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.
So he did his derivations and proofs on the blackboard, and just wrote down the result in his notebook and then erased the blackboard.
He knew he didn't have room for anything else in his notebooks.
Source: https://www.quantamagazine.org/three-puzzles-inspired-by-ram...
There is a method behind it. The formulas are derived from the study of relatively simple ODEs, in this example f'(x) = x.f(x) + 1 and g'(x) = x.g(x) − 1. The latter was solved 100 years prior by Jacobi. While the solutions are non-trivial, they are fairly compact, accessible to a high schooler. The presentation linked from the blog is pretty good at unveiling the magic.
This is to say that, with proper guidance, the kid down the street can too become as cool as Ramanujan.
https://math.ucr.edu/home/baez/ramanujan/ramanujan_whittier_...
Surely there would be a method for coming up with such assertions (maybe only known to him though)
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...""
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_...
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?
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.
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.
https://www.quora.com/How-did-Strassen-derive-his-matrix-mul...
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.
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.
"Good mathematicians see analogies between theorems or theories. the very best ones see analogies between analogies." Stefan Banach.
https://golem.ph.utexas.edu/category/2020/08/chasing_the_tai...
https://golem.ph.utexas.edu/category/2020/09/chasing_the_tai...
My favorite Ramanujan’s formula: 1 + 2 + 3 + 4 + ⋯ = −1/12
[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.
[1] https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...
[2] https://medium.com/cantors-paradise/the-ramanujan-summation-...
Who is the author of the paper?
> 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."
