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Continued fraction expansion of π and its convergents

To calculate the convergents of π we may set a = ⌊ π ⌋ = 3, define u = 1 / π − 3 ≈ 7.0625 and a = ⌊ u ⌋ = 7, u = 1 / u − 7 ≈ 15.9966 and a = ⌊ u ⌋ = 15, u = 1 / u − 15 ≈ 1.0034. Continuing like this, one can determine the infinite continued fraction of π as

The fourth convergent of π is = 355 / 113 = 3.14159292035..., sometimes called Milü, which is fairly close to the true value of π.

Let us suppose that the quotients found are, as above,. The following is a rule by which we can write down at once the convergent fractions which result from these quotients without developing the continued fraction.

The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, 3 / 1. Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, 22 / 7, which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small. Thus, the third quotient being 15, we have for our numerator (22 × 15 = 330) + 3 = 333, and for our denominator, (7 × 15 = 105) + 1 = 106. The third convergent, therefore, is 333 / 106. We proceed in the same manner for the fourth convergent. The fourth quotient being 1, we say 333 times 1 is 333, and this plus 22, the numerator of the fraction preceding, is 355; similarly, 106 times 1 is 106, and this plus 7 is 113.In this manner, by employing the four quotients, we obtain the four fractions:

To sum up, the pattern is Numerator i = Numerator (i − 1) ⋅ Quotient i + Numerator (i − 2) {\displaystyle {\text{Numerator}}_{i}={\text{Numerator}}_{(i-1)}\cdot {\text{Quotient}}_{i}+{\text{Numerator}}_{(i-2)}} Denominator i = Denominator (i − 1) ⋅ Quotient i + Denominator (i − 2) {\displaystyle {\text{Denominator}}_{i}={\text{Denominator}}_{(i-1)}\cdot {\text{Quotient}}_{i}+{\text{Denominator}}_{(i-2)}}

These convergents are alternately smaller and larger than the true value of π, and approach nearer and nearer to π. The difference between a given convergent and π is less than the reciprocal of the product of the denominators of that convergent and the next convergent. For example, the fraction 22 / 7 is greater than π, but 22 / 7 − π is less than 1 / 7 × 106 = 1 / 742 (in fact, 22 / 7 − π is just more than 1 / 791 = 1 / 7 × 113).

The demonstration of the foregoing properties is deduced from the fact that if we seek the difference between one of the convergent fractions and the next adjacent to it we shall obtain a fraction of which the numerator is always unity and the denominator the product of the two denominators. Thus the difference between 22 / 7 and 3 / 1 is 1 / 7, in excess; between 333 / 106 and 22 / 7, 1 / 742, in deficit; between 355 / 113 and 333 / 106, 1 / 11978, in excess; and so on. The result being, that by employing this series of differences we can express in another and very simple manner the fractions with which we are here concerned, by means of a second series of fractions of which the numerators are all unity and the denominators successively be the product of every two adjacent denominators. Instead of the fractions written above, we have thus the series:

The first term, as we see, is the first fraction; the first and second together give the second fraction, 22 / 7; the first, the second and the third give the third fraction 333 / 106, and so on with the rest; the result being that the series entire is equivalent to the original value.