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> Arrange the given block, if necessary, so that no ciphers [zeros] occur in its interior.
I forgot that cipher used to have a different meaning: zero, via Arabic. In some languages it means digit.
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.
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).
The word used to mean "empty" (and not algebraic zero) in both Arabic and Sanskrit.
https://www.open.ac.uk/blogs/MathEd/index.php/2022/08/25/the...
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.
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?
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…
And the English word 'algorithm' comes from Al Khwarizmi's name[1].
1: https://en.wikipedia.org/wiki/Al-Khwarizmi#:~:text=His%20nam...
Ah, but you need to dig deeper, my friend.
"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.
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).
0 is a full loop!
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).
imza is signature while şifre is password. I imagine the conflation occurred because signatures are used like passwords for authentication...
Hmm i don’t think that one is related in Turkish — i only know of “imza” as signature, but there could also be other variants.
In Romanian:
- cifru -> cipher
- cifră -> digit
In Spanish:
- cifrar -> to encipher
- cifra -> digit
In Tamil, it still means a zero. It's usually pronounced like 'cyber' though, because Tamil doesn't have the 'f'/'ph' sound natively.
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?
I think it means HN is full of misleading ideas.
Isn’t the implication that cipher is a loanword? So language relatedness is irrelevant?
We use “arabic” numerals around the world. So use of an Arabic loan word is unsurprising.
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"
Arabic -> Tamil <- Arabic - Sanskrit
Buddy English has no "shared root" with Japanese but we still say sushi.
What does it mean when someone creates a new account for posting contradictory comments?
Dutch too: "Cijfer", German, "Ziffer", French: "Chifre", Spanish: "Cifra".
Swedish: "Siffra"
That explains Major Zero and his organisation Cipher in the metal gear series
Terrence Tao blogged about this.
https://terrytao.wordpress.com/2017/08/28/dodgson-condensati...
> Dodgson’s original paper from 1867 is quite readable, surprisingly so given that math notation and terminology changes over time.
Given that Jabberwocky is also quite readable, we shouldn't be too astonished.
> 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 think in this case "readable" means "comprehensive", which maybe doesn't apply quite as much to Jabberwocky (albeit by design).
Wow, I never realized the cofactor method wasn’t the only one.
I loathed it and it put me off wanting to get into advanced matrix topics.
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:
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...
> 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.
When I'm not cognitively depleted from over working and kids I'd really like to sit down and read this properly.
And just like back in university I know how how calculate Determinants but have no clue what one would actually use it for.
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.
Rotations have determinant 1, but not all matrices of determinant 1 in the connected component of the identity are rotations
3blue1brown is your friend
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).
HN title filter cut off the initial "How".
You can manually edit it back in.
“Drop the ‘how.’ It’s cleaner.”
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.