Some ways to use the continued fractions

If the numerator and the segment of the given fraction are very large numbers, sometimes it is easier to take fractions with simpler numbers convergent to its real meaning. So it is necessary to change the given fraction for the continued fraction and to use instead of it a value of one of other convergents. For example, if the real value between the circle and its diameter is between the fractions 3,1415926 and 3,1415927, if we express the required multiple π (or otherwise pi), to find the values of π easy to write and convergent to the real value, first of all let’s convert the previous fractions into the form of the continued fraction.

Then

If we take only mutually common multiples of two continued fractions, we will find that π is equal to this fraction: . So the first convergents of π would be the following.

A scholar Archimed found the convergent value of π is , hence when we take it instead of π our error will be less than . If we take instead of π, our error will be greater than this fraction:

As the conergent fractions of Archimed and Meshiy are at the even places, values of both actually are greater than π.

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