Tag: continued fractions


Approximating Pi (Happy Pi Day)

Posted in Mathematics

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Favorite Pi Approximations

What's your favorite \(\pi\) approximation?

Some of my favorite approximations of \(\pi\) come from Ramanujan-Sato series. These are mathematical series that generalize from a remarkable formula for \(\pi\) given by Srinivasa Ramanujan, an Indian mathematician:

$$ \pi^{-1} = \dfrac{\sqrt{8}}{99^2} \sum_{k \geq 0} \dfrac{ (4k)! }{ \left( 4^k k! \right)^4 } \dfrac{ 1103 + 26390k }{ 99^{4k} } $$

This completely novel formula opened up new branches of mathematics and provided a whole new class of \(\pi\) approximations (the Ramanujan-Sato series) and approximations that are extremely accurate, making them very useful for computer applications. (Each term of …



Tags:    pi    continued fractions    number theory    mathematics    python    irrational numbers   


Computing Square Roots: Part 2: Using Continued Fractions

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Continued Fractions

Let's start part 2 of our discussion of computing square roots by talking about continued fractions. When we first learn mathematics, we learn to count in the base 10 system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We can construct representations of all of the integers using these 10 digits, by arranging them in a different order. So, for example, saying 125 is equivalent to saying …