The infinite radicals of Ramanujan


What if I asked you to find the numerical value of

\displaystyle \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{\cdots}}}}}} ?

If you have difficulty figuring out what this thing is, don’t worry. You’re not the only one. This question is one the problems posed by Srinivasa Ramanujan, one of the most brilliant and, well, mysterious mathematicians of all time. In fact, it was head enough that Ramanujan had to give the answer a few months later.

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Tweet time!


I’ve been using twitter for about five years, and I wondered if my use of it changed over time, and more precisely, linked to my wake/sleep cycle. That’s fortunately kind of simple to check because you can simply request your whole Twitter archive, delivered as a plain CSV File! Let’s see how we can juice it.

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Square Roots (Part V)


Last week we examined the complexity of obtaining k in the decomposition n=2^k+b for some integer n. This week, we’ll have a look at how we can use it to speed-up square root extraction. Recall, we’re interested in k because

2^k \leqslant 2^k+b < 2^{k+1},

with 0 \leqslant b < 2^k, which allows us to get easy bounds on \sqrt{n}. Better, we also have that

\sqrt{2^k} \leqslant \sqrt{2^k+b} \leqslant \sqrt{2^{k+1}},

and we know how to compute \sqrt{2^k}=2\frac{k}{2} (somewhat efficiently! Let’s combine all this and see how it speeds up things.

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Square Roots (part IV)


In a previous post, we noticed that

\sqrt{2^k} \leqslant \sqrt{n} = \sqrt{2^k+b} \leqslant \sqrt{2^{k+1}},

where 0 \leqslant b < 2^k, could be used to kick-start Newton's (or another) square root finding algorithm. But how fast can we find k and b in this decomposition?

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Regula Falsi


The regular falsi, or method of false position, is a method to solve an equation in one unknown, from an initial “guess”. Guesses are progressively refined, by a rather simple method as we will see, until the exact answer is reached or until convergence is reached. The method is useful when you don’t really know how to divide by arbitrary values (as it was the case in Ancient Times) or when the equation is cumbersome.

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