Recurring Decimals are a particular kind of series, which arise from the expression of a fraction as a decimal. If the denominator of the fraction, when it is in its lowest terms, contains any other prime factors than 2 and 5, it cannot be expressed exactly as a decimal; but after a certain point a definite series of figures will constantly recur. The interest of these series is, however, mainly theoretical.
116. Continued Products.—Instead of being expressed as the sum of a series of terms, a number may be expressed as the product of a series of factors, which become successively more and more nearly equal to 1. For example,
3.1416 = 3 × 10472⁄10000 = 3 × 1309⁄1250 = 3 × 22⁄21 × 2499⁄2500 = 3(1 + 1⁄21)(1 − 1⁄2500).
Hence, to multiply by 3.1416, we can multiply by 31⁄7, and subtract 1⁄2500 (= .0004) of the result; or, to divide by 3.1416, we can divide by 3, then subtract 1⁄22 of the result, and then add 1⁄2499 of the new result.
117. Continued Fractions.—The theory of continued fractions (q.v.) gives a method of expressing a number, in certain cases, as a continued product. A continued fraction, of the kind we are considering, is an expression of the form
where b, c, d, ... are integers, and a is an integer or zero. The expression is usually written, for compactness, a + 1/b+ 1/c+ 1/d+ &c. The numbers a, b, c, d, ... are called the quotients.
Any exact fraction can be expressed as a continued fraction, and there are methods for expressing as continued fractions certain other numbers, e.g. square roots, whose values cannot be expressed exactly as fractions.
The successive values, a/1, (ab + 1)/b, ..., obtained by taking account of the successive quotients, are called convergents, i.e. convergents to the true value. The following are the main properties of the convergents.
(i) If we precede the series of convergents by 0⁄1 and 1⁄0, then the numerator (or denominator) of each term of the series 0⁄1, 1⁄0, a/1, (ab + 1)/b ..., after the first two, is found by multiplying the numerator (or denominator) of the last preceding term by the corresponding quotient and adding the numerator (or denominator) of the term before that. If a is zero, we may regard 1/b as the first convergent, and precede the series by 1⁄0 and 0⁄1.