Elements of Algebra

CHAP. I. ADDITIONS. 493 2, 1, $ \frac{3587785776203}{1142027682075} $ $ \frac{5371151992734}{1709690779483} $ 1, 15, $ \frac{8959937768937}{2851718461558} $ $ \frac{139755218526789}{44495467702853} $ 3, 13, $ \frac{428224593349304}{136308121570117} $ $ \frac{5706674931067741}{1816498048114374} $ 1, 4, $ \frac{6134899525417045}{1952799169684491} $ $ \frac{30246273033735921}{9627657726852335} $ 2, 6, $ \frac{66627445593888887}{212081746213891679} $ $ \frac{430010946591069243}{1368767354078873409} $ 6, 1, $ \frac{2646693125139304345}{8424685874265383207} $ $ \frac{3076704071730373588}{979345322993700547} $ These fractions will therefore be alternately less and greater than the real ratio of the circumference to the diameter; that is to say, the first $ \frac{3}{1} $ will be less, the second $ \frac{22}{7} $ greater, and so on; and each of them will approach nearer the truth than can be done by any other fraction expressed in simpler terms; or, in general, having a denominator less than that of the succeeding fraction: so that we may be assured that the fraction $ \frac{3}{1} $ approaches nearer the truth than any other fraction whose denominator is less than 7; also the fraction $ \frac{22}{7} $ approaches nearer the truth than any other fraction whose denominator is less than 106; and so of others. With regard to the error of each fraction, it will always be less than unity divided by the product of the denominator of that fraction, by the denominator of the following fraction. Thus, the error of the fraction $ \frac{3}{1} $ will be less than $ \frac{1}{1 \times 7} $, that of the fraction $ \frac{22}{7} $ will be less than $ \frac{1}{7 \times 106} $, and so on. But, at the same time, the error of each fraction will be greater than unity divided by the product of the denominator of that fraction, into the sum of this denominator, and of the denominator of the succeeding fraction; so that the error of the fraction $ \frac{3}{1} $ will be greater than $ \frac{1}{1 \times 8} $, that of the fraction $ \frac{22}{7} $ greater than $ \frac{1}{7 \times 113} $, and so on (Art. 14). If we now wish to separate the fractions that are less than the ratio of the circumference to the diameter, from those which are greater, by inserting the proper intermediate fractions, we may form two series of fractions, the one in-