topic entry on real numbers

topic entry on real numbers

Introduction

The

real number

system may be conceived as an attempt to fill in the
gaps in the

rational number

system. These gaps first became apparent
in connection with the

Pythagorean theorem

, which requires one to extract
a

square root

in order to find the third side of a

right triangle

two of
whose sides are known. Hypossos, a student of Pythagoras, showed that there
is no rational number whose square is exactly 2. In particular, this means
that there is no rational number which may be used to describe the length
of the

diagonal

of a square the length of whose sides is rational. This
result ruined the philosophical program of Pythagoras, which was to describe
everything in terms of whole numbers (or ratios of whole numbers) and,
according to legend, resulted in Hypossos drowning himself.

Eventually

,
geometers reconciled themselves to the

existence

of

irrational

magnitudes
and Eudoxos devised his

method of exhaustion

which allowed one to prove
results about irrational magnitudes by considerations of rational magnitudes
which are smaller and larger than the the irrational magnitude in question.

Centuries later, Descartes showed how it is systematically possible to
reduce questions of

geometry

to

algebra

. This brought up the issue of
irrational numbers again — if one is going to reformulate everything
in terms of algebra, then one cannot have recourse to defining magnitudes
geometrically, but have to find some sort of number which can adequately
represent things like the

hypotenuse

of a square with rational sides.
At first, such problems of logical

consistency

were swept under the rug,
but eventually mathematicians realized that their subject needed to be
put on a firm

logical foundation

. In particular, Dedekind solved this
difficulty by defining the real numbers as a certain type of

partition

of
the set of rational numbers which he termed a cut and defining

operations

on these numbers, such as

addition

,

subtraction

,

multiplication

, and

division

in terms of operations on these partitions.

Index of entries on real numbers

The below list presents entries on real numbers in an order
suitable for studying the subject.

1.

Rational numbers

2.

Axiomatic

definition

of the real numbers.

3.

Constructions of real numbers (advanced):

(a)

Dedekind cuts

(b)

Cauchy sequences

(

http://

planetmath

.org/RealNumber

)

(c)

Characterization

of real numbers (

http://planetmath.org/EveryOrderedFieldWithTheLeastUpperBoundPropertyIsIsomorphicToTheRealNumbers

)

(d)

Reals not

isomorphic

to

p

-adic numbers (

http://planetmath.org/NonIsomorphicCompletionsOfMathbbQ

)

4.

commensurable numbers

5.

positive

6.

Inequalities for real numbers

(

http://planetmath.org/InequalityForRealNumbers

)

7.

index of inequalities

8.

rational numbers are real numbers

9.

interval

10.

nested interval theorem

11.

Real numbers are

uncountable

(

http://planetmath.org/CantorsDiagonalArgument

)

12.

Archimedean property

13.

Operations for real numbers

(a)

infimum and supremum for real numbers

(b)

minimal and maximal number

(c)

absolute value

(d)

square root

(e)

fraction power

14.

Topic entry on

algebraic

and

transcendental numbers

(

http://planetmath.org/TheoryOfAlgebraicNumbers

)

(a)

Irrational number (

http://planetmath.org/Irrational

)

(b)

Transcendental number

(c)

Algebraic number

(

http://planetmath.org/AlgebraicNumber

)

15.

Particular real numbers

(a)

natural log base

(b)

pi

(c)

Mascheroni constant

(d)

golden ratio

Generalizations

There are many

generalizations

of real numbers. These include
the

complex numbers

, quaternions,

extended real numbers

,

hyperreal numbers

(

http://planetmath.org/

Hyperreal

),
and

surreal numbers

.  Of course the field

ℝ

has many other

field extensions

, e.g. the field

ℝ

⁢

(

x

)

of the

rational functions

.

Title

topic entry on real numbers

Canonical

name

TopicEntryOnRealNumbers

Date of creation

2013-03-22 15:40:38

Last modified on

2013-03-22 15:40:38

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

22

Author

matte (1858)

Entry type

Topic

Classification

msc 54C30

Classification

msc 26-00

Classification

msc 12D99

Related topic

GelfandTornheimTheorem

Related topic

PositivityInOrderedRing

Related topic

TopicEntryOnRationalNumbers

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