8 ms·

Ramanujan's Easiest Formula

215 points by johncarlosbaez 6y ago

justforfunhere 6y ago
It's a puzzle within a puzzle as the post author describes and describes in detail in the presentation he has made. Took me back to my college days. Here's the direct link to the presentation the post author links to in the blog https://math.ucr.edu/home/baez/ramanujan/ramanujan_whittier_web.pdf https://math.ucr.edu/home/baez/ramanujan/ramanujan_whittier_...
- Sniffnoy 6y ago
  And here are direct links to the pair of posts where he discussed it on the n-Category Cafe: https://golem.ph.utexas.edu/category/2020/08/chasing_the_tail_of_the_gaussi.html https://golem.ph.utexas.edu/category/2020/08/chasing_the_tai... https://golem.ph.utexas.edu/category/2020/09/chasing_the_tail_of_the_gaussi_1.html https://golem.ph.utexas.edu/category/2020/09/chasing_the_tai...
- ddxxdd 6y ago
  I am not a fan of these types of proofs. Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. In other words, this presentation doesn't seem to teach me how to think like a mathematician, it seems more like showing me how mathematicians can find solutions that nobody else could ever find in a million years. I guess that's the point when the topic is about Ramanujan, maybe?
  - TheGallopedHigh 6y ago
    Unfortunately sometimes mathematics just uses tricks to do certain things. As you move along in your math learning you pick up a bag of tricks to deal with certain situations that you’ve seen before.
  - jan_Inkepa 6y ago
    When you see some infinite nested pattern, you try to capture its symmetries in some way. Differential equations are one such way. Or looking for symmetries. Induction can also help, or looking for fixed points. They can all give information. It’s a bag of tools. One could go and then look at the differential equation in more detail and relate it geometrically to the series and maybe learn some cool things (but that would be a posteriori, which you don’t want!) Also, the differential equation doesn’t come from nowhere/is not random, it’s derived in the presentation from differentiating the function and seeing that it still resembles the original function in some way, allowing you to describe it with a differential equation. What he describes as a trick with solving the differential equation can be explained - if you have f’(x)=A(x)f(x)+b, that’s a strong hint there’s an exponential there somewhere; if A is x, then the chain rule hints that you have and x^2 in the exponential, etc... A lot of it (depending on what mathematician) can boil down to pattern matching and having a big-enough bag of tricks.
  - dan-robertson 6y ago
    The source of the derivation of the continued fraction part is https://mobile.twitter.com/duetosymmetry/status/1302021471749971970 https://mobile.twitter.com/duetosymmetry/status/130202147174... and it was an attempt to simplify a derivation due to Jacobi. Perhaps he explains how he comes up with it, though you’ll need to be able to read Latin. Possibly this is a relatively standard technique for solving differential equations with continued fractions which people no longer use.
  - MaxBarraclough 6y ago
    I'm reminded of this answer, regrettably on Quora, about how mathematicians have an unfortunate habit of tearing down the scaffolding they used to build their results. https://www.quora.com/How-did-Strassen-derive-his-matrix-multiplication-algorithm/answer/Robert-Smith-9 https://www.quora.com/How-did-Strassen-derive-his-matrix-mul...
  - sn41 6y ago
    There is a lot of pattern-based theorem proving in the works of Ramanujan, Jacobi and Euler. This is beautiful mathematics, but this is not how usual mathematical research is done. Thinking like a mathematician often involves generalization, finding analogies between seemingly different theorems etc. "Good mathematicians see analogies between theorems or theories. the very best ones see analogies between analogies." Stefan Banach.
  - seppel 6y ago
    > I am not a fan of these types of proofs. Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. Indeed. But without knowing what really happend here, I suspect the following: Somebody (I guess Ramanujan), came up with the random differential equation, then found two equal solutions that looked structurally different. So he asked how you can prove that they are equal. So, I'm not sure here, but in general, this is how many mathematical puzzles are created: You come up with a random proof by starting somewhere in the middle that then transforming your equation in two totally different directions. If you now throw away the middle part, you have a hard to prove fact.
  - gus_massa 6y ago
    > he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly Yep. One of the main differences of advanced Math is that the problem gives no clues about the solution. You look at the problem, you pick one of the standard tricks from your backpack of trick, and try to hit the problem as hard as you can. Sometimes the trick solves the problem. Sometimes the trick simplifies the problem. (Sometimes it is not obvious that it is a simplification). Sometimes the trick does nothing, so you just pick another trick from your backpack of tricks... If that doesn't work, you call a friend that has another backpack full of tricks ... The idea is that in a Math BS or PhD you see a lot of tricks, and get some advice about where each one can be useful. Transforming an infinite sum to a function is an standard trick. Transforming that to a differential equation is not so standard, but I've seen it before. And sometimes no trick solves the problem, so you must invent a new trick. After a few year, if the trick is useful in other problems it will become popular and it will be added to the standard curriculum of a major in Math, or to the advanced classes for PhD, or just be a standard trick in a small niche.
    - kkwteh 6y ago
      There are two ways to solve a math problem and you described one of them: hit the nut with a hammer as hard as you can until it breaks. Grothendieck preferred the second method of slowly raising the ocean and soaking the nut for a few years, until one day it opens all by itself.
      - jan_Inkepa 6y ago
        Homeopathic approaches can feel just as arbitrary, though. “How did the mathematician know to take a bazillion tiny steps to get here?”, “Which steps in this process are the ones that do anything?”, and “Why can’t they just get to the point?!” are all questions one might ask oneself while in the midst of reading such material.
    - mrdmnd 6y ago
      This is the best comment I've ever seen describing the actual practice of mathematics.
  - scythe 6y ago
    >Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. If you're not used to working with continued fractions, the techniques are very counterintuitive. None of the usual calculus tricks for infinite sequences work normally. It's probably not best to read that part as an introduction to the theory of continued fractions. As Baez himself admitted, even he had a hard time understanding Laplace's proof. The rest of the development, I assume, was not so confounding. It's a little of both: here's something you might understand, there's some black magic.
anythingdude321 6y ago
aahhh just another one of Ramanujan's random identities that he divined from the minds of the gods. reading these has been a reliable way for mathematicians to humble themselves since 1900. he often wouldn't (or wasn't able to) prove them. they were just assertions that, much more often than not, turned out to be accurate.
- Crazyontap 6y ago
  I don't understand what you mean. He couldn't have just come up with that huge equation on the page out of thin air? Surely there would be a method for coming up with such assertions (maybe only known to him though)
  - shmerl 6y ago
    According to him it wasn't a method, but an insight. So if you like to phrase it that way, you could say it he came up with it "out of thin air".
  - dharma1 6y ago
    "Ramanujan said that, throughout his life, he repeatedly dreamed of a Hindu goddess known as Namagiri. She presented him with complex mathematical formulas over and over, which he could then test and verify upon waking. Once such example was the infinite series for Pi: Describing one of his many insightful math dreams, Ramanujan said: "While asleep I had an unusual experience. There was a red screen formed by flowing blood as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of results in elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing...""
    - emmanueloga_ 6y ago
      Reminds me of TempleOS, "created by American programmer Terry A. Davis, who developed it alone over the course of a decade after a series of prophetic episodes that he later described as a revelation from God."
    - 6y ago
deleted 6y ago
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fireattack 6y ago
The article says "Cauchy’s two-page paper" but the slides linked in the article says "Jacobi wrote a 2-page paper". Who is the author of the paper?
- deleted 6y ago
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- dan-robertson 6y ago
  Jacobi
lanevorockz 6y ago
He wrote it when he was merely 2 years old.
- dan-robertson 6y ago
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jarenmf 6y ago
Interesting relevant paper https://arxiv.org/abs/math/0607694 https://arxiv.org/abs/math/0607694
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MrXOR 6y ago
Thank you John. My favorite Ramanujan’s formula: 1 + 2 + 3 + 4 + ⋯ = −1/12
- RHSeeger 6y ago
  That one is kind of silly.
  - gus_massa 6y ago
    It is weird, very weird. A very nice explanation of the proof is in a video by Mathologer "Ramanujan: Making sense of 1+2+3+... = -1/12 and Co." https://www.youtube.com/watch?v=jcKRGpMiVTw https://www.youtube.com/watch?v=jcKRGpMiVTw [spoiler alert] You can extend finite summations to infinite summations. The first extensions are nice and you get intuitive results, and are the extensions studied in Calculus in the university. But these extensions are not enough for this sum. You must make more bold extensions, and the results are not as intuitive, and not useful outside some special applications.
    - RHSeeger 6y ago
      Honestly, I don't buy the proof. When you take a formula that's valid for a certain range of inputs and just use it outside that range anyways, you're no longer in the land of proof. At that point, you're in the land of "if we extend out mathematical system, we get this". And that's fine, but you don't wind up with "sum(1...inf) === -1/12", you wind up with "in this other mathematical system, sum....".
      - gus_massa 6y ago
        I'm not going to blame you. As the video explain, this is not a straightforward and obvious generalization of summation. I think that the generalization that use averages are fine, and some analytic continuations using power series are fine. Analytic functions are just too good to ignore them. It is not another mathematical system. It it just the extension of our mathematical system. Anyway, to get a result for this sum, you must drop too many of the obvious and expected properties of summation. For example if you add a zero in front of the sum, the result changes :(. There was a nice blog post I can't find, but there is a hint of the problem in https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Failure_of_stable_linear_summation_methods https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B... . So the result is ... polemic. Anyway, in some courses of a Math degree, one of the important ideas is that you have many possible definitions of convergence. For example, if you have a sequence of functions, you have https://en.wikipedia.org/wiki/Uniform_absolute-convergence https://en.wikipedia.org/wiki/Uniform_absolute-convergence and https://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_space) https://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_spac... and many many many more. So you don't write lim f_n -> f, you must write lim_{something} f_n -> f where something explain which definition of convenience you are using. The trick here is to hide the nasty problem of the multiple definitions of convergence in the dots. So the correct statement of the problem is 1+2+3+4+..._{with a weird convergence}=-1/12 but it doesn't look as nice as 1+2+3+4+...=-1/12
29athrowaway 6y ago
Is it there an alternative notation for continued fractions in the style of pi notation?
- groupdeterminac 6y ago
  You'll sometimes see it like so, written with a capital K
29athrowaway 6y ago
Ramanujan's number, 1729: > I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
jason_slack 6y ago
A few videos that talk about Ramanujan's genius https://www.youtube.com/watch?v=eURuh9aS3eo https://www.youtube.com/watch?v=eURuh9aS3eo https://www.youtube.com/watch?v=XFsuRxospbU https://www.youtube.com/watch?v=XFsuRxospbU

It's a puzzle within a puzzle as the post author describes and describes in detail in the presentation he has made. Took me back to my college days. Here's the direct link to the presentation the post author links to in the blog https://math.ucr.edu/home/baez/ramanujan/ramanujan_whittier_web.pdf https://math.ucr.edu/home/baez/ramanujan/ramanujan_whittier_...

Sniffnoy 6y ago
And here are direct links to the pair of posts where he discussed it on the n-Category Cafe: https://golem.ph.utexas.edu/category/2020/08/chasing_the_tail_of_the_gaussi.html https://golem.ph.utexas.edu/category/2020/08/chasing_the_tai... https://golem.ph.utexas.edu/category/2020/09/chasing_the_tail_of_the_gaussi_1.html https://golem.ph.utexas.edu/category/2020/09/chasing_the_tai...
ddxxdd 6y ago
I am not a fan of these types of proofs. Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. In other words, this presentation doesn't seem to teach me how to think like a mathematician, it seems more like showing me how mathematicians can find solutions that nobody else could ever find in a million years. I guess that's the point when the topic is about Ramanujan, maybe?
- TheGallopedHigh 6y ago
  Unfortunately sometimes mathematics just uses tricks to do certain things. As you move along in your math learning you pick up a bag of tricks to deal with certain situations that you’ve seen before.
- jan_Inkepa 6y ago
  When you see some infinite nested pattern, you try to capture its symmetries in some way. Differential equations are one such way. Or looking for symmetries. Induction can also help, or looking for fixed points. They can all give information. It’s a bag of tools. One could go and then look at the differential equation in more detail and relate it geometrically to the series and maybe learn some cool things (but that would be a posteriori, which you don’t want!) Also, the differential equation doesn’t come from nowhere/is not random, it’s derived in the presentation from differentiating the function and seeing that it still resembles the original function in some way, allowing you to describe it with a differential equation. What he describes as a trick with solving the differential equation can be explained - if you have f’(x)=A(x)f(x)+b, that’s a strong hint there’s an exponential there somewhere; if A is x, then the chain rule hints that you have and x^2 in the exponential, etc... A lot of it (depending on what mathematician) can boil down to pattern matching and having a big-enough bag of tricks.
- dan-robertson 6y ago
  The source of the derivation of the continued fraction part is https://mobile.twitter.com/duetosymmetry/status/1302021471749971970 https://mobile.twitter.com/duetosymmetry/status/130202147174... and it was an attempt to simplify a derivation due to Jacobi. Perhaps he explains how he comes up with it, though you’ll need to be able to read Latin. Possibly this is a relatively standard technique for solving differential equations with continued fractions which people no longer use.
- MaxBarraclough 6y ago
  I'm reminded of this answer, regrettably on Quora, about how mathematicians have an unfortunate habit of tearing down the scaffolding they used to build their results. https://www.quora.com/How-did-Strassen-derive-his-matrix-multiplication-algorithm/answer/Robert-Smith-9 https://www.quora.com/How-did-Strassen-derive-his-matrix-mul...
- sn41 6y ago
  There is a lot of pattern-based theorem proving in the works of Ramanujan, Jacobi and Euler. This is beautiful mathematics, but this is not how usual mathematical research is done. Thinking like a mathematician often involves generalization, finding analogies between seemingly different theorems etc. "Good mathematicians see analogies between theorems or theories. the very best ones see analogies between analogies." Stefan Banach.
- seppel 6y ago
  > I am not a fan of these types of proofs. Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. Indeed. But without knowing what really happend here, I suspect the following: Somebody (I guess Ramanujan), came up with the random differential equation, then found two equal solutions that looked structurally different. So he asked how you can prove that they are equal. So, I'm not sure here, but in general, this is how many mathematical puzzles are created: You come up with a random proof by starting somewhere in the middle that then transforming your equation in two totally different directions. If you now throw away the middle part, you have a hard to prove fact.
- gus_massa 6y ago
  > he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly Yep. One of the main differences of advanced Math is that the problem gives no clues about the solution. You look at the problem, you pick one of the standard tricks from your backpack of trick, and try to hit the problem as hard as you can. Sometimes the trick solves the problem. Sometimes the trick simplifies the problem. (Sometimes it is not obvious that it is a simplification). Sometimes the trick does nothing, so you just pick another trick from your backpack of tricks... If that doesn't work, you call a friend that has another backpack full of tricks ... The idea is that in a Math BS or PhD you see a lot of tricks, and get some advice about where each one can be useful. Transforming an infinite sum to a function is an standard trick. Transforming that to a differential equation is not so standard, but I've seen it before. And sometimes no trick solves the problem, so you must invent a new trick. After a few year, if the trick is useful in other problems it will become popular and it will be added to the standard curriculum of a major in Math, or to the advanced classes for PhD, or just be a standard trick in a small niche.
  - kkwteh 6y ago
    There are two ways to solve a math problem and you described one of them: hit the nut with a hammer as hard as you can until it breaks. Grothendieck preferred the second method of slowly raising the ocean and soaking the nut for a few years, until one day it opens all by itself.
    - jan_Inkepa 6y ago
      Homeopathic approaches can feel just as arbitrary, though. “How did the mathematician know to take a bazillion tiny steps to get here?”, “Which steps in this process are the ones that do anything?”, and “Why can’t they just get to the point?!” are all questions one might ask oneself while in the midst of reading such material.
  - mrdmnd 6y ago
    This is the best comment I've ever seen describing the actual practice of mathematics.
- scythe 6y ago
  >Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. If you're not used to working with continued fractions, the techniques are very counterintuitive. None of the usual calculus tricks for infinite sequences work normally. It's probably not best to read that part as an introduction to the theory of continued fractions. As Baez himself admitted, even he had a hard time understanding Laplace's proof. The rest of the development, I assume, was not so confounding. It's a little of both: here's something you might understand, there's some black magic.

anythingdude321 6y ago

aahhh just another one of Ramanujan's random identities that he divined from the minds of the gods. reading these has been a reliable way for mathematicians to humble themselves since 1900. he often wouldn't (or wasn't able to) prove them. they were just assertions that, much more often than not, turned out to be accurate.

Crazyontap 6y ago
I don't understand what you mean. He couldn't have just come up with that huge equation on the page out of thin air? Surely there would be a method for coming up with such assertions (maybe only known to him though)
- shmerl 6y ago
  According to him it wasn't a method, but an insight. So if you like to phrase it that way, you could say it he came up with it "out of thin air".
- dharma1 6y ago
  "Ramanujan said that, throughout his life, he repeatedly dreamed of a Hindu goddess known as Namagiri. She presented him with complex mathematical formulas over and over, which he could then test and verify upon waking. Once such example was the infinite series for Pi: Describing one of his many insightful math dreams, Ramanujan said: "While asleep I had an unusual experience. There was a red screen formed by flowing blood as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of results in elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing...""
  - emmanueloga_ 6y ago
    Reminds me of TempleOS, "created by American programmer Terry A. Davis, who developed it alone over the course of a decade after a series of prophetic episodes that he later described as a revelation from God."
  - 6y ago

deleted 6y ago

[deleted]

fireattack 6y ago

The article says "Cauchy’s two-page paper" but the slides linked in the article says "Jacobi wrote a 2-page paper". Who is the author of the paper?

deleted 6y ago
[deleted]
dan-robertson 6y ago
Jacobi

lanevorockz 6y ago

He wrote it when he was merely 2 years old.

dan-robertson 6y ago
27

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jarenmf 6y ago

Interesting relevant paper https://arxiv.org/abs/math/0607694 https://arxiv.org/abs/math/0607694

deleted 6y ago

[deleted]

deleted 6y ago

[deleted]

MrXOR 6y ago

Thank you John. My favorite Ramanujan’s formula: 1 + 2 + 3 + 4 + ⋯ = −1/12

RHSeeger 6y ago
That one is kind of silly.
- gus_massa 6y ago
  It is weird, very weird. A very nice explanation of the proof is in a video by Mathologer "Ramanujan: Making sense of 1+2+3+... = -1/12 and Co." https://www.youtube.com/watch?v=jcKRGpMiVTw https://www.youtube.com/watch?v=jcKRGpMiVTw [spoiler alert] You can extend finite summations to infinite summations. The first extensions are nice and you get intuitive results, and are the extensions studied in Calculus in the university. But these extensions are not enough for this sum. You must make more bold extensions, and the results are not as intuitive, and not useful outside some special applications.
  - RHSeeger 6y ago
    Honestly, I don't buy the proof. When you take a formula that's valid for a certain range of inputs and just use it outside that range anyways, you're no longer in the land of proof. At that point, you're in the land of "if we extend out mathematical system, we get this". And that's fine, but you don't wind up with "sum(1...inf) === -1/12", you wind up with "in this other mathematical system, sum....".
    - gus_massa 6y ago
      I'm not going to blame you. As the video explain, this is not a straightforward and obvious generalization of summation. I think that the generalization that use averages are fine, and some analytic continuations using power series are fine. Analytic functions are just too good to ignore them. It is not another mathematical system. It it just the extension of our mathematical system. Anyway, to get a result for this sum, you must drop too many of the obvious and expected properties of summation. For example if you add a zero in front of the sum, the result changes :(. There was a nice blog post I can't find, but there is a hint of the problem in https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Failure_of_stable_linear_summation_methods https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B... . So the result is ... polemic. Anyway, in some courses of a Math degree, one of the important ideas is that you have many possible definitions of convergence. For example, if you have a sequence of functions, you have https://en.wikipedia.org/wiki/Uniform_absolute-convergence https://en.wikipedia.org/wiki/Uniform_absolute-convergence and https://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_space) https://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_spac... and many many many more. So you don't write lim f_n -> f, you must write lim_{something} f_n -> f where something explain which definition of convenience you are using. The trick here is to hide the nasty problem of the multiple definitions of convergence in the dots. So the correct statement of the problem is 1+2+3+4+..._{with a weird convergence}=-1/12 but it doesn't look as nice as 1+2+3+4+...=-1/12

29athrowaway 6y ago

Is it there an alternative notation for continued fractions in the style of pi notation?

groupdeterminac 6y ago
You'll sometimes see it like so, written with a capital K

29athrowaway 6y ago

Ramanujan's number, 1729: > I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

jason_slack 6y ago

A few videos that talk about Ramanujan's genius https://www.youtube.com/watch?v=eURuh9aS3eo https://www.youtube.com/watch?v=eURuh9aS3eo https://www.youtube.com/watch?v=XFsuRxospbU https://www.youtube.com/watch?v=XFsuRxospbU