The taxi cab story is easily a top-5 math story, and is quintessential Ramanujan. Has there been a genius of his kind since? Maybe Terry Tao, but his work also lacks the ease and lack of machinery that Ramanujan had. Truly amazing.
What are the other 4 top math stories? For me one of them has to be of Évariste Galois[1], who, legend has it, hastily wrote fragments of his last mathematical discoveries on his shirt sleeves before fighting the duel that would end his life. [1] - https://en.wikipedia.org/wiki/%C3%89variste_Galois
Ramanujan Surprises Again (2015)
61–70 of 96 posts
Re: Ramanujan Surprises Again (2015)
#62Earlier quoted context omitted.
Another fun fact: 1 + 7 + 2 + 9 = 19 ; 19 × 91 = 1729
This kind of coincidence is just cute, it doesn't imply anything useful mathematically right?
(something like treating the digits as coefficients of a series of powers of the number base, defining a function f(b,n) that returns the sum of the digits of n in base b, setting that equal to the product of the forward and reverse representation of that result in base b, and seeing if the equation looks interesting)
E.g. you can use a similar technique to show why the finger counting method of multiplying by 9 works, or why multiples of 3 have digits that sum to a multiple of 3 (same for 9)
Re: Ramanujan Surprises Again (2015)
#63Earlier quoted context omitted.
It never occurred to me that 7% of 50 is equal to 50% of 7 perhaps because they are equally easy to calculate? Multiply 7 by 5 and fix the decimal point.
What about 17% of 50? 17*5 isn’t so simple anymore, whereas 50% of 17 is 8.5. To be fair, with most shortcuts, it’s possible to construct difficult cases (17% of 23 is difficult in either order) but where it applies (when one of the pairs is a common percentage), exploiting commutativity can be quite useful. Plus the mental overhead of remembering the rule is extremely minimal.
Re: Ramanujan Surprises Again (2015)
#64Earlier quoted context omitted.
Yes definitely: Alexandre Grothendieck. And Terrence Tao can’t sit at his table (yet?). But honestly it’s kind of a silly game to rank mathematicians this way.
Grothendieck and lack of machinery do not belong in the same sentence. But yes, it is kind of silly.
They most certainly do! Grothendieck’s work is heavy on definitions, but the essence of his work is that the right definitions obviate (and are seen to be right because they obviate) the need for heavy machinery. See the famous quote, taken from Wikipedia (https://en.wikiquote.org/wiki/Alexander_Grothendieck#Quotes_...) because that’s the first place I found it:
> I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance.
Re: Ramanujan Surprises Again (2015)
#65Earlier quoted context omitted.
A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one…
I think most of us are impressed by computational parlor tricks (and indeed raw computational intelligence in general -- being able to process information and compute quickly and accurately), but for me, genius goes beyond that. Genius is about having rare and useful insights that the rest of us are incapable of, and that a computer is unable to easily replicate. For instance, there was this thing on Twitter recently…
Re: Ramanujan Surprises Again (2015)
#66He credited his work to his family goddess. From wikipedia: "A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said the mathematical knowledge he displayed was revealed to him by his family goddess. "An equation for me has no meaning," he once said, "unless it expresses a thought of God.""
Mathematics are the best expression of the transcendental divine. Pythagoras and Plato had the same perspective.
Thomas Paine wrote on his point of view in "The Age of Reason" [0]:
> Each of those churches show certain books, which they call revelation, or the word of God. The Jews say, that their word of God was given by God to Moses, face to face; the Christians say, that their word of God came by divine inspiration: and the Turks say, that their word of God (the Koran) was brought by an angel from Heaven. Each of those churches accuse the other of unbelief; and for my own part, I disbelieve them all.
> As it is necessary to affix right ideas to words, I will, before I proceed further into the subject, offer some other observations on the word revelation. Revelation, when applied to religion, means something communicated immediately from God to man.
> No one will deny or dispute the power of the Almighty to make such a communication, if he pleases. But admitting, for the sake of a case, that something has been revealed to a certain person, and not revealed to any other person, it is revelation to that person only. When he tells it to a second person, a second to a third, a third to a fourth, and so on, it ceases to be a revelation to all those persons. It is revelation to the first person only, and hearsay to every other, and consequently they are not obliged to believe it.
> It is a contradiction in terms and ideas, to call anything a revelation that comes to us at second-hand, either verbally or in writing. Revelation is necessarily limited to the first communication- after this, it is only an account of something which that person says was a revelation made to him; and though he may find himself obliged to believe it, it cannot be incumbent on me to believe it in the same manner; for it was not a revelation made to me, and I have only his word for it that it was made to him.
> ...
> But some, perhaps, will say: Are we to have no word of God- no revelation? I answer, Yes; there is a word of God; there is a revelation.
> THE WORD OF GOD IS THE CREATION WE BEHOLD and it is in this word, which no human invention can counterfeit or alter, that God speaketh universally to man.
> …
> It is only in the CREATION that all our ideas and conceptions of a word of God can unite. The Creation speaketh an universal language, independently of human speech or human language, multiplied and various as they may be. It is an ever-existing original, which every man can read. It cannot be forged; it cannot be counterfeited; it cannot be lost; it cannot be altered; it cannot be suppressed. It does not depend upon the will of man whether it shall be published or not; it publishes itself from one end of the earth to the other. It preaches to all nations and to all worlds; and this word of God reveals to man all that is necessary for man to know of God.
> Do we want to contemplate his power? We see it in the immensity of the Creation. Do we want to contemplate his wisdom? We see it in the unchangeable order by which the incomprehensible whole is governed! Do we want to contemplate his munificence? We see it in the abundance with which he fills the earth. Do we want to contemplate his mercy? We see it in his not withholding that abundance even from the unthankful. In fine, do we want to know what God is? Search not the book called the Scripture, which any human hand might make, but the Scripture called the Creation.
> The only idea man can affix to the name of God is that of a first cause, the cause of all things. And incomprehensible and difficult as it is for a man to conceive what a first cause is, he arrives at the belief of it from the tenfold greater difficulty of disbelieving it. It is difficult beyond description to conceive that space can have no end; but it is more difficult to conceive an end. It is difficult beyond the power of man to conceive an eternal duration of what we call time; but it is more impossible to conceive a time when there shall be no time.
> In like manner of reasoning, everything we behold carries in itself the internal evidence that it did not make itself Every man is an evidence to himself that he did not make himself; neither could his father make himself, nor his grandfather, nor any of his race; neither could any tree, plant, or animal make itself; and it is the conviction arising from this evidence that carries us on, as it were, by necessity to the belief of a first cause eternally existing, of a nature totally different to any material existence we know of, and by the power of which all things exist; and this first cause man calls God.
> ...
> That which is now called natural philosophy, embracing the whole circle of science, of which astronomy occupies the chief place, is the study of the works of God, and of the power and wisdom of God in his works, and is the true theology.
---
Re: Ramanujan Surprises Again (2015)
#67Earlier quoted context omitted.
A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one…
This reminds me of the von Neumann fly puzzle story: https://en.wikipedia.org/wiki/John_von_Neumann#Cognitive_abi...
Re: Ramanujan Surprises Again (2015)
#68Earlier quoted context omitted.
What was the quote about Feynman? That he loved to cultivate anecdotes about himself or something similar? Makes a lot of his stories make a lot more sense, too.
i recall him explaining several shortcuts one can use to solve problems in seemingly impossible speeds by drawing on a breadth of experience from similar problems that you have memorized or are easy to compute and interpolating. its still genius but not in the sense of actually being able to do huge calculations in ones head the way a computer would.
Re: Ramanujan Surprises Again (2015)
#69He credited his work to his family goddess. From wikipedia: "A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said the mathematical knowledge he displayed was revealed to him by his family goddess. "An equation for me has no meaning," he once said, "unless it expresses a thought of God.""
Recently visited the "Ramanujan Museum":
* Where we could see his family goddess - Sri Namakkal Namagiri Thayaar who he credited for his works:
http://casualwalker.com/wp-content/uploads/2019/08/Ramanujan...
* His desk where he made his early mathematical findings:
http://casualwalker.com/wp-content/uploads/2019/08/Ramanujan...
* Really sadden to see how a Ramanujan's wife lived her life after Ramanujan's demise:
http://casualwalker.com/wp-content/uploads/2019/08/Ramanujan...
* Check the full photo coverage about Ramanujan and his fascinating works at:
http://casualwalker.com/museum-for-the-man-who-knew-infinity...
Re: Ramanujan Surprises Again (2015)
#70Ramanujan also claimed 1 + 2 + 3 + ... = -1/12. How does that work? Who can explain this to me?
I think this is basically a “shock value” interpretation of a more subtle statement. Obviously adding strictly-positive numbers does not result in a negative number under normal arithmetic. Check the numberphile video and you may be simultaneously irritated and disappointed.
[0] https://www.youtube.com/watch?v=YuIIjLr6vUA [1] https://www.youtube.com/watch?v=jcKRGpMiVTw