theory of rational and irrational numbers

theory of rational and irrational numbers

The following entry is some sort of index of articles in PlanetMath about the basic

theory of rational and irrational numbers

, and it should be studied together with its

complement

: the

theory of algebraic and transcendental numbers

(

http://planetmath.org/TheoryOfAlgebraicNumbers

). The reader should follow the links in each bullet-point to learn more about each topic. For a somewhat deeper approach to the subject, the reader should read about

Algebraic Number Theory

. In this entry we will concentrate on the properties of the

complex numbers

and the

extension

ℂ

/

ℚ

.

There is also a

topic entry on rational numbers

.

1

Basic Definitions

1.

A number is said to be

rational

if it can be expressed as a

quotient

of

integers

(with non-zero

denominator

). The

set of all rational numbers

forms a field, denoted by

ℚ

.

2.

Such rational numbers, which are not integers, may be expressed as sum of

partial fractions

(the denominators being powers of distinct

prime numbers

).

3.

The

real numbers

are the set of all possible

decimal expansions

(where we don’t allow any expansion to end in all

9

’s). For the formal definition please see the entry real number (

http://planetmath.org/RealNumber

). The real numbers form a field, usually denoted by

ℝ

.

4.

A real number is said to be

irrational

(

http://planetmath.org/IrrationalNumber

) if it is not rational, i.e. it cannot be expressed as a quotient of integers.  The decimal expansion is non-periodic for any irrational, but periodic for any rational number.

5.

For example,

2

is irrational.

6.

Commensurable numbers

have a rational ratio.  See also

http://planetmath.org/node/12150

sine at irrational

multiples

of full angle.

2

Small Results

1.

The field

ℚ

is, up to an

isomorphism

,

subfield

(

prime subfield

) in every field where no sum of unities can be 0.  One may also say that

ℚ

is the least

field of numbers

.

2.

2

is irrational (

http://planetmath.org/SquareRootOf2IsIrrationalProof

). Similarly

d

is irrational as long as

d

∈

ℕ

is not a

perfect square

.

3.

The

sum of two square

roots of

positive

squarefree

integers (

http://planetmath.org/UsingThePrimitiveElementOfBiquadraticField

) is irrational.

4.

Rational and irrational

:  the sum,

difference

,

product

(

http://planetmath.org/Ring

) and quotient of two non-zero real numbers, from which one is rational and the other irrational, is irrational.

5.

There exists

real functions

, which are

continuous at

any irrational but

discontinuous

at any rational number (e.g. the Dirichlet’s

function

).

6.

Every irrational (and also rational) number is a limit of sequence of rational numbers (see real numbers (

http://planetmath.org/RealNumber

)).  An example:  the

sequence

(

1

+

1

1

)

1

,

(

1

+

1

2

)

2

,

(

1

+

1

3

)

3

,

…

converges

to the number

e

.

7.

The number e is irrational (this is not as difficult to prove as it is to show that e is

transcendental

). In fact, if

r

∈

ℚ

∖

{

0

}

then

e

r

is also irrational (

http://planetmath.org/ErIsIrrationalForRinmathbbQsetminus0

). There is an easier way to show that e is not a quadratic irrational.

8.

Every real

transcendental number

(such as

e

) is irrational, but not all irrational numbers are transcendental — some (such as

2

) are

algebraic

.

9.

“Most”

logarithms

(

http://planetmath.org/RationalBriggsianLogarithmsOfIntegers

) of

positive integers

are irrational (and transcendental).

10.

If

a

n

is irrational then

a

is irrational (see here (

http://planetmath.org/IfAnIsIrrationalThenAIsIrrational

)).

11.

A surprising fact: an irrational to an irrational power can be rational.

12.

π

and

π

2

are irrational (

http://planetmath.org/PiAndPi2AreIrrational

).

3

BIG Results

The irrational numbers are, in general, “easily” understood. The BIG

theorems

appear in the theory of transcendental numbers. Still, there are some

open problems

:  is Euler’s

constant

irrational? is

π

+

e

rational?

Title

theory of rational and irrational numbers

Canonical name

TheoryOfRationalAndIrrationalNumbers

Date of creation

2013-03-22 15:14:10

Last modified on

2013-03-22 15:14:10

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

15

Author

alozano (2414)

Entry type

Topic

Classification

msc 11R04

Related topic

TheoryOfAlgebraicNumbers

Related topic

AlgebraicNumberTheory

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