isometry 283

One can establish a number of forward-inverse

identities for the trigonometric functions. As examples

we have:

For instance, if a= cos–1 x, then cos a= x= . Thus

angle aappears in a right triangle with hypotenuse 1

and adjacent leg of length x. By P

YTHAGORAS

’

S THEO

-

REM

, the length of the opposite leg is and so

sin a= = , establishing the first rela-

tion. The remaining identities are proved similarly.

See also

INVERSE HYPERBOLIC FUNCTIONS

.

irrational number Any number that cannot be

expressed as a

RATIO

of two integers is called an

irrational number. As the study of

RATIONAL NUM

-

BERS

shows, the irrational numbers are precisely

those numbers whose decimal expansions do not ter-

minate or fall into a repeating cycle of values. For

example, the number with the decimal expansion

0.113133133313333133333133… is irrational. The

study of rational numbers also shows that, in a very

real sense, “most” numbers are irrational.

A famous proof, often attributed to Hippasus of

Metapontum (ca. 470

B

.

C

.

E

.), shows that √

–

2 is irra-

tional. T

HEODORUS OF

C

YRENE

(ca. 465–398

B

.

C

.

E

.)

established the same result geometrically, and also

showed that the numbers √

–

3 through to √

–

17 (exclud-

ing √

–

4, √

–

9, and √

–

16) are irrational. The

FUNDAMEN

-

TAL THEOREM OF ARITHMETIC

can be used to prove

that the mth root of a positive integer nis rational if,

and only if, nis already the mth power of an integer.

(If m

√

–

n= for some integers aand b, then am= nbm.

Writing each of a, b, and nas a product of primes, and

noting that the primes that consequently appear on the

left side of this equation must match those that appear

on the right, we conclude that each prime factor of n

appears in na multiple of mtimes. This establishes

that n= cmfor some integer c.) The same reasoning

shows that a number such as log25 is irrational. (If

log25 = , then 2a= 5b, contradicting the fundamental

theorem of arithmetic.)

Truncating the decimal expansion of an irrational

number produces a rational arbitrarily close to that

irrational number. For example, 1.4, 1.41, 1.414, … is

a sequence of rational numbers converging to √

–

2=

1.41421356…

In 1737 L

EONHARD

E

ULER

established that the

number eis irrational, and in 1761 J

OHANN

H

EINRICH

L

AMBERT

(1728–77) proved the irrationality of π. No

one to this day knows whether or not the numbers 2e,

πe, and π√

–

2are irrational. (It is known that eπand e· π

are irrational.) Surprisingly, the rationality or irra-

tionality of E

ULER

’

S CONSTANT

γis still not known.

It is possible for an irrational number raised to an

irrational power to be rational. For example, if x=

(√

–

2)√

–

2turns out to be rational, then we have an exam-

ple of such a phenomenon. If x, on the other hand, is

not rational, then it is irrational and x√

–

2= (√

–

2)√

–

2)√

–

2=

(√

–

2)2= 2 is an example of what we seek. (Unfortu-

nately this indirect line of reasoning does not indicate

which of the two possibilities actually occurs.)

See also

ALGEBRAIC NUMBER

;

CONTINUED FRAC

-

TION

;

E

;

NUMBER

;

REAL NUMBERS

;

SURD

;

TRANSCEN

-

DENTAL NUMBER

.

isolated point (acnode) A point that satisfies the

equation of a curve but is not on the main arc of the

curve is called an isolated point. For example, the curve

has y2= x3– x2has x= 0, y= 0 as a solution, with no

other solution near this position. The point (0,0) is an

isolated point for the equation.

See also

DOUBLE POINT

.

isometry (congruence transformation) A

GEOMETRIC

TRANSFORMATION

, such as a translation, rotation, or a

reflection, that preserves the distances between points

in space is called an isometry. Isometries thus have the

property of preserving the shape and size of geometric

a

–

b

a

–

b

√1 – x2

√1 – x2

–––––

1

√1 – x2

x

–

1