continued fraction

continued fraction

Given a

sequence

of positive real numbers

(

a

n

)

n

≥

1

, with

a

0

any

real number

. Consider the sequence

c

1

=

a

0

+

1

a

1

c

2

=

a

0

+

1

a

1

+

1

a

2

c

3

=

a

0

+

1

a

1

+

1

a

2

+

1

a

3

c

4

=

…

The limit

c

of this sequence, if it exists, is called the value or limit of the

infinite continued fraction

with

convergents

(

c

n

)

, and is denoted by

a

0

+

1

a

1

+

1

a

2

+

1

a

3

+

…

or by

a

0

+

1

a

1

+

⁢

1

a

2

+

⁢

1

a

3

+

⁢

…

In the same way, a

finite sequence

(

a

n

)

1

≤

n

≤

k

defines a finite sequence

(

c

n

)

1

≤

n

≤

k

.

We then speak of a finite continued fraction with value

c

k

.

An archaic word for a

continued fraction

is

anthyphairetic ratio

.

If the

denominators

a

n

are all (

positive

)

integers

, we speak of a

simple

continued fraction. We then use the notation

q

=

⟨

a

0

;

a

1

,

a

2

,

a

3

,

…

⟩

or, in the finite case,

q

=

⟨

a

0

;

a

1

,

a

2

,

a

3

,

…

,

a

n

⟩

.

It is not hard to prove that any

irrational number

c

is the value of a unique

infinite

simple continued fraction

. Moreover, if

c

n

denotes its

n

th convergent, then

c

-

c

n

is an alternating sequence and

|

c

-

c

n

|

is decreasing (as well as convergent to zero). Also, the value of an infinite simple continued fraction is perforce
irrational.

Any

rational number

is the value of two and only two finite continued fractions; in one of them, the last denominator is 1. E.g.

43

30

=

⟨

1

;

2

,

3

,

4

⟩

=

⟨

1

;

2

,

3

,

3

,

1

⟩

.

These two conditions on a real number

c

are

equivalent

:

1.

c

is a root of an

irreducible

quadratic polynomial

with
integer

coefficients

.

2.

c

is irrational and its simple
continued fraction is “

eventually periodic

”; i.e.

c

=

⟨

a

0

;

a

1

,

a

2

,

…

⟩

and, for some integer

m

and some integer

k

>

0

, we have

a

n

=

a

n

+

k

for all

n

≥

m

.

For example, consider the

quadratic equation

for the

golden ratio

:

x

2

=

x

+

1

or equivalently

x

=

1

+

1

x

.

We get

x

=

1

+

1

1

+

1

x

=

1

+

1

1

+

1

1

+

1

x

and so on.
If

x

>

0

, we therefore expect

x

=

⟨

1

;

1

,

1

,

1

,

…

⟩

which indeed can be proved. As an exercise, you might like to look for a continued fraction expansion of the

other

solution

of

x

2

=

x

+

1

.

Although

e

is

transcendental

, there is a surprising pattern in its simple continued fraction expansion.

e

=

⟨

2

;

1

,

2

,

1

,

1

,

4

,

1

,

1

,

6

,

1

,

1

,

8

,

1

,

1

,

10

,

…

⟩

No pattern is apparent in the expansions some other well-known transcendental

constants

, such as

π

and Apéry’s constant

ζ

⁢

(

3

)

.

Owing to a kinship with the Euclidean division algorithm, continued fractions arise naturally in

number theory

. An
interesting example is the Pell

diophantine equation

x

2

-

D

⁢

y

2

=

1

where

D

is a nonsquare integer

>

0

. It turns out that if

(

x

,

y

)

is any solution of the

Pell equation

other than

(

±

1

,

0

)

, then

|

x

y

|

is a convergent to

D

.

22

7

and

355

113

are well-known

rational approximations

to

π

, and indeed both are convergents to

π

:

3.14159265

⁢

…

=

π

=

⟨

3

;

7

,

15

,

1

,

292

,

…

⟩

3.14285714

⁢

…

=

22

7

=

⟨

3

;

7

⟩

3.14159292

⁢

…

=

355

113

=

⟨

3

;

7

,

15

,

1

⟩

=

⟨

3

;

7

,

16

⟩

For one more example, the distribution of leap years in the 4800-month cycle of the Gregorian calendar can be interpreted (loosely speaking) in terms of the continued fraction expansion of the number of days
in a solar year.

Title

continued fraction

Canonical name

ContinuedFraction

Date of creation

2013-03-22 12:47:12

Last modified on

2013-03-22 12:47:12

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

29

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11Y65

Classification

msc 11J70

Classification

msc 11A55

Synonym

chain fraction

Related topic

FareySequence

Related topic

AdjacentFraction

Related topic

SternBrocotTree

Related topic

FareyPair

Defines

anthyphairetic ratio

Defines

simple continued fraction

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