(ii) Each convergent is a fraction in its lowest terms.
(iii) The convergents are alternately less and greater than the true value.
(iv) Each convergent is nearer to the true value than any other fraction whose denominator is less than that of the convergent.
(v) The difference of two successive convergents is the reciprocal of the product of their denominators; e.g. (ab + 1)/b − a/1 = 1/(1·b), and (abc + c + a)/(bc + 1) − (ab + 1)/b = −1/b(bc + 1).
It follows from these last three properties that if the successive convergents are p1/1, p2/q2, p3/q3, ... the number can be expressed in the form p1(1 + 1/p1q2) (1 − 1/p2q3) (1 + 1/p3q4) ..., and that if we go up to the factor 1 ± 1/(pnqn + 1) the product of these factors differs from the true value of the number by less than ±{1/(qnqn + 1).
In certain cases two or more factors can be combined so as to produce an expression of the form 1 ± 1/k, where k is an integer. For instance, 3.1415927 = 3(1 + 1⁄3.7) (1 − 1⁄22.106) (1 + 1⁄333.113) ...; but the last two of these factors may be combined as (1 − 1⁄22.113). Hence 3.1415927 = 3⁄1 · 22⁄21 · 2485⁄2486 ...
XII. Applications
(i.) Systems of Measures.[1]
118. Metric System.—The metric system was adopted in France at the end of the 18th century. The system is decimal throughout. The principal units of length, weight and volume are the metre, gramme (or gram) and litre. Other units are derived from these by multiplication or division by powers of 10, the names being denoted by prefixes. The prefixes for multiplication by 10, 102, 103 and 104 are deca-, hecto-, kilo- and myria-, and those for division by 10, 102 and 103 are deci-, centi- and milli-; the former being derived from Greek, and the latter from Latin. Thus kilogramme means 1000 grammes, and centimetre means 1⁄100 of a metre. There are also certain special units, such as the hectare, which is equal to a square hectometre, and the micron, which is 1⁄1000 of a millimetre.
The metre and the gramme are defined by standard measures preserved at Paris. The litre is equal to a cubic decimetre. The gramme was intended to be equal to the weight of a cubic centimetre of pure water at a certain temperature, but the equality is only approximate.