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How the 2nd and 3rd lines in the tables at the bottom of Euclidian Algorithm page were created.
The first two values in the second line are: 0 and 1, and the first two values in the 3rd line are: 1 and 0 These are starting values and are always used.
The following numbers in both the second and third lines are generated by exactly the same procedure. (I will describe the procedure in terms of the second line.) The Continued Fraction values in the order they were generated are used as column headers to the right of the starting pair.
The values in the second line are generated one at a time from left to right. The steps are:
A good practice, for you, would be to find the first few approximations for Pi starting from something like: 3.1416 which we know is not Pi, but only an approximation. Remember, the mathematical process described here will generate approximations for what you give it. Right or Wrong!
The square root of 2 is an easy one. The continued fraction representation of Square Root of 2 is (1,2,2 . . .) the 2's repeat for ever. Knowing the CF you may want to generate a few fractional approximations, before you look at mine.