Click here for audio of Episode 504.
That definitely gives away the fact that it's a radio program.
I agree that this isn't a remarkably efficient algorithm, but it is novel (it's completely unintuitive to the average reader), and it has a sound basis in mathematics. For the people who employed it, it was an efficient algorithm, or else they wouldn't have continued to use it. Nobody is proposing you're not as smart because you don't do math this way (although frankly, I doubt if you would have derived this by yourself, under the same conditions).
On a side note, I strongly recommend 'Guns, Germs and Steel'. I've been knocked for mentioning it before because it's a pop-sci book, but it explains why cultures 'progressed' at different rates because of largely environmental factors. This might help you overcome your ridiculously defensive bias against African cultures.
This algorithm allows people to multiply two numbers if all they can do is multiply and divide by 2, and add.
> It couldn't have been efficient since it is patently idiotic.
And yet is it so efficient it is how computers multiply.
Yes, and the algorithm of making N groups of M and then counting allows people to multiply if all they can do is count. And they will do it far faster than the shaman every time.
>And yet is it so efficient it is how computers multiply.
No, it isn't.
So why didn't they?
My strongest guess is that it's the stones. In any base > 1 the algorithm is efficient. In unary (counting stones), it inefficient to the point of being nonsensical.
So my guess is that this was not used by counting stones, but with some form of positional number system, and that in the retelling, stones have been added as a way to make it sound more "tribal".
Edit: Alternatively, it may be the idea that they're doing this exactly. If the doubling side is done by rough estimation (eyeballing the size of the piles), it might be faster.
But it breaks down when multiplying much larger numbers, because the number of holes you need increases by N. With this addition system, you only need log(N) holes.
The system exists to remove cumbersome aspects of multiplying large numbers by counting. Consider 34x34. While one fellow is out digging 34 holes, making sure not to make a mistake, or finding a piece of wood that has 34 holes marked in it, the other guy never needs more than 10 holes if his numbers are less than 1024. This keeps his working area small, and he can see all of it at the same time. It's less cumbersome this way.
As described, the Ethiopian Method would take far more stones and far more time. The larger the numbers you are working with, the larger the discrepancy.
But let's say it was. In saying that the shaman knows how to put two sets of 7 stones in a particular pile, it is implying that the shaman also knows how to put 34 sets of 7 stones in a particular pile. He also knows how to count them, so the problem is already solved. What he is doing is a bit more "rube goldberg" than novel. It may have parallels with some real algorithm for multiplication, but it is certainly not a very direct representation of it.
(https://en.wikipedia.org/wiki/Multiplication_algorithm#Peasa...)
It's a big difference: 1000 * 1000 would require 1000000 stones if done with the "Ehiopian algorithm", but it can be done with less than 1000 stones by using normal binary multiplication.
This algorithm is totally pointless if you're not going to work with a positional notation for the number; I am sure it was never used in the way described, with stones. Anyone capable of understanding the problem in the first place - 34 goats, 7 pieces per goat - would count out the price - here is 7 for the first goat, 7 for the next, etc. - and tell the shaman to take a hike. This story amounts to calling Ethiopians idiots for taking an algorithm suited for positionally represented numbers - or at least numbers in some kind of a concise representation - and using it in a totally unnecessary way with numbers represented as groups of stones, and not understanding the pointlessness of all the drudgery.
John H. Lienhard is Professor Emeritus of Mechanical Engineering and History, getting his PhD from UC Berkeley.
> I fail to see the ingenuity,
You can't see something ingenious with probably illiterate people using a different base to multiply numbers? As per the Wikipedia link posted by vilhem_s "The method as interpreted by conversion to binary is therefore still in wide use today as implemented by binary multiplier circuits in modern computer processors."
Close enough for me. Frankly I doubt that he honestly believed every word he wrote. People in high places have to routinely perform public worship of the PC god, "all cultures are equal", etc.
Regarding the "ingenuity" see me reply above: https://news.ycombinator.com/item?id=6502665
> So how do we and that Ethiopian shaman differ? Very little, I reckon. > Very little indeed. Of course, I wouldn't be surprised if he makes fewer mistakes than we do.
The article clearly dumbs down (or misunderstands)the algortihm, insulting the audience's intelligence to make a feel-good claim about the arithmetic talent of "uncivilized" Ethiopians, and is accidentally racist in the process.
While it's a nice trick, it's a circuitous route to get to the answer, and requires considerably more stones than the number you're trying to count to. Simply laying out X stones per Y items and counting them uses no methods that the shaman's system doesn't already use, requires less stones, and doesn't require any 'good' or 'evil' silliness.
Besides, if a trick works, but it isn't understood why it works, then the users of it lose the 'ingenious' tag, methinks.
The fact that we think it is just a trick does not mean the Ethiopians did not understand it.
Also there are "apocalypse numbers" http://mathworld.wolfram.com/ApocalypseNumber.html , "beast numbers" http://mathworld.wolfram.com/BeastNumber.html , "leviathan numbers" http://mathworld.wolfram.com/LeviathanNumber.html , and "legion's numbers" http://mathworld.wolfram.com/LegionsNumbers.html among others.
Or maybe that the language translation is imprecise.
Sure. Me laying in bed listening to radio 4 is not ingenious, even though there's some ingenuity in creating radio (and the industrial infrastructure to get a radio to me).
Of course, no-one is yelling about ZOMG PC CONSPIRACY when I say how amazing it is that I can buy a radio for a few dollars.
>>Reality doesn't always conform to the PC narrative of how all civilizations are equal. Unless you are a sociology major that is.
I'm not sure what your point is. True positional numerals have a long cultural history that doesn't lend itself easily to an analysis of which culture performed math better.
If you squint your eyes a bit, you can even imagine where this comes from. To avoid creating too many holes, you start with a group of 7 stones, then double them, then again, until you "double" them 34 times. Obviously 34 is not a power of two, so you need some extra machinery to make it work for non powers of two.
Its an interesting Sunday morning to rack my brain trying to figure out which algorithm "wins" when parallel processed.
Obviously 34 shamans working together could each count out seven dollars/rocks, but someone needs to count up all those rocks.
There's an obvious token passing algorithm involving three shamans and I believe 7+34=41 rocks and 4 holes where one does a long count of the total and count++ every time upstream hands him a rock, then upstreams upstream does the same. So one dude counts to 34 using 34 rocks one time, the next dude counts to 7 (and does it 34 times) using 7 rocks, and the last dude counts every time dude 2 hands him a rock and never resets to zero (at least until done) giving him a total.
You can tell scientists invented the original algorithm not engineers just by a simple observation of actually implementing the algo... "Yo shaman come over and mult 7 by 34 for me" "OK but how many rocks will I need to multiply this... do me a favor and whip out your iphone and google 7 asterisk 34 so I know a lower bound of how many rocks to bring to implement this algo.."
The worst part of the whole discussion is it reads like a really extraordinarily bad stereotypical tech interview question. "So, I see you've got extensive experience on the job with Scala and Mysql and play framework, thats all very nice, but a major problem we often have here at BigCorp is sometimes we want to measure precisely 4 oz of rum but we only have a 3 oz and 5 oz cup and we were wondering what you thought of that ..." You know some kind of space alien anthropologist is going to hear this is how those ancient "humans" used to select their programmers and the space aliens who hear pop sci stories about human programmer interviews are going to think 1) We were insane/stupid 2) They're superior. Theres not much other reason for the whole story.
Good thing for the shaman's job security that there are more numbers that can be multiplied than 7 and 34, then. Try multiplying 34x34.
For large multiplications, using the method described uses a logarithmically smaller working space in a society where positional numbers didn't exist and paper was rare, expensive, or nonexistent.
How did you generalize from an Ethiopian shaman's unique method of calculation to the superiority of civilizations?
I also take issue with the automatism of accusing "political correctness". Nobody says civilizations are equal, there are numerous obvious differences. Somehow whenever I dig deeper and ask about what those people think, I get to hear/read profound racist undertones. The "PC" curse is really most common as a defense to accusations of racism.
The actual problem arises when people go on to make value-judgements along ethnical boundaries (racism). And then they go even further, using these not necessarily well founded judgements to pose "scientific facts" about how one ethnicity is genetically superior (because they believe all differences between two sets of humans have to come down to genetics/evolution)and thus are somehow entitled to more wealth than another ethnicity.
Their is no right to wealth and no birth right to it. That does not however imply a redistribution of wealth in the slightest way.
There are strong factors which contributed to the economic rise of Europe "first" and which have nothing to do with human genetics. Agriculture benefits from certain climates and soil conditions (ask the mayans about their collapse...). It also benefits a lot from domesticateable animals, and most animals are not nearly as well suited as for example horses and cows. Then the coastlines and rivers in Europe have shown to be much more conducive to commerce (before industrialization) than in Africa or America.
Systems are complex. Don't simplify them to justify judgements.
I said nothing about genetic overall superiority although it is plainly evident to anyone not blinded by PC that some peoples are genetically more likely to produce sublime music or advanced mathematical frameworks / complex works of art or advanced technology just like other peoples are genetically more likely to be stronger, better jumpers, runners, sturdier and so on. Environment certainly played a role but after many hundreds or even thousands of generations the genetic makeup of a people changes as well (as an aggregate statistical entity).
I've no doubt these genetic differences were non-existent when we were all together in the primordial soup but we've moved past that phase and these differences now exist. I am saying nothing about overall superiority and certainly nothing about rights of individuals.
If you make the extraordinary claim that the genetic makeup of a people played no role in the difference of achievements between cultures the onus is on you to prove a negative. Explain why we take it for granted that people of Samoan heritage are huge but it's somehow racist to argue that hey, that DNA thing may be affecting brain cells as well and not just bones / muscle tissue.
Individuals have genes that make them sublime musicians or athletes or mathematicians or painters. Cultures have different patterns which set the tone of that music and sports and math and art.
Since you're hedging with "peoples are genetically different" instead of "people are genetically different" I wonder if you've had some kind of formal rhetorical training.
This method in practice would have used objects that represent larger numbers(1,5,10) and with those a practitioner only needs to know how to add, subtract and multiply or divide by 2. All of which are operations which can be performed quickly. It certainly isn't true that you would need as many stones as the final result.
But note that this is different from what John H. Lienhard explicitly says: "the hole contains 224 stones". His version is completely pointless. None of us are objecting to the algorithm itself - yes, it is equivalent to long multiplication in binary, it has been known since ancient times, and there is nothing wrong with it. However, Lienhart has a) apparently misunderstood the point of the algorithm, and b) ascribed it to Ethiopian village "shamans", which gives one the mental image of Ethiopians standing around and watching a shaman manipulate hundreds of stones, instead of simply counting out the price - 7 for the first goat, 7 for the second, etc. This probably never happened, and would be stupid if it did; and yet people are coming here to defend this as an example of "Ethiopian ingenuity".
What kind of "shamans" is Ethiopia supposed to have, anyway? Aren't they a Siberian thing? Ethiopia has one of the most ancient Christian traditions in the world, you know. I agree with some other posters that this whole thing is some kind of a politically correct cringe which somehow tries to compliment Ethiopians for something they have nothing to do with, and only ends up insulting them. And every reader's intelligence, to boot.
Erm, no. Ethiopia has the kind of shamanic spiritual healers/wise men that are quite common within the various cultures that exist in that country.
You are flailing at windmills here. Sure the country has a long history of Christianity; however that part of the world has a long history of overlapping animistic traditions too.
The Russian Peasant Algorithm optimizes this for the binary case by noticing that the inner multiply is always by 0 or 1 so it's trivial and never carries, removing the inner loop entirely and allowing you to use only one accumulator.
1. http://ed.ted.com/lessons/the-magic-of-vedic-math-gaurav-tek...
2. http://www.openthemagazine.com/article/art-culture/the-fraud...
Paper! Seriously, this was 1960? Even ancient Sumerians were using paper to do math. I don't understand why in 1960 anyone would seriously still be doing math with holes and rocks.