real number

real number

1

Definition

There are several

equivalent

definitions

of

real number

, all in common use. We give one definition in detail and mention the other ones.

A

Cauchy sequence

of

rational numbers

is a

sequence

{

x

i

}

,

i

=

0

,

1

,

2

,

…

of rational numbers with the

property

that, for every rational number

ϵ

>

0

, there exists a

natural number

N

such that, for all natural numbers

n

,

m

>

N

, the

absolute value

|

x

n

-

x

m

|

satisfies

|

x

n

-

x

m

|

<

ϵ

.

The set

ℝ

of

real numbers

is the set of

equivalence classes

of Cauchy sequences of rational numbers, under the

equivalence relation

{

x

i

}

∼

{

y

i

}

if the

interleave sequence

of the two sequences is itself a Cauchy sequence.
The real numbers form a ring, with

addition

and

multiplication

defined by

•

{

x

i

}

+

{

y

i

}

=

{

(

x

i

+

y

i

)

}

•

{

x

i

}

⋅

{

y

i

}

=

{

(

x

i

⋅

y

i

)

}

There is an

ordering relation

on

ℝ

, defined by

{

x

i

}

≤

{

y

i

}

if either

{

x

i

}

∼

{

y

i

}

or there exists a natural number

N

such that

x

n

<

y

n

for all

n

>

N

. This definition is

well-defined

and does not depend on the choice of Cauchy sequences used to represent the equivalence classes.

One can prove that the real numbers form an

ordered field

and that they satisfy the

Dedekind completeness property

(also known as the

least

upper bound

property

): For every nonempty subset

S

⊂

ℝ

, if

S

has an upper bound then

S

has a

lowest upper bound

. It is also true that every ordered field with the least upper bound property is

isomorphic

to the real numbers.

Alternative definitions of the set of real numbers include:

1.

Equivalence classes of decimal sequences (sequences consisting of natural numbers between 0 and 9, and a single

decimal point

), where two decimal sequences are equivalent if they are identical, or if one has an

infinite

tail of 9’s, the other has an infinite tail of 0’s, and the leading portion of the first sequence is one lower than the leading portion of the second.

2.

Dedekind cuts

of rational numbers (that is, subsets

S

of

ℚ

with the property that, if

a

∈

S

and

b

<

a

, then

b

∈

S

).

3.

The real numbers can also be defined as the unique (up to

isomorphism

) ordered field satisfying the least upper bound property, after one has proved that such a field exists and is unique up to isomorphism.

2

Completeness

The main reason for introducing the reals is that the reals contain all limits.
More technically, the reals are

complete

(in the sense of

metric spaces

or

uniform spaces

, which is a different sense than the Dedekind completeness of
the order in the previous

section

). This means the following:

A sequence

(

x

n

)

of real numbers is called a Cauchy sequence if for any

ε

>

0

there exists an

integer

N

(possibly depending on

ε

) such that the
distance

|

x

n

-

x

m

|

is less than

ε

provided that

n

and

m

are
both

greater

than

N

. In other
words, a sequence is a Cauchy sequence if its

elements

x

n

eventually

come
and remain arbitrarily close to each other.

A sequence

(

x

n

)

converges

to the limit

x

if for any

ε

>

0

there exists an integer

N

(possibly depending on

ε

) such that the
distance

|

x

n

-

x

|

is less than

ε

provided that

n

is greater
than

N

. In other words, a sequence has limit

x

if its elements eventually
come and remain arbitrarily close to

x

.

It is easy to see that every

convergent sequence

is a Cauchy sequence. Now the
important fact about the real numbers is that the

converse

is true:

Every Cauchy sequence of real numbers is

convergent

.

That is, the reals are complete.

Note that the rationals are not complete. For example, the sequence

1

,

1.4

,

1.41

,

1.414

,

1.4142

,

1.41421

,

…

is Cauchy but it does not
converge to a rational number. (In the real numbers, in contrast, it converges
to the

square root

of

2

.)

The

existence

of limits of Cauchy sequences is what makes

calculus

work and is
of great practical use. The standard numerical test to determine if a sequence
has a limit is to test if it is a Cauchy sequence, as the limit is typically
not known in advance.

For example the standard series of the exponential function

e

x

=

∑

n

=

0

∞

x

n

n

!

converges to a real number because for every

x

the sums

∑

n

=

N

M

x

n

n

!

can be made arbitrarily small by choosing

N

sufficiently large. This proves
that the sequence is Cauchy, so we know that the sequence converges even if we
don’t know ahead of time what the limit is.

3

“The complete ordered field”

The real numbers are often described as “the complete ordered field,” a phrase
that can be interpreted in several ways.

First, an order can be

lattice

complete. It’s easy to see that no ordered field
can be lattice complete, because it can have no largest element (given any
element

z

,

z

+

1

is larger), so this is not the sense that is meant.

Additionally, an order can be Dedekind-complete, as defined in the Definitions section. The uniqueness result at the end of that section justifies using the
word “the” in the phrase “complete ordered field” when this is the sense of
“complete” that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction
starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.

These two notions of completeness ignore the field

structure

. However, an

ordered group

(and a field is a group under the

operations

of addition and

subtraction

) defines a uniform structure, and uniform structures have a notion
of completeness (

topology

); the description in the Completeness section above
is a special case. (We refer to the notion of completeness in uniform spaces
rather than the related and better known notion for metric spaces, since the
definition of metric space relies on already having a

characterisation

of the
real numbers.) It is not true that

ℝ

is the only uniformly complete
ordered
field, but it is the only uniformly complete Archimedean field, and indeed one
often hears the phrase “complete Archimedean field” instead of “complete
ordered field.” Since it can be proved that any uniformly complete Archimedean
field must also be

Dedekind complete

(and vice versa, of course), this
justifies using “the” in the phrase “the complete Archimedean field.” This
sense of completeness is most closely related to the construction of the reals
from Cauchy sequences (the construction carried out in full in this article),
since it starts with an Archimedean field (the rationals) and forms the uniform

completion

of it in a standard way.

But the original use of the phrase “complete Archimedean field” was by David
Hilbert, who meant still something else by it. He meant that the real numbers
form the largest Archimedean field in the sense that every other Archimedean
field is a

subfield

of

ℝ

. Thus

ℝ

is “complete” in the
sense that nothing
further can be added to it without making it no longer an Archimedean field.
This sense of completeness is most closely related to the construction of the
reals from

surreal numbers

, since that construction starts with a

proper class

that contains every ordered field (the surreals) and then selects from it the
largest

Archimedean

subfield.

This article contains material from the

http://en.wikipedia.org/wiki/Real_numbers

Wikipedia article on Real numbers which is incorporated herein under the terms of the

http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License

GNU Free Documentation License.

Title

real number

Canonical

name

RealNumber

Date of creation

2013-03-22 11:52:22

Last modified on

2013-03-22 11:52:22

Owner

djao (24)

Last modified by

djao (24)

Numerical id

23

Author

djao (24)

Entry type

Definition

Classification

msc 54C30

Classification

msc 26-00

Classification

msc 12D99

Synonym

real

Synonym

ℝ

Related topic

DedekindCuts

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