
Ramanujan Surprises Again (2015) - tmbsundar
https://plus.maths.org/content/ramanujan
======
nneonneo
An interesting coincidence: it was recently (2019) discovered that the fastest
way to multiply two n-bit integers, in time O(n log n), involves
1729-dimensional Fourier transforms: [https://hal.archives-
ouvertes.fr/hal-02070778](https://hal.archives-ouvertes.fr/hal-02070778). It
is quite surprising that the asymptotically best way to perform such an
elementary operation should be tied to Ramanujan’s famous taxicab number.

(Technically, it works for any number of dimensions >= 1729, but the proof
fails for dimensions less than that. Future work might bring the bound down,
or better explain why that bound is necessary.)

~~~
user2994cb
In fact, there seems to be a lot of interesting things about 1729:
[https://en.wikipedia.org/wiki/1729_(number)](https://en.wikipedia.org/wiki/1729_\(number\))

~~~
pixelpoet
I love how the article starts with the most boring facts about 1729:

> 1729 is the natural number following 1728 and preceding 1730.

~~~
russellbeattie
Heh. I've been reading HN for long enough to never be surprised by the
capability of incredibly pedantic people to be incredibly pedantic.

~~~
JoeAltmaier
We had to have a home somewhere :) And this is it.

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Vinceo
He 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.""

~~~
dr_dshiv
Mathematics are the best expression of the transcendental divine. Pythagoras
and Plato had the same perspective.

~~~
unlinked_dll
Funny to use the word transcendental there, since the Pythagoreans held ratios
to be divine but couldn't figure out irrational numbers, like pi. They had
trouble squaring that circle.

~~~
dr_dshiv
That's why I think Pythagoras refused to write down his doctrines. He knew
there was more to be empirically discovered -- and he was wary of how text
could become dogma. The divine he uncovered was based on a mathematical,
harmonious cosmos; but he recognized it was beyond understanding in a
lifetime. That's why Pythagorean mysticism is compatible with modern science
-- he didn't write anything down!

2000 years later, Kepler had faith in a harmonious cosmos, and charged his
model of harmony so it could fit the evidence. He elipsed the circles, instead
of squaring them.

Fun fact #1: it is impossible to square a circle [1]

Fun fact #2: the Pythagoreans conducted the first attested scientific
experiment in Western history (according to a recent PhD thesis at UMich [2])

[1]
[https://en.m.wikipedia.org/wiki/Squaring_the_circle](https://en.m.wikipedia.org/wiki/Squaring_the_circle)

[2]
[https://deepblue.lib.umich.edu/handle/2027.42/150050](https://deepblue.lib.umich.edu/handle/2027.42/150050)

------
v64
Great read! When you first hear the taxicab number story, your initial
impression is to be struck by Ramanujan's innate calculating capability. It's
interesting to find out that the real coincidence here is that Hardy rode in a
taxicab whose number had happened to show up in Ramanujan's investigations of
Fermat's last theorem.

~~~
hnews_account_1
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's smarter
than I was etc. Then I just ... grew up, I guess. And I remembered this story
by Feynman and I realised that despite his absolutely undoubtable genius, he'd
have appeared godlike to me if I was his classmate back in the day.

Ramanujan's brain worked even faster by most accounts. He dreamed in math, I
think. So there are multiple stories where people ask him a puzzle and he'll
answer with an equation that solves it for the entire family of problems that
the puzzle could come from.

~~~
dorchadas
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.

~~~
gameswithgo
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.

~~~
vidarh
Often it's also simply just that people are not _used to_ thinking about more
efficient ways of solving a problem.

There's a (quite possibly apocryphal) story about Niels Henrik Abel in primary
school, where his teacher supposedly wanted time to do some grading and
assigned the students the busywork of adding up all numbers from 1 to 100.
Abel supposedly quickly found the well known formula n(n+1)/2 and gave the
teacher the answer within minutes, and the teacher supposedly believed he'd
somehow "cheated" because he could not imagine any of them could figure it
out.

I have no idea if the story is real (I grew up in Norway, so Abel was a
popular subject for stories like this) - it was told to me in high school by a
maths teacher after giving us the modified task of seeing if we could find any
shortcuts to doing the sums, and seeing what we'd come up with. I found the
formula quickly, but at that age that's nothing special, especially not when
prompted to find an alternative solution.

But the overall idea the teacher was trying to get us to understand was how to
pause and think about how to decompose a problem rather than just picking the
most obvious alternative, and learning to be "lazy" in the sense of
relentlessly looking for an easier way to do things is a large part of what
got me into software development..

~~~
fingerlocks
When I heard this story it was about Gauss.

And I looked it up- Yes, the same possibly apocryphal story is on his
Wikipedia page:
[https://en.m.wikipedia.org/wiki/Carl_Friedrich_Gauss](https://en.m.wikipedia.org/wiki/Carl_Friedrich_Gauss)

~~~
vidarh
Interesting. Not surprised this is the kind of story people might have adapted
rather freely to sound more familiar to a local audience...

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dannykwells
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.

~~~
pmoriarty
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](https://en.wikipedia.org/wiki/%C3%89variste_Galois)

~~~
dboreham
Fun mathematical tourism stop: go up to the top of the Tour Montparnasse. Look
straight down. Galois is buried in the cemetery below. Nobody knows where, but
he's down there somewhere.

------
dang
Discussed at the time:
[https://news.ycombinator.com/item?id=10518452](https://news.ycombinator.com/item?id=10518452)

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perseusprime11
I always found Ramanujan very intriguing. He operates on a dimension that is
unknown to most of us. Makes me wonder if he is a great yogi or a time
traveler.

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jackconnor
Fantastic article that explains the math (and physics) very clearly.

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skunkworker
Interesting read, The title should have 2015 in it though.

~~~
dang
Added. Thanks!

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foo101
Ramanujan also claimed 1 + 2 + 3 + ... = -1/12.

How does that work? Who can explain this to me?

~~~
asfarley
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.

~~~
stan_rogers
Burkard Polster's (Mathologer) videos on the subject are probably going to be
more useful than Numberphile. Numberphile merely presents a trick; Mathologer
points out _both_ that it's patent nonsense [0] and that it's _useful_ patent
nonsense [1].

[0]
[https://www.youtube.com/watch?v=YuIIjLr6vUA](https://www.youtube.com/watch?v=YuIIjLr6vUA)
[1]
[https://www.youtube.com/watch?v=jcKRGpMiVTw](https://www.youtube.com/watch?v=jcKRGpMiVTw)

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rkhacker
Don't we think that the credit for the number 1729 should belong to Hardy, for
he took the cab and mentioned that number to Ramanujan. Of course, Ramanujan
could see beauty in every number and would have produced something equally
beautiful for some other number Hardy could utter.

