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Multiplying and dividing with Hindu-Arabic numerals and with Roman ones
- adictator 10y agoYou know what grinds my gears? When people like Neil DeGrasse Tyson think that there existed an Islamic golden age because we (the Western world) borrowed and still use things such as "Arabic" numerals. Neil has never once expressed that the golden age happened in the Indian subcontinent (Afghanistan to Eastern parts of India). You'll see that in most of his talks in the Cosmos show etc. In fact, he ignores (or is ignorant of the fact) that it was the Islamic invasions that stunted that golden period by brutal massacres of the local peaceful people (Hindu's, Jains, Buddhists etc). So much for the "golden age of Islam". I'm almost convinced that Neil has Islamic ancestors which is what he's subconsciously trying to project as part of the "golden age of Islam"! PS: I respect the man for so many other things, but he says this so often that it's become a major turn off.
- TazeTSchnitzel 10y agohttps://en.wikipedia.org/wiki/Islamic_Golden_Age https://en.wikipedia.org/wiki/Islamic_Golden_Age
- adictator 10y agoWikipedia? Ok, I have this from a PhD: (surely that counts marginally higher than random wikipedia edited articles): https://youtu.be/CLybCo1cWfc https://youtu.be/CLybCo1cWfc There never was a golden age in the land of the deserts, especially after Islam took hold. I mean look at some of the things the Quran says. Any golden age was just in people's minds. Also, if there was one, what happened? Why are there so few scientific breakthroughs in the regions that are largely Islamic?
- fractaled 10y agoFWIW, Bill Warner's PhD is in math & physics.
- adictator 10y agoYes, that's correct. Not sure what your point is.
- ajkjk 10y agoObviously it is that he has no special authority on this subject.
- adictator 10y agoAh, as opposed to the wikipedia authors? According to you they had a special authority on this subject, never mind any bias?
- jacobolus 10y agoWith 20–40 minutes of effort, someone could easily track down a large pile of peer-reviewed scholarly literature examining the contributions of Arab, Persian, Seljuk, etc. mathematicians and scientists of 800–1600 CE, and putting it in historical / scientific context. But I doubt you really care much, or you would do the same searching for yourself. So it’s not worth the time.
- adictator 10y agoIf I didn't care then I would not have made the post. And that goes the same for you. I think you're making this a battle between who is right - you or me. For me it is a battle between what is truth and what is a blatant lie. For me it is an anomaly that Islam rose in the desert and has always been a catalyst for death, destruction and chaos, yet suddenly there was this golden period with scientific accomplishments etc, and again suddenly all that vanished until this very day! That's a bit too much for my logical mind to accept. Maybe it is the nature of the people living in the harsh environment or maybe because Islam is the youngest of the Abrahamic religions - either way, people generally are giving Islam a free pass and not examining the inconsistencies in this narrative leading up to the Islamic Golden Age. For me it seems this was a great way for Islam to show that even though there are problems, they have had a great & tolerant past that was rich with scientific advancements. And we are naively consuming & being fed all that as the undeniable truth.
- userbinator 10y agoYou now add the results from the right hand column leaving out those were the number on the left is even i.e. rows 2, 4 and 5. In other words, it's just the shift-and-add algorithm: https://en.wikipedia.org/wiki/Multiplication_algorithm#Shift_and_add https://en.wikipedia.org/wiki/Multiplication_algorithm#Shift... ...which can really be done in any number base (and is even "easier" in base 2), so I'm still not convinced. All we need to carry out the multiplication is the ability to multiply and divide by two! Somewhat simpler than the same operation in the Hindu-Arabic number system! Maybe it is "simpler" but certainly not faster, because it glosses over the fact that division by 2 of Roman numerals is itself not exactly straightforward. Roman numerals are a mix of positional and unary-ish "repeated symbol" --- that's what makes them hard to manipulate.
- jacobolus 10y agoBoth doubling and halving on a base ten counting board (e.g. of the style commonly used in Europe up until at least the 16th century; Roman counting boards were presumably recognizably similar, though not much physical evidence remains) are very fast and easy. To halve any even group of pebbles, just remove half of them. To halve a lone pebble resting on a line (representing some power of ten), just move it downward into the adjacent space. To halve a number in a space (representing 5 times a power of ten), replace it by 2 pebbles on the line below, and one pebble in the space below that. These operations can be done very quickly and fluidly with a bit of practice. Maybe not quite as fast as operating a soroban, but not inordinately much slower either. Historically nobody performed arithmetic calculations by writing down a bunch of intermediate steps on scratch paper using Roman numerals (for one thing, paper wasn’t available in the time of the Romans and both papyrus and parchment were expensive). Roman numerals are just a way of recording the finished answers to calculations done on an abacus (counting board).
- chopin 10y ago>Historically nobody performed arithmetic calculations by writing down a bunch of intermediate steps on scratch paper When I was in elementary school we used a slate and chalk. I think this was quite common when paper wasn't abundant.
- labster 10y agoperl6 -e 'say Ⅶ * 2' 14 Nothing hard about this nowadays.
- rokhayakebe 10y agoWhich one requires more computing power and/pr steps (if that is a right question to ask) 72 or VII 2
- linschn 10y ago$ perl6 -e 'say VII * 2' ===SORRY!=== Error while compiling -e Two terms in a row at -e:1 ------> say VII *⏏ 2 expecting any of: infix infix stopper postfix statement end statement modifier statement modifier loop EDIT: One should use https://en.wikipedia.org/wiki/Numerals_in_Unicode#Roman_numerals_in_Unicode https://en.wikipedia.org/wiki/Numerals_in_Unicode#Roman_nume...
- my_first_acct 10y agoIf you want some up-to-date (and very in-depth) information on ancient Mesopotamian arithmetic, you might be interested in the paper "Floating Calculation in Mesopotamia" [1]. By "floating" the author means what I would call floating-point. If I understand the paper correctly, there were actually two base-60 systems: one for counting (addition and subtraction) and a different one for multiplying and dividing. [1] http://cdli.ucla.edu/pubs/cdlp/cdlp0005_20160501.pdf http://cdli.ucla.edu/pubs/cdlp/cdlp0005_20160501.pdf
- Taniwha 10y agoWe (the West) were screwed over by those who brought Arabic numbers into Europe ... They should have reversed their order as they did it ... It makes addition and subtraction easier if you work them right to left as in the original Arabic
- arjie 10y agoHow? Isn't that essentially adding and subtracting from the most significant to least significant? Could you demonstrate why it's simpler with an example? 1532 - 879 will be adequate.
- TimonKnigge 10y agoNot OP, but I guess it's easier because you work from left to right. At least, I find subtraction/addition easier to do from least to most significant digit, since the results of the less significant digits affect the more significant digits, but not the other way around. EDIT: To be clear, going from left to right is not inherently 'easier' but it would be more consistent with our direction of writing.
- arjie 10y agoIsn't that what we actually do, though? Everyone is taught to subtract or add least to most. Why would we want to flip anything.
- TimonKnigge 10y agoBecause then 'least' is on the left, and you work from left to right, in the same way that you read from left to right.
- arjie 10y agoAh I see. Thought there was more to it consider the "screwed over". Bit dramatic, I thinnk, that fellow.
- Strilanc 10y agoI think it's a fair point that the reason dividing MCMLXVI by XXXIX seems daunting is mostly due to lack of familiarity with the notation. Roman numerals jump up in value by factors of 5 then 2, which is 10, so translation to/from base-10 Arabic numerals can be done inline... assuming you ignore the god-awful reverse-order-means-subtraction rule, and the fact that you don't have any letters for fractional numbers or huge numbers. The post seems to imply you can't do the doubling algorithm with Arabic numerals, but of course that's wrong. It also associates roman numerals with abacus computations, which is probably historically sensible, but to me the columns of the pictured abacus are literally base-10 positional notation and so more akin to Arabic numerals. For the abacus to "really" be like roman numerals, you'd have to remove one bead from each section of the pictured abacus. Representing four (or one, for the top sections) would be done by moving a bead to the next grove in sequence. You'd be able to tell it was a bead from the previous section because every section would be required by law to use a different kind of bead.
- jacobolus 10y agoIt’s not clear how much the “hand abacus” pictured, with its sliding counters in grooves, was used in practice. Most calculations in Roman times were probably done with a free-form counting board with some kind of tokens (for instance “calculi”, i.e. pebbles). We don’t have a lot of surviving evidence though, since a wood/cloth/drawing-on-the-ground counting board with pebbles for counters isn’t exactly the most permanent artifact. The way such counting boards worked in Europe in the Middle Ages was to put pebbles on horizontal lines representing powers of ten, or in the spaces between the lines representing five times some power of ten. The Roman ones probably weren’t too different.
- shripadk 10y agoThank you for this: "I would point out, as I have already done in my nineteenth-century style over long title, that one should call them Hindu-Arabic numerals, as although we appropriated them from the Islamic Empire, they in turn had appropriated them from the Indians, who created them." "The Hindu-Arabic number system developed sometime in the early centuries CE and our first written account of it is from the Indian mathematician, Brahmagupta, in his Brāhmasphuṭasiddhānta (“Correctly Established Doctrine of Brahma“) written c. 628 CE. It came into Europe via Al-Khwārizmī’s treatise, On the Calculation with Hindu Numerals from 825 CE, which only survives in the 12th-century Latin translation, Algoritmi de numero Indorum." You wouldn't believe me if I said so but in Indian schools we were taught it to be "Arabic numerals" with no context of the origin of the modern-day numeral system. I'm not sure if this has changed in the recent years but that is what I was taught anyways. As a child I assumed that the origin of the modern-day numeral system was thanks to the Arabs. It was this whitewashing of anything "Hindu" that really surprises me (especially in a country like India with 70% Hindu populace). It's almost as if Hindus never contributed anything to the field of Science. It was quite later in my life that I was able to read up various articles online and realize that most Mathematical concepts originated from India.
- shripadk 10y agoMy feelings about Indian education are re-affirmed in this article: http://www.thehindu.com/features/metroplus/The-Indian-number-system/article13997406.ece http://www.thehindu.com/features/metroplus/The-Indian-number... “Akkangalude Charithram deals with everything about numbers; the science and history of digits, ancient culture and tradition which led to the discovery of the number system, and how the political, social and scientific fabric steered India towards the magic world of numbers. “This book which shows the history, science and uniqueness of the Indian number system can help instill in students a love for Mathematics. The sad part is, many Indians are still unaware that the decimal system originated around their homeland,” said Paulose Jacob.
- jacobolus 10y ago“Hindu” is even somewhat misleading, as plenty of early Indian mathematicians were Jain or Buddhist. Likewise “Arabic” is misleading, considering Islamic mathematicians working with such numerals were from a wide variety of ethnic backgrounds, such as Al-Khwarizmi who was Persian. Anyway, if you’re interested in Indian mathematics, check out this lecture https://www.youtube.com/watch?v=2HmMzhZ8zJg https://www.youtube.com/watch?v=2HmMzhZ8zJg (Sadly I don’t know of any other good online lectures about the subject.)
- meitham 10y agoDecimal numbers along with zero ٠١٢٣٤٥٦٧٨٩ were invented in India. The arabs used them until early days of Islam. Then due to fraud issues with the zero, being just a dot that can easily be overwritten to anything else, the arabs set to invent their own symbols. The new symbol is angles based, zero being a circle has no angles, 1 has a single angle and so on ... The symbols 0123456789 is yet to be adopted by many Arab countries because it was invented post writings of Quran, where it uses the original Hindi symbols.
- tn13 10y agoThere is nothing Arabic about the numerals. The numerals are pretty much Indian. May be the west learned about them from Arab travelers but Arabs haven't contributed those.
- partycoder 10y agoA proficient abacus user can learn how to do math quickly by exploiting muscle memory and imagining the abacus rather than physically using it. https://www.youtube.com/watch?v=Px_hvzYS3_Y https://www.youtube.com/watch?v=Px_hvzYS3_Y
- rahil31 10y agoHindu is the follower of Hinduism , much as muslim is the follower of islam .... but Hindi is a language much as Arabic is one .... It should be Hindi-Arabic numerals :)
- pavanky 10y agoArabic and Hindu also refer to civilizations. The name doesn't come from the languages. Besides Hindi is relatively new. It did not even exist as a language when the number system was invented.
- arjie 10y agoNothing could have demonstrated the advantage of modern numerals any more than this has. What an awfully cumbersome algorithm shift-and-add is to hand execute. With things like this (where conversion is trivial), internal use of Roman numerals would be widespread if they were advantageous. They were not, previously. and now the numeral system we use is irrelevant with ubiquitous computers except for readability . I think the evidence is quite strong that Roman numerals are a poorer numeral system.