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# How do you convert an irrational numbers continued fraction into its convergents? watch

1. For example, Pi's continued fraction is [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84]
And its initial convergents are 22/7, 333/106, 355/113, 103993/33102...

With 22/7 being the best approximation for pi with a denominator smaller than or equal to 7, 333/106 being the best approximation for pi with denominators smaller than or equal to 106, and 355/113.

You can get continued fractions in a sort of way like shown below:
Pi = 3 + 0.14159...
= 3+ 1/7.062513...
= 3+ 1/(7 + 1/15.996594...)
= 3 + 1/(7 + 1/(15 + 1/1.003417...)
= 3 + 1/(7 + 1/(15 +1/(1 + 1/292.0654...)

And so on.

It is also interesting to note that e's continued fractions are [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12...] and initial convergents 5/2, 8/3, 11/4, 19/7, 73/32...

And just for good measure Gamma has continued fractions of [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1...] and initial convergents of 1, 1/2, 3/5, 4/7, 11/19, 15/26, 71/123, 228/395...

As an AS math student this obviously isn't part of my course but I'm quite interested in it so if someone can please help thx.

Edit: The pattern to e's continued fractions... why does e's continued fractions behave like this?

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Updated: January 25, 2015
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