How Lewis Carroll computed determinants (2023)
johndcook.com
johndcook.com
I forgot that cipher used to have a different meaning: zero, via Arabic. In some languages it means digit.
- cifru -> cipher
- cifră -> digit
- cifrar -> to encipher
- cifra -> digit
I think it means HN is full of misleading ideas.
What does it mean when someone creates a new account for posting contradictory comments?
Sushi is now an English word. So is hummus, etc.
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.
We use “arabic” numerals around the world. So use of an Arabic loan word is unsurprising.
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.
https://en.wikipedia.org/wiki/Hindu-Arabic_numeral_system
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
https://en.wikipedia.org/wiki/Cipher#Etymology
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..
https://thebetterindia.com/270912/chaturbhuj-temple-in-gwali...
World's oldest known evidence of Mathematical Zero and numerals - ancient inscription on wall of Chaturbhuj temple in Gwalior, India.
https://www.glam.ox.ac.uk/article/carbon-dating-finds-bakhsh...
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.
https://www.manchester.ac.uk/about/news/indians-predated-new...
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.
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 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.
Arabic -> Tamil <- Arabic - Sanskrit
https://en.wikipedia.org/wiki/Hindu-Arabic_numeral_system
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).
https://www.open.ac.uk/blogs/MathEd/index.php/2022/08/25/the...
https://www.etymonline.com/word/cipher
https://en.wikipedia.org/wiki/Cipher#Etymology
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..
https://thebetterindia.com/270912/chaturbhuj-temple-in-gwali...
World's oldest known evidence of Mathematical Zero and numerals - ancient inscription on wall of Chaturbhuj temple in Gwalior, India.
https://www.glam.ox.ac.uk/article/carbon-dating-finds-bakhsh...
Bakhshali manuscript (stored in Oxford) from ancient India/Bharat - is the world's oldest text having Mathematical Zero and equations.
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?
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
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.
1: https://en.wikipedia.org/wiki/Al-Khwarizmi#:~:text=His%20nam...
"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.
https://web.archive.org/web/20181118154937/http://library.al...
https://en.wikipedia.org/wiki/Robert_of_Chester
https://archive.org/details/robertofchesters00khuw
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.
https://terrytao.wordpress.com/2017/08/28/dodgson-condensati...
Given that Jabberwocky is also quite readable, we shouldn't be too astonished.
The conventions of literature have changed a lot less than math notation and terminology have since 1867.
I loathed it and it put me off wanting to get into 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:
Dardan Hajriza, "New Method to Compute the Determinant of a 3x3 Matrix," International Journal of Algebra, Vol. 3, 2009, no. 5, 211 - 219. https://www.m-hikari.com/ija/ija-password-2009/ija-password5...
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
> 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.
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.
- 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.
You can manually edit it back in.
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.