continued fraction 99

That this process does not terminate proves that √

–

2 is

not a fraction. In a similar way, one establishes:

and √

–

6 = [2,

—

2,4], showing that these quantities are also

irrational. The number ealso has an infinite continued-

fraction representation: e= [2,1,2,1,1,4,1,1,6,1,1,8,1,

1,10,…], as does the golden mean .

Mathematicians have proved that finite continued

fractions in standard form have a curious property:

Reversing the order of the integers that appear

in a finite continued fraction produces a new

fraction with the same numerator as the origi-

nal quantity.

For example, as we have seen, [3,14,1,2,1,7] =

1,402/457, and one calculates that:

equals the fraction 1,402/181.

Continued fractions were systematically studied by

L

EONHARD

E

ULER

(1707–83), and he was the first to

formally introduce them in a written text. J

OSEPH

-

L

OUIS

L

AGRANGE

(1736–1813) extended much of

Euler’s work. Continued fractions have proved to be

very useful in solving a large selection of D

IOPHANTINE

EQUATION

s. They also provide excellent rational

approximations to irrational numbers. For example,

terminating the continued fraction representation for

√

–

2 after a finite number of steps yields good approxi-

mations to the square root of 2:

This particular sequence of fractions, generated by the

formula , was used by Theon of Smyrna as

early as the first century

C

.

E

. It holds some mysterious

properties. For example, every second term of the

sequence corresponds to a P

YTHAGOREAN TRIPLE

:

and every other term, rounding the numerator and

denominator each down to the half, yields

FIGURATE

NUMBERS

that are both square and triangular: