Continued Fraction 

Call the extra x.
Now we can take $\frac{1}{x}.$ That too is a whole number, plus a bit left over. Again, that extra bit might be zero, in which case we're done, but it might not be. Subtract off the whole number, 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 $\pi.$ 