String Theorists Accidentally Find a New Formula for Pi
scientificamerican.com
scientificamerican.com
I don’t really get the logic here. If you can pick any number for the parameter and get the same result, why does it matter? Why is it even there? They even simplify it the formula without the parameter right after. What’s stopping anyone from writing:
2 = 2 + λ - λ
What it means is, first, try prove that formula does not depends on λ. It is not obvious on first glance it is. (You may try to d/dλ and proof that equals 0.)
Second, change λ, it gives you a different infinite series, while each converge to π, they converge at a different rate.
In fact, part of the game of computing digits of π is to use different formula, where each converges to π eventually, but they have different converging “speed”, in the sense that giving n=100 terms, some gives you much more accurate digits. Also, since each term has a different computing “speed” too, ie using a computer to compute a term might be much slower in one series, that even if it has better converging “speed”, would be very slow to compute.
The art of breaking world record then is to find a balance between the 2, such that overall it is fastest.
Regardless of that though, discovering a whole family of series converging to π is in itself interesting. A special case reducing it to something we already know is not too surprising, but a good entry point for a story.
I don't see "2 = 2 + λ - λ" in the article anywhere. The formula without λ (set to infinity, which sets the dependent factors to zero) is Madhava's formula. What is demonstrated is that some values of λ result in more rapid convergence than not having it.
It's about convergence: you never actualize the value of pi without computing an infinite seqence, which you will never do. It is about approximation.
As a more poetic retort, rhetorically I will ask where does h go when you compute a derivative? Why can't you just set it to zero? It's a placeholder for "something", what that something is exactly is of nobody's concern but God.
After cancelling it from everywhere we can, we set it to zero.
> Why can't you just set it to zero?
Before cancelling it as much as we can? Because generally we most of the time would end up with something like 0/0.
It's just that you're not very interested in formulas to compute 2. They've just given us a bunch of different ways to compute pi.
Most of them aren't interesting. They show that there's an interesting case where λ=infinity; it condenses down to a nice well-known formula. It would be interesting to evaluate it at λ=0.
Are there any other interesting cases? Probably not. But it does give opportunities to recognize "Oh, if I set λ up like thus-and-such, it resembles that formula over there". That's how you end up finding connections between seemingly unrelated fields -- like the way ellitic curves resulted in proving Fermat's last theorem.