Continued FractionYou are currentlynot logged in Click here to log in 

Take any positive real number. By rounding down we can see that it consists of an integer plus a bit left over. That extra bit might be zero, in which case we're done, but it might not be. If you subtract off the integer, the bit left over is between 0 and 1. It might be 0, but it can't be 1.
Call the extra x.
Now we can take $\frac{1}{x}.$ That too is an integer plus a bit left over. Again, that extra bit might be zero, in which case we're done, but it might not be. Again, subtract off the integer, and repeat.
$\pi$  =  3  +  0.1415926... 
1/0.1415926...  =  7  +  0.0625133... 
1/0.0625133...  =  15  +  0.9965944... 
1/0.9965944...  =  1  +  0.0034172... 
1/0.0034172...  =  292  +  0.6345908... 
We can write this as $\pi=[3;7,15,1,292,...]$
Cutting this off at different stages gives us rational approximations.
As you can see, the large number in the expansion causes a sudden jump in the terms used in the rational approximation.
However, a large number in the continued fraction implies a small error in the previous step. That means that cutting off the continued fraction just before a large number will give an unreasonably good approximation.
Hence $\pi\approx\frac{355}{113}$
Using this technique gives an approximation with an error that is "best" given a limit on the size of the denominator.
 So we have $\frac{41}{29}$ and $\frac{99}{70}$ as approximations to $\sqrt{2}.$ 
Each term is the product of the quotient above it and the term on its left,
plus the term two to the left. Here it is for $\pi$
 So 333=15*22+3 and 113=1*106+7, and we get $\frac{355}{113}$ as an approximation of $\pi.$ 
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