Encyclopaedia Britannica, 11th Edition, "Constantine Pavlovich" to "Convention" / Volume 7, Slice 2

by Various

The value of such a fraction is the positive root of a quadratic equation whose coefficients are real and of which one root is negative. Since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number. Conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction. Thus

2 - √3 =	1		1		1		1		1
	3	+	1	+	2	+	1	+	2	+ ...,

√28 = 5 +	1		1		1		1		1		1		1		1
	3	+	2	+	3	+	10	+	3	+	2	+	3	+	10	+ ...

The second case illustrates a feature of the recurring continued fraction which represents a complete quadratic surd. There is only one non-recurring partial quotient a₁. If b₁, b₂, ..., b_n is the cycle of recurring quotients, then b_n = 2a₁, b₁ = b_n-1, b₂ = b_n-2, b₃ = b_n-3, &c.

In the case of a recurring continued fraction which represents √N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and p_nr/q_nr the nr^th convergent, then p²_nr -Nq²_nr = (-1)^nr, whence, if n is odd, integral solutions of the indeterminate equation x² - Ny² = ±1 (the so-called Pellian equation) can be found. If n is even, solutions of the equation x² -Ny² = +1 can be found.

The theory and development of the simple recurring continued fraction is due to Lagrange. For proofs of the theorems here stated and for applications to the more general indeterminate equation x² -Ny² = H the reader may consult Chrystal’s Algebra or Serret’s Cours d’Algèbre Supérieure; he may also profitably consult a tract by T. Muir, The Expression of a Quadratic Surd as a Continued Fraction (Glasgow, 1874).

The General Continued Fraction.

1. The Evaluation of Continued Fractions.—The numerators and denominators of the convergents to the general continued fraction both satisfy the difference equation u_n = a_nu_n-1 + b_nu_n-2. When we can solve this equation we have an expression for the n^th convergent to the fraction, generally in the form of the quotient of two series, each of n terms. As an example, take the fraction (known as Brouncker’s fraction, after Lord Brouncker)

1		1²		3²		5²		7²
1	+	2	+	2	+	2	+	2	+ ...

Here we have