1911 Encyclopædia Britannica/Continued Fractions

by Arthur Ernest Jolliffe

CONTINUED FRACTIONS. In mathematics, an expression of the form

a1 ±	b2
	a2 ±	b3
		a3 ±	b4
			a4 ±	b5
				a5 ±	. . .,

where a₁, a₂, a₃, . . . and b₂, b₃, b₄, . . . are any quantities whatever, positive or negative, is called a “continued fraction.” The quantities a₁ . . ., b₂ . . . may follow any law whatsoever. If the continued fraction terminates, it is said to be a terminating continued fraction; if the number of the quantities a₁ . . ., b₂ . . . is infinite it is said to be a non-terminating or infinite continued fraction. If b₂/a₂, b₃/a₃ . . ., the component fractions, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to be recurring or periodic. It is obvious that every terminating continued fraction reduces to a commensurable number.

The notation employed by English writers for the general continued fraction is

a1 ±	b2		b3		b4		. . .
	a2	±	a2	±	a2	±

Continental writers frequently use the notation

a1 ±

b2

±

b3

±

b4

± . . ., or a1 ±

b2

±

b3

±

b4

± . . .

a2

a3

a4

a2

a3

a4

The terminating continued fractions

a1,   a1 +

b2

, a1 +

b2

b3

,   a1 +

b2

b3

b4

, . . .

a2

a2

+

a3

a2

+

a3

+

a4

reduced to the forms

a1	,	a1a2 + b2	,	a1a2a3 + b2a3 + b2a1	,	a1a2a3a4 + b2a3a4 + b3a1a4 + b4a1a2 + b2b4	, . . .
1		a2		a2a3 + b3		a2a3a4 + a4b3 + a2b4

are called the successive convergents to the general continued fraction.

Their numerators are denoted by p₁, p₂, p₃, p₄. . .; their denominators by q₁, q₂, q₃, q₄. . .

We have the relations

p_n = a_np_n−1 + b_np_n−2,    q_n = a_nq_n−1 + b_nq_n−2.

In the case of the fraction

a1 −	b2		b3		b4		. . .,
	a2	−	a3	−	a4	−

we have the relations p_n = a_np_n−1 − b_np_n−2,    q_n= a_nq_n−1 − b_nq_n−2.

Taking the quantities a₁ . . ., b₂ . . . to be all positive, a continued fraction of the form

a1 +	b2		b3		. . .,
	a2	+	a3	+

is called a continued fraction of the first class; a continued fraction of the form

b2		b3		b4		. . .
a2	−	a3	−	a4	−

is called a continued fraction of the second class.

A continued fraction of the form

a1 +	1		1		1		. . .,
	a2	+	a3	+	a4	+

where a₁, a₂, a₃, a₄ . . . are all positive integers, is called a simple continued fraction. In the case of this fraction a₁, a₂, a₃, a₄ . . . are called the successive partial quotients. It is evident that, in this case,

p₁, p₂, p₃ . . .,   q₁, q₂, q₃ . . .,

are two series of positive integers increasing without limit if the fraction does not terminate.

The general continued fraction

a1 +	b2		b3		b4		. . .
	a2	+	a3	+	a4	+

is evidently equal, convergent by convergent, to the continued fraction

a1 +	λ2b2		λ2λ3b3		λ3λ4b4		. . .,
	λ2a2	+	λ3a3	+	λ4a4	+

where λ₂, λ₃, λ₄, . . . are any quantities whatever, so that by choosing λ₂b₂ = 1,    λ₂λ₃b₃ = 1, &c., it can be reduced to any equivalent continued fraction of the form

a1 +	1		1		1		. . .,
	d2	+	d3	+	d4	+

 

Simple Continued Fractions.

1. The simple continued fraction is both the most interesting and important kind of continued fraction.

Any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity. A non-terminating simple continued fraction must be incommensurable.

In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the last partial quotient, a_n as a_n − 1 + 1/1.

The numerators and denominators of the successive convergents obey the law p_nq_n−1 − p_n−1q_n = (−1)ⁿ, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are:—

The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.

Every even convergent is greater than every odd convergent; every odd convergent is less than, and every even convergent greater than, any following convergent.

Every convergent is nearer to the value of the whole fraction than any preceding convergent.

Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent.

The difference between the continued fraction and the n^th convergent is less than 1/q_nq_n+1 and greater than a_n+2/q_nq_n+2. These limits may be replaced by the following, which, though not so close, are simpler, viz. 1/q²_n and 1/q_n(q_n + q_n+1)

Every simple continued fraction must converge to a definite limit; for its value lies between that of the first and second convergents and, since

pn	~	pn−1	=	1	,    Lt.	pn	= Lt.	pn−1	,
qn		qn−1		qnqn−1		qn		qn−1

so that its value cannot oscillate.

The chief practical use of the simple continued fraction is that by means of it we can obtain rational fractions which approximate to any quantity, and we can also estimate the error of our approximation. Thus a continued fraction equivalent to π (the ratio of the circumference to the diameter of a circle) is

3 +	1		1		1		1		1		1		. . .
	7	+	15	+	1	+	292	+	1	+	1	+

of which the successive convergents are

3	,	22	,	333	,	355	,	103993	, &c.,
1		7		106		113		33102

the fourth of which is accurate to the sixth decimal place, since the error lies between 1/q₄q₅ or .0000002673 and a₆/q₄q₆ or .0000002665.

Similarly the continued fraction given by Euler as equivalent to 1/2(e − 1) (e being the base of Napierian logarithms), viz.

1		1		1		1		1
1	+	6	+	10	+	14	+	18	+ . . .,

may be used to approximate very rapidly to the value of e.

For the application of continued fractions to the problem “To find the fraction, whose denominator does not exceed a given integer D, which shall most closely approximate (by excess or defect, as may be assigned) to a given number commensurable or incommensurable,” the reader is referred to G. Chrystal’s Algebra, where also may be found details of the application of continued fractions to such interesting and important problems as the recurrence of eclipses and the rectification of the calendar (q.v.).

Lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a + 1, put x = a + 1/y and form the equation in y; if the equation in y has a root between b and b + 1, put y = b + 1/z, and so on. Such a method is, however, too tedious, compared with such a method as Homer’s, to be of any practical value.

The solution in integers of the indeterminate equation ax + by = c may be effected by means of continued fractions. If we suppose a/b to be converted into a continued fraction and p/q to be the penultimate convergent, we have aq − bp = +1 or −1, according as the number of convergents is even or odd, which we can take them to be as we please. If we take aq−bp = +1 we have a general solution in integers of ax + by = c, viz. x = cq − bt, y = at − cp; if we take aq − bp = −1, we have x = bt − cq, y = cp − at.

An interesting application of continued fractions to establish a unique correspondence between the elements of an aggregate of m dimensions and an aggregate of n dimensions is given by G. Cantor in vol. 2 of the Acta Mathematica.

Applications of simple continued fractions to the theory of numbers, as, for example, to prove the theorem that a divisor of the sum of two squares is itself the sum of two squares, may be found in J. A. Serret’s Cours d’Algèbre Supérieure.

2. Recurring Simple Continued Fractions.—The infinite continued fraction

a1 +

1

1

1

1

1

1

1

1

1

1

a2

+

a3

+ . . . +

an

+

b1

+

b2

+ . . . +

bn

+

b1

+

b2

+ . . . +

bn

+

b1

+ . . .,

where, after the n^th partial quotient, the cycle of partial quotients b₁, b₂, . . ., b_n recur in the same order, is the type of a recurring simple continued fraction.

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

u_n+1 = 2u_n + (2n−1)²u_n−1,

whence

u_n+1 − (2n + 1)u_n = −(2n − 1){u_n − (2n − 1)u_n−1},

and we readily find that

pn	= 1 −	1	+	1	–	1	+ . . . ±	1	,
qn		3		5		7		2n + 1

whence the value of the fraction taken to infinity is 1/4π.

It is always possible to find the value of the n^th convergent to a recurring continued fraction. If r be the number of quotients in the recurring cycle, we can by writing down the relations connecting the successive p’s and q’s obtain a linear relation connecting

p_nr+m,    p_(n−1)r+m,    p_(n−2)r+m

in which the coefficients are all constants. Or we may proceed as follows. (We need not consider a fraction with a non-recurring part). Let the fraction be

a1		a2		ar		a1
b1	+	b2	+ . . . +	br	+	b1	+ . . .

Let un ≡	pnr+m	; then un =	a1		a2		ar	,
	qnr+m		b1	+	b2	+ . . . +	br + un1

leading to an equation of the form Au_nu_n−1 + Bu_n + Cu_n−1 + D = 0, where A, B, C, D are independent of n, which is readily solved.

2. The Convergence of Infinite Continued Fractions.—We have seen that the simple infinite continued fraction converges. The infinite general continued fraction of the first class cannot diverge for its value lies between that of its first two convergents. It may, however, oscillate. We have the relation p_nq_n−1 − p_n−1q_n = (−1)ⁿb₂b₃. . .b_n, from which

pn	–	pn−1	= (−1)n	b2b3 . . . bn	,
qn		qn−1		qnqn−1

and the limit of the right-hand side is not necessarily zero.

The tests for convergency are as follows:

Let the continued fraction of the first class be reduced to the form

d1 +	1		1		1
	d2	+	d3	+	d4	+ . . .,

then it is convergent if at least one of the series d₃ + d₅ + d₇ + . . ., d₂ + d₄ + d₆ + . . . diverges, and oscillates if both these series converge.

For the convergence of the continued fraction of the second class there is no complete criterion. The following theorem covers a large number of important cases.

“If in the infinite continued fraction of the second class a_n ≥ b_n + 1 for all values of n, it converges to a finite limit not greater than unity.”

3. The Incommensurability of Infinite Continued Fractions.—There is no general test for the incommensurability of the general infinite continued fraction.

Two cases have been given by Legendre as follows:—

If a₂, a₃, . . ., a_n, b₂, b₃, . . ., b_n are all positive integers, then

I. The infinite continued fraction

b2		b3		bn
a2	+	a3	+ . . . +	an	+ . . .

converges to an incommensurable limit if after some finite value of n the condition a_n ≮ b_n is always satisfied.

II. The infinite continued fraction

b2		b3		bn
a2	−	a3	− . . . −	an	− . . .

converges to an incommensurable limit if after some finite value of n the condition $a_{n}\geq b_{n}+1$ is always satisfied, where the sign > need not always occur but must occur infinitely often.

Continuants.

The functions p_n and q_n, regarded as functions of a₁, . . ., a_n, b₂, . . ., b_n determined by the relations

p_n = a_np_n−1 + b_np_n−2,
q_n = a_nq_n−1 + b_nq_n−2,

with the conditions p₁ = a₁, p₀ = 1; q₂ = a₂, q₁ = 1, q₀ = 0, have been studied under the name of continuants. The notation adopted is

pn = K(		b2, . . ., bn	),
	a1,	a2, . . ., an

and it is evident that we have

qn = K(		b3, . . ., bn	).
	a2,	a3, . . ., an

The theory of continuants is due in the first place to Euler. The reader will find the theory completely treated in Chrystal’s Algebra, where will be found the exhibition of a prime number of the form 4p + 1 as the actual sum of two squares by means of continuants, a result given by H. J. S. Smith.

The continuant K (  b₂, b₃, . . ., b_n
a₁, a₂, a₃, . . ., a_n) is also equal to the determinant

a1  −1  0  0      0

b2  a2  −1  0      0

0  b3  a3  −1      —

0  0  b4  a4      —

.  .  .  b5    u  0

.  .  .  .    −1  0

.  .  .  .    an−1  −1

0 0 0 —   bn an

from which point of view continuants have been treated by W. Spottiswoode, J. J. Sylvester and T. Muir. Most of the theorems concerning continued fractions can be thus proved simply from the properties of determinants (see T. Muir’s Theory of Determinants, chap. iii.).

Perhaps the earliest appearance in analysis of a continuant in its determinant form occurs in Lagrange’s investigation of the vibrations of a stretched string (see Lord Rayleigh, Theory of Sound, vol. i. chap. iv.).

The Conversion of Series and Products into Continued Fractions.

1. A continued fraction may always be found whose n^th convergent shall be equal to the sum to n terms of a given series or the product to n factors of a given continued product. In fact, a continued fraction

b1		b2		bn
a1	+	a2	+ . . . +	an	+ . . .

can be constructed having for the numerators of its successive convergents any assigned quantities p₁, p₂, p₃, . . ., p_n, and for their denominators any assigned quantities q₁, q₂, q₃, . . ., q_n . . .

The partial fraction b_n/a_n corresponding to the n^th convergent can be found from the relations

p_n = a_np_n−1 + b_np_n−2,   q_n = a_nq_n−1 + b_nq_n−2 ;

and the first two partial quotients are given by

b₁ = p₁,    a₁ = q₁,    b₁a₂ = p₂,    a₁a₂ + b₂ = q₂.

If we form then the continued fraction in which p₁, p₂, p₃, . . ., p_n are u₁, u₁ + u₂, u₁ + u₂ + u₃, . . ., u₁ + u₂ + . . ., u_n, and q₁, q₂, q₃, . . ., q_n are all unity, we find the series u₁ + u₂ + . . ., u_n equivalent to the continued fraction

u1		u2 ⁄ u1			u3 ⁄ u2			un ⁄ un−1
1	−	1 +	u2	–	1 +	u3	– . . . −	1 +	un
			u1			u2			un−1

which we can transform into

u1		u2		u1u3		u2u4		un−2un	,
1	−	u1 + u2	−	u2 + u3	−	u3 + u4	− . . . −	un−1 + un

a result given by Euler.

2. In this case the sum to n terms of the series is equal to the n^th convergent of the fraction. There is, however, a different way in which a Series may be represented by a continued fraction. We may require to represent the infinite convergent power series a₀ + a₁x + a₂x² + . . . by an infinite continued fraction of the form

β0		β1x		β2x		β3x
1	−	1	−	1	−	1	− . . .

Here the fraction converges to the sum to infinity of the series. Its n^th convergent is not equal to the sum to n terms of the series. Expressions for β₀, β₁, β₂, . . . by means of determinants have been given by T. Muir (Edinburgh Transactions, vol. xxvii.).

A method was given by J. H. Lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series. It is practically identical with that of finding the greatest common measure of two polynomials. As an instance leading to results of some importance consider the series

F(n,x) = 1 +	x	+	x²	+ . . .
	(γ + n)1!		(γ + n)(γ + n + 1)2!

We have

F(n + 1,x) − F(n,x) = −	x	F(n + 2,x),
	(γ + n)(γ + n + 1)2!

whence we obtain

F(1,x)	=	1		x ⁄ γ(γ + 1)		x ⁄ (γ + 1)(γ + 2)
F(0,x)		1	+	1	+	1	+ . . .,

which may also be written

γ		x		x
γ	+	γ + 1	+	γ + 2	+ . . .

By putting ± x² ⁄ 4 for x in F(0,x) and F(1,x), and putting at the same time γ = 1 ⁄ 2, we obtain

tan x =

x

x²

x²

x²

tanh x =

x

x²

x²

x²

1

−

3

−

5

−

7

− . . .

1

+

3

+

5

+

7

+ . . .

These results were given by Lambert, and used by him to prove that π and π² incommensurable, and also any commensurable power of e.

Gauss in his famous memoir on the hypergeometric series

F(α, β, γ, x) = 1 +	α · β	x +	α(α + 1)β(β + 1)	x² + . . .
	1 · γ		1 · 2 · γ · (γ + 1)

gave the expression for F(α, β + 1, γ + 1, x) ÷ F(α, β, γ, x) as a continued fraction, from which if we put β = 0 and write γ − 1 for γ, we get the transformation

1 +

α

x +

α(α + 1)

x2 +

α(α + 1)(α + 2)

x3 + . . . =

1

β1x

β2x

γ

γ(γ + 1)

γ(γ + 1)(γ + 2)

1

−

1

−

1

− . . .

where

β1 =	α	,   β3 =	(α + 1)γ	, . . .,   β2n−1 =	(α + n − 1)(γ + n − 2)	,
	γ		(γ + 1)(γ + 2)		(γ + 2n – 3)(γ + 2n − 2)

β2 =	γ − α	,   β4 =	2(γ + 1 − α)	, . . .,   β2n =	n(γ + n − 1 − α)	.
	γ(γ + 1)		(γ + 2)(γ + 3)		(γ + 2n – 2)(γ + 2n − 1)

From this we may express several of the elementary series as continued fractions; thus taking α = 1, γ = 2, and putting x for −x, we have

log(1 + x) =

x

12x

12x

22x

22x

32x

32x

1

+

2

+

3

+

4

+

5

+

6

+

7

+ . . .

Taking γ = 1, writing x ⁄ α for x and increasing α indefinitely, we have

ex =

1

x

x

x

x

x

1

−

1

+

2

−

3

+

2

−

5

+ . . .

For some recent developments in this direction the reader may consult a paper by L. J. Rogers in the Proceedings of the London Mathematical Society (series 2, vol. 4).

 

Ascending Continued Fractions.

There is another type of continued fraction called the ascending continued fraction, the type so far discussed being called the descending continued fraction. It is of no interest or importance, though both Lambert and Lagrange devoted some attention to it. The notation for this type of fraction is

			b4 +	b5 +
		b3 +		a5
	b2 +		a4
a1 +		a3
	a2

It is obviously equal to the series

a1 +	b2	+	b3	+	b4	+	b5	+ . . .
	a2		a2a3		a2a3a4		a2a3a4a5

 

Historical Note.

The invention of continued fractions is ascribed generally to Pietro Antonia Cataldi, an Italian mathematician who died in 1626. He used them to represent square roots, but only for particular numerical examples, and appears to have had no theory on the subject. A previous writer, Rafaello Bombelli, had used them in his treatise on Algebra (about 1579), and it is quite possible that Cataldi may have got his ideas from him. His chief advance on Bombelli was in his notation. They next appear to have been used by Daniel Schwenter (1585–1636) in a Geometrica Practica published in 1618. He uses them for approximations. The theory, however, starts with the publication in 1655 by Lord Brouncker of the continued fraction

1		12		32		52
1	+	2	+	2	+	2	+ . . .

as an equivalent of π ⁄ 4. This he is supposed to have deduced, no one knows how, from Wallis’ formula for 4 ⁄ π viz.

3 . 3 . 5 . 5 . 7 . 7 . . .

2 . 4 . 4 . 6 . 6 . 8 . . .

John Wallis, discussing this fraction in his Arithmetica Infinitorum (1656), gives many of the elementary properties of the convergents to the general continued fraction, including the rule for their formation. Huygens (Descriptio automati planetarii, 1703) uses the simple continued fraction for the purpose of approximation when designing the toothed wheels of his Planetarium. Nicol Saunderson (1682–1739), Euler and Lambert helped in developing the theory, and much was done by Lagrange in his additions to the French edition of Euler’s Algebra (1795). Moritz A. Stern wrote at length on the subject in Crelle’s Journal (x., 1833; xi., 1834; xviii., 1838). The theory of the convergence of continued fractions is due to Oscar Schlömilch, P. F. Arndt, P. L. Seidel and Stern. O. Stolz, A. Pringsheim and E. B. van Vleck have written on the convergence of infinite continued fractions with complex elements.

References.—For the further history of continued fractions we may refer the reader to two papers by Gunther and A. N. Favaro, Bulletins di bibliographia e di storia delle scienze mathematische e fisicke, t. vii., and to M. Cantor, Geschichte der Mathematik, 2nd Bd. For text-books treating the subject in great detail there are those of G. Chrystal in English; Serret’s Cours d`algèbre supérieure in French; and in German those of Stern, Schlömilch, Hatterdorff and Stolz. For the application of continued fractions to the theory of irrational numbers there is P. Bachmann’s Vorlesungen über die Natur der Irrationalzahnen (1892). For the application of continued fractions to the theory of lenses, see R. S. Heath’s Geometrical Optics, chaps. iv. and v. For an exhaustive summary of all that has been written on the subject the reader may consult Bd. 1 of the Encyklopädie der mathematischen Wissenschaften (Leipzig).  (A. E. J.)