Fun fact: zero and numerals were not invented by the Arabs. The Arabs learnt the concept & use of mathematical zero, numerals, decimal system, mathematical calculations, etc. from the ancient Hindus/Indians.
And from the Arabs, the Europeans learnt it.
Persian scholar Al Khwarizmi translated and used the Hindu/Indian numerals (including concept of mathematical zero) and "Sulba Sutras" (Hindu/Indian methods of mathematical problem solving) into the text Al-Jabr, which the Europeans translated as "Algebra" (yup, that branch of mathematics that all schoolkids worldwide learn from kindergarten).
Origin trivia: Originating from the Sanskrit word for zero शून्य (śuṇya), via the Arabic word صفر (ṣifr), the word "cipher" spread to Europe as part of the Arabic numeral system during the Middle Ages.
Fun fact: The Sanskrit word for mathematical zero and emptiness/voidness is the same: Shunya (शून्य). In fact, mathematicians are of the opinion that ancient Indians were among the first to understand the concept of mathematical zero because they understood the meaning of empty/void (Shunyata). Dhyana (meditation by focusing on voidness/stillness, away from random intrusive thoughts) is an aspect of Yoga (world's oldest active fitness discipline).
Another fun fact: The world's oldest recorded cipher (as an example of cryptography/ encryption) is the ancient Indian epic Ramayana by Maharshi Valmiki. It has 24000 verses (Sanskrit shlokas), and the first syllable (akshara) of each 1000th verse/shloka forms a series of 24 syllables that form the sacred Sri Gayatri Mantra.
Proofs of oldest records mathematical zero being of Indian origin, are available..
Yeah I believe modern trigonometry and the terms sine and cos also trace their origins to Sanskrit through Arabic. It's a shame that ancient/medieval India contributed so much to science and math but hasn't been able to innovate in centuries past :(
The word "Trigonometry" itself is of Sanskrit origin: Tri (three) + kona (corner/angle) + niti (measure). The word "meter" or "metre" is from Sanskrit "Miti" (मिती) (meaning: measure/measurement). The decimal system of weights and counting we all know so well is of Indian origin too.
India was enslaved and exploited for centuries. By the people who stole its ideas and claimed them as their own.
But greatness can only be suppressed for a while, sooner or later, it will show itself.
The world will heal from its wounds, and the truths shall surface again.
India is #5 world economy now, by the way, and will become #3 before the end of this decade. Not bad for a nation that was still a slave just a few decades ago.
Did you know..
Ancient India (subcontinent) was world #1 economy for thousands of years? Guess who made it poor?
I mean i would never dispute that India was enslaved, but I think characterizing it as they were enslaved and all there ideas were stolen is a stretch. If they were so innovative and advanced how were they enslaved? Western institutions were a major advancement, things like the caste system held back India and did not come from the west. Those systems were certainty perpetuated by the west but also be Indian leadership as well the same way slavery held the US back but still was spurred on by southern leadership.
Adopting western institutions is a large reason Japan become the dominant Asian force leading up to ww2.
I think your probably correct they will become the 3rd largest economy but they also have the second largest economy that makes a huge difference. What makes Americas economy insane is they have been about 25-50% of world GDP for the last 100 years despite being less than 5% of the population. In terms of an efficient economy they are a large way to go still but I think they will become very wealth because I agree the country is full of smart people
Q.> I mean i would never dispute that India was enslaved, but I think characterizing it as they were enslaved and all there ideas were stolen is a stretch. If they were so innovative and advanced how were they enslaved?
A.: It's because the ancient Indians focused mostly on scientific and cultural progress, while their enemies focused on warfare and destruction.
That's why ancient India built and shared the world's first universities, but the Turkish/Arabic invaders (led by Khilji) from the desert raided, destroyed and looted those priceless vast knowledge repositories.
It is always easier to destroy, it is much harder to build. It is easier to shoot a gun to kill, it is harder to build a library or a home.
Ancient Indians shared so much information to the world, but instead of thanks, the world took so much. Because it is easier to hate when you are jealous of someone's achievements and prosperity (ancient India was world's #1 economy for thousands of years, and had the most fertile lands and biggest rivers).
Q.>What makes Americas economy insane is they have been about 25-50% of world GDP for the last 100 years despite being less than 5% of the population.
A.>The super economies of America, Europe and UK were not built upon their own merits, it was all done by invading, looting and enslaving half the world, especially India, Asia and Africa. Read up on colonial history first.
It is easy to build a skyscraper or a beige in USA or UK or Europe, if you have tons of money that was looted by selling the tons of food & goods stolen from the mouths & hands of millions of Indians that starved and died on the streets of the most fertile land in the world, due to artificial famines deliberately caused by evil governance during colonial enslavement.
Churchill killed more Hindus, than Hitler killed Jews.
The colonial powers have blood on their hands.
Search Google Images for "Great Bengal Famines", "Great Madras Famines", "Great Decdan Famines". You will not get sleep after seeing those horrific images from history that has been suppressed at most schools of the world.
And before you argue about smartness, you should first find out why Wikipedia has suppressed that fact that Arabs never invented any numerals or decimal system or algebra or trigonometry or calculus -- it was all copied and translated from the ancient Indians. But Wikipedia doesn't credit the numerals to be of Indian/Hindu origin and invention.
Once you understand why and how even in modern era, the powers that be, are still suppressing India because they are afraid of India rising to be great again, then all your arguments will fall by the wayside.
Start with love of the domain and a culture of respect working in it, then move to a love of the status and respect, then a focus on those instead of the domain…
"Algorithm" is derived from Al-Khwarizmi, but only because he translated the ancient Indian/Hindu "Sulba Sutras" texts into Persian, especially in his "Al Jabr" text.
"Sulba Sutra" literally means "method of problem solving". So the Sukna Sutras were all basically Algorithms - different ways to solve mathematical and scientific problems.
In fact, Al Khwarizmi himself borrowed the title of the original Indian/Hindu texts for his translations and he even acknowledged their Indian/Hindu origin. That's why the meaning of the full title of the Al Jabr book is "The Concise Book of Calculation by Restoration and Balancing" (because that's how algebraic equations are understood and solved).
This Al Jabr book (based on Hindu methods of problem solving and algebraic equations) got translated and understood by British and Europeans, so they simply named this new (new to them) branch of Mathematics as Algebra (derived from "Al Jabr").
SOURCE: the British scholar "Robert of Chester" who translated the Al Jabr book to Latin (during 1876-2956, published in 1915, under book title "Algebra of al-Khowarizmi") documented that the ancient Indians knew the algebraic equations BEFORE Al Khwarizmi. Not only that, Robert also confirmed the ancient Indians knew and used the Pythagorean triangle theorem before the time of Pythagoras.
You can check and read these evidences for yourself please. Sources are linked below.
Trivia: in 1974, IBM released an advertisement, in which it gave the credit of Algebra to most ancient Indian mathematicians.
In the advertisement, IBM had explained 'How India gave the world the logic of indeterminate equations' by naming three prominent historic mathematicians: Aryabhata, Bhaskara and Brahmagupta, who developed the concept of Algebra and gave meaning to something (Zero) which was termed to be meaningless before.
The IBm ad proclaimed:
"History owes a debt to three Indian mathematicians of 1500 years ago who developed Algebra to give meaning to the meaningless. Bhaskara, who originated the radical signs. Brahmagupta, who created the symbols. Aryabhata, who worked out the first equations. A search that continues today in new directions with newer tools, among them, a machine that helps man in more ways than any other inventions in history: the computer. We are proud that IBM introduced the manufacturing of computers and other data processing equipment in India, which are helping the nation meet the challenge of building a new tomorrow," reads the IBM advertisement.
IBM's advertisement also features an excerpt 'The Poetry of Algebra' from the book Indian Wisdom by Sir Monier Monier-Williams.
* Among the several contributions made by Aryabhata, he discovered the nine planets and found out the correct number of days in a year i.e. 365.
* Brahmagupta made one of the most significant contributions to mathematics when he introduced zero(0), which once stood for “nothing”.
* Bhāskara declared that any number divided by zero is infinity and that the sum of any number and infinity is also infinity.
* 2000 years before Fibinacci, the Indian scholar Pingala discovered and documented by 200 BC the series we today call as Fibonacci series. Pingala wrote the Chandahśāstra, a treatise on prosody — poetic meter. To study Sanskrit meters, he analyzed long and short syllables, generating combinations using what we would now call binary patterns and recursive enumeration.
And in doing so, he uncovered (and documented) the interesting series we today call as Fibonacci series.
lol I never made that connection — in Turkish, zero is sıfır, which does sound a lot like cipher. Also, password is şifre, which again sounds similar. Looking online, apparently the path is sifr (Arabic, meaning zero) -> cifre (French, first meaning zero, then any numeral, then coded message) -> şifre (Turkish, code/cipher)
Nice! Imagine the second meaning going back to Arabic and now it's a full loop! It can even override the original meaning given enough time and popularity (not especially for "zero", but possibly for another full-loop word).
The Turkish password word may be the same used for signature? I suspect so, because in Greek we have the Greek word for signature but also a Turkish loan word τζίφρα (djifra).
I wasn't sure how encipher is in Romanian (it's not common), it's "a cifra". The infinitive in Romanian puts "a" in front of the verb, so it's very close to Spanish.
Spouse of a linguist here. That is absolutely not true. To summarize a LOT, there are multiple languages that share common roots, which linguists classify into language "families". If you go to https://en.wikipedia.org/wiki/List_of_language_families#Spok... and sort the list by number of current speakers (which adds up to far more than the population of the world because so many people speak two or more languages), you'll find the top five language families are Indo-European (which includes most European languages, including English), Sino-Tibetan (which includes Chinese), Atlantic-Congo (which includes Bantu and many other languages spoken in Africa, most of which you probably won't have heard of unless you're a linguist or you live in Africa), Afroasiatic (which includes Arabic), and Austronesian (which includes Tagalog, which you might know by the name Filipino).
It might be possible to claim that the Afroasiatic languages are all derived from Arabic, but the only influence that the Arabic language has had on Indo-European languages such as English is via loanwords (like algebra, for example). This does not make English a derivative of Arabic any more than Japanese (which has borrowed several English words such as カメラ, "kamera", from camera) is a derivative of English. Borrowing a word, or even a few dozen words, from another language does not make it a derivative. English, while it gleefully borrows loanwords from everywhere, is derived from French and German (or, to be more accurate, from Anglo-Norman and Proto-Germanic).
Can I also add that "Arabic numbers" - the numbers we use today, are actually of Indian origin, the Arabs translated the Indian logic/math texts into Arabic, and Western society used the Arabic translations (and additions like those of "Algorithm")
I have it on consumer-grade authority that the Indians got them in turn from the Shang dynasty, decimal since ca.1200BCE. Thus proving conclusively that numeral systems naturally travel deasil. Ne'er let thine diʒits, goe widdershins.
Also as long as we are going down the terminology nerd rabbit hole: it's Arabic numerals, not numbers. Numbers refers to the abstract concept, numerals refers to the method one uses to write them down.
Yeah - I quoted that to show that it was normal usage rather than technical correctness - I also did the same for the name that I didn't have the correct spelling for as I wrote the comment - not sure if I should update it (with your input) or to leave it and let people work down the thread
It's still better than believing we were apes who didn't know how to speak then we invented languages all the way to Chinese. One of us has to be ignorant that's for sure.
When someone says "it still means zero" about Tamil when responding to comments about Arabic, two languages which have no shared root and little similarity, what does that mean?
Cipher and "Arabic" numerals are not just loan words, they are loan concepts - from India. They originated from ancient India, because the Arabs adopted and translated those ideas and texts from the original texts written by ancient Indians/Hindus.
Fact - Origin of Numerals and Mathematical Zero:
Mathematical zero and numerals were not discovered/invented by the Arabs. There is no such thing as "Arabic" numerals because the Arabs did NOT invent numerals or cipher (to represent emptiness and/or mathematical zero).
The Arabs learnt the concept & use of mathematical zero, numerals, decimal system, complex mathematical calculations (including the subjects we call today as Algebra, Calculus, Trignometry), etc. from the ancient Hindus/Indians. And from the Arabs, the Europeans learnt it all.
Persian scholar Al Khwarizmi translated and used the Hindu/Indian numerals (including concept of mathematical zero) and "Sulba Sutras" (Hindu/Indian methods of mathematical problem solving) into the text Al-Jabr, which the Europeans translated as "Algebra" (yup, that branch of mathematics that all schoolkids worldwide learn from kindergarten).
Origin trivia: Originating from the Sanskrit word for zero शून्य (śuṇya), via the Arabic word صفر (ṣifr), the word "cipher" spread to Europe as part of the Arabic numeral system during the Middle Ages.
https://www.etymonline.com/word/cipher
Fun fact: The Sanskrit word for mathematical zero and emptiness/voidness is the same: Shunya (शून्य). In fact, mathematicians are of the opinion that ancient Indians were among the first to understand the concept of mathematical zero because they understood the meaning of empty/void (Shunyata). Dhyana (meditation by focusing on voidness/stillness, away from random intrusive thoughts) is an aspect of Yoga (world's oldest active fitness discipline).
Another fun fact: The world's oldest recorded cipher (as an example of cryptography/ encryption) is the ancient Indian epic Ramayana by Maharshi Valmiki. It has 24000 verses (Sanskrit shlokas), and the first syllable (akshara) of each 1000th verse/shloka forms a series of 24 syllables that form the sacred Sri Gayatri Mantra.
Proofs of oldest records mathematical zero being of Indian origin, are available..
Bakhshali manuscript (stored in Oxford) from ancient India/Bharat - is the world's oldest text having Mathematical Zero and equations.
World should know the vital mathematical concepts & representations of numerals, decimal system, binary system, algebra, calculus, trigonometry, etc. we know and use today are originated from Indian/Hindu texts and scholars.
Both Isaac Newton and Gottfried Leibniz claimed to be The Father of Calculus, but reality is that they likely learnt it from maths-savvy Jesuit missionaries who themselves simply carried the knowledge of Calculus to Europe from its source: The Kerala School of Mathematics from Malabar, India.
Wikipedia used to mention that the "Arabic" numerals are originated from "Hindu" numerals, but I see that origin has been removed from Wikipedia. It is sad when historical truths are hidden from the world, and mistruths are propagated instead.
By the way, Tamil and Sanskrit are the oldest extant (active) languages in the world.
While the use of zero and of the positional writing system for numbers have become widespread in Europe after taking them from the Arabs, which had previously taken them from India, these were already known in the Ancient World, both in Ancient Greece and even earlier in Assyria and Babylonia.
However, in the Ancient World the use of zero and positional numbers was restricted to some special applications, e.g. in astronomical tables, and it was unknown for most of the people.
The most novel feature of the Indian system was the application of the positional principle to decimal numbers, instead of sexagesimal numbers, and not the use of zero, which did not differ much from how it was used earlier.
The most novel feature of the Indian numerals is the use of the rounded zero symbol for the mathematical zero, so we all know it so well.
The Indian numerals also covered all the basic numeral digits and fit perfectly into the decimal system (which was also invented by the ancient Indians - they primarily used it for measuring weights, especially for currency/trade). The word meter/metre (from Sanskrit "miti") is also of Indian origin. The mathematical zero also fits in perfectly with binary system, also an ancient Indian invention.
The ancient Babylonians did use a dot/period as symbol of zero, but there is no information on whether they also associated zero with voidness/emptiness which the ancient Indians certainly did.
It can be argued that the ancient Babylonians and Indians independently discovered the concept of mathematical zero, and rest of the world learnt such basic concepts from them gradually,
Interestingly, while even modern science+mathematics only uses big numbers to a certain extent, the ancient Indian Jain's & Hindus were doing computations of up to 10^32! Hindu cosmology even calculates time up to 10^15, and knows about multiverse, whereas modern science calculates Time only upto billions of years (10^9) and only recently started acknowledging the possibility of multiverse (as it is only explanation of what existed before the Big Bang), I
i.e. Time is cyclical, and universes are birthed (Big Bang), grow (expand), decay (collapse) and shrink back to the Infinitesimal Dot again).
So it is a shame that ancient India's contributions to mathematics and other fields (e.g., geography, surgeries, medical tools, metallurgy, etc.) are unknown and ignored by most of the world, and the credits for such knowledge were stolen.
Did you know?..
India built and managed the world's first universities, in Takshashila and Nalanda, which has lots of diverse subjects/disciplines being taught and researched. The Arabs/Turks later invaded, looted and destroyed these amazing universities and their priceless treasure trove of books (the libraries were so huge that the arson fires burnt for months). The ruins of these ancient pioneering repositories of knowledge still stand as mute witnesses to their glorious knowledge-sharing past.
The original comment was about one language that borrowed cipher from Arabic (i.e. English) where the word no longer means zero. So my comment was about a different language that also borrowed the word cipher (i.e. Tamil) where it still retains that meaning.
So is Gemini. but from it I gather there might be something interesting about a word that "loops back" (geographically) but evolutionarily speaking it was a reworking of _independent_ discoveries of "emptiness"
Eyeballing the Wikipedia page, and out of the only scripts I could read, I counted 72 languages that used a direct transcription of "sushi". It isn’t as much a superpower as a thing languages do in general.
I don't think determinants play a central role in modern advanced matrix topics.
Luckily for me I read Axler's "Linear Algebra Done Right" (which uses determinant-free proofs) during my first linear algebra course, and didn't concern myself with determinants for a very long time.
Edit: Beyond cofactor expansion everyone should know of at least one quick method to write down determinants of 3x3 matrices. There is a nice survey in this paper:
> I don't think determinants play a central role in modern advanced matrix topics.
Not true at all. It's integral to determinantal stochastic point processes, commute distances in graphs, conductance in resistor networks, computing correlation via linear response theory, enumerating subgraphs, representation theory of groups, spectral graph theory... I am sure many more
The 4th edition of Linear Algebra Done Right has a much improved approach to determinants themselves (still relegated to the end, where it should be). From the list of improvements:
> New Chapter 9 on multilinear algebra, including bilinear forms, quadratic forms, multilinear forms, and tensor products. Determinants now are defned using a basis-free approach via alternating multilinear forms.
The basis-free definition is really rather lovely.
As another poster has also said, the determinant of a matrix provides 2 very important pieces of information about the associated linear transformation of the space.
The sign of the determinant tells you whether the linear transformation includes a mirror reflection of the space, or not.
The absolute value of the determinant tells you whether the linear transformation preserves the (multi-dimensional) volume (i.e. it is an isochoric transformation, which changes the shape without changing the volume), or it is an expansion of the space or a compression of the space, depending on whether the absolute value of the determinant is 1, greater than 1 or less than 1.
To understand what a certain linear transformation does, one usually decomposes it in several kinds of simpler transformations (by some factorization of the matrix), i.e. rotations and reflections that preserve both size and shape (i.e. they are isometric transformations), isochoric transformations that preserve volume but not shape, and similitude transformations (with the scale factor computed from the absolute value of the determinant), which preserve shape, but not volume. The determinant provides 2 of these simpler partial transformations, the reflection and the similitude transformation.
Suppose you have (let's say) a 3x3 matrix. This is a linear transformation that maps real vectors to real vectors. Now let's say you have a cube as input with volume 1, and you send it into this transformation. The absolute value of the determinant of the matrix tells you what volume the transformed cube will be. The sign tells you if there is a parity reversal or not.
Here are three reasons you want to be able to calculate the volume change for arbitrary parallelpipeds:
- If det M = 0, then M is not invertible. Knowing this is useful for all kinds of reasons. It means you cannot solve an equation like Mx = b by taking the inverse ("dividing") on both sides, x = M \ b. It means you can find the eigenvalues of a matrix by rearranging Mx = λx <--> (M-λI)x = 0 <--> det M-λI = 0, which is a polynomial equation.
- Rotations are volume-preserving, so the rotation group can be expressed as the matrices where det M = 1 (well, the component connected to the identity). This is useful for theoretical physics, where they're playing around with such groups and need representations they can do things with.
- In information theory, the differential entropy (or average amount of bits it takes to describe a particular point in a continuous probability distribution) increases if you spread out the distribution, and decreases if you squeeze it together by exactly log |det M| for a linear transformation. A nonlinear transformation can be linearized with its gradient. This is useful for image compression (and thus generation) with normalizing flow neural networks.
Form a quadratic equation to solve for the eigenvalues x of a 2x2 matrix (|A - xI| = 0). The inverse of a matrix can be calculated as the classical adjugate multiplied by the reciprocal of the determinant. Use Cramer's Rule to solve a system of linear equations by computing determinants. Reason that if x is an eigenvalue of A then A - xI has a non-trivial nullspace (using the mnemonic |A - xI| = 0).
It gives it a different implication. As I read it, an article titled "Lewis Carroll Computed Determinates" has three possible subjects:
1. Literally, Carroll would do matrix math. I know, like many on HN, that he was a mathematician. So this would be a dull and therefore unlikely subject.
2. Carroll invented determinates. This doesn't really fit the timeline of math history, so I doubt it.
3. Carroll computed determinates, and this was surprising. Maybe because we thought he was a bad mathematician, or the method had recently been invented and we don't know how he learned of it. This is slightly plausible.
4. (The actual subject). Carroll invented a method for computing determinates. A mathematician inventing a math technique makes sense, but the title doesn't. It'd be like saying "Newton and Leibnitz Used Calculus." Really burying the lede.
Of course, this could've been avoided had the article not gone with a click-bait style title. A clearer one might've been "Lewis Carroll's Method for Calculating Determinates Is Probably How You First Learned to Do It." It's long, but I'm not a pithy writer. I'm sure somebody could do better.
"How Lewis Carroll Computed Determinates" is fine and not clickbait because it provides all the pertinent information and is an accurate summary of its contents. Clickbait would be "you would never guess how this author/mathematician computed determinants" since it requires a clickthrough to know who the person is. How is perfectly fine IMO to have in the title because I personally would expect the How to be long enough to warrant a necessary clickthrough due to the otherwise required title length.
I forgot that cipher used to have a different meaning: zero, via Arabic. In some languages it means digit.
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