axiomatic definition of the real numbers

axiomatic definition of the real numbers

Axiomatic definition of the real numbers

The real numbers consist of a set

ℝ

together with

mappings

+

:

ℝ

×

ℝ

→

ℝ

and

⋅

:

ℝ

×

ℝ

→

ℝ

and a

relation

<

⊆

ℝ

×

ℝ

satisfying
the following conditions:

1.

(

ℝ

,

+

)

is an

Abelian group

:

(a)

For

a

,

b

,

c

∈

ℝ

, we have

a

+

b

=

b

+

a

,

(

a

+

b

)

+

c

=

a

+

(

b

+

c

)

,

(b)

there exists an

element

0

∈

ℝ

such that

a

+

0

=

a

for all

a

∈

ℝ

,

(c)

every

a

∈

ℝ

has an

inverse

(

-

a

)

∈

ℝ

such that

a

+

(

-

a

)

=

0

.

2.

(

ℝ

∖

{

0

}

,

⋅

)

is an Abelian group:

(a)

For

a

,

b

,

c

∈

ℝ

, we have

a

⋅

b

=

b

⋅

a

,

(

a

⋅

b

)

⋅

c

=

a

⋅

(

b

⋅

c

)

,

(b)

there exists an element

1

∈

ℝ

∖

{

0

}

such that

a

⋅

1

=

a

for all

a

∈

ℝ

,

(c)

every

a

∈

ℝ

∖

{

0

}

has an inverse

a

-

1

∈

ℝ

such that

a

-

1

⋅

a

=

1

.

3.

The

operation

⋅

is

distributive

over

+

: If

a

,

b

,

c

∈

ℝ

, then

a

⋅

(

b

+

c

)

=

a

⋅

b

+

a

⋅

c

,

(

b

+

c

)

⋅

a

=

b

⋅

a

+

c

⋅

a

.

4.

(

ℝ

,

<

)

is a

total order

:

(a)

(

transitivity

) if

c

∈

ℝ

,

a

<

b

, and

b

<

c

, then

a

<

c

,

(b)

(

trichotomy

) precisely one of the below alternatives
hold:

a

<

b

,

a

=

b

,

b

<

a

.

For convenience we make the following notational definitions:

a

>

b

means

b

<

a

,

a

≤

b

means either

a

<

b

or

a

=

b

, and

a

≥

b

means either

b

<

a

or

a

=

b

.

5.

The operations

+

and

⋅

are

compatible

with the order

<

:

(a)

If

a

,

b

,

c

∈

ℝ

and

a

<

b

, then

a

+

c

<

b

+

c

.

(b)

If

a

,

b

,

c

∈

ℝ

with

a

<

b

and

0

<

c

, then

a

⁢

c

<

b

⁢

c

.

6.

ℝ

has the least

upper bound

property: If

A

⊂

ℝ

,
then an element

M

∈

ℝ

is an for

A

if

a

<

M

,

for all

⁢

a

∈

A

.

If

A

is non-empty, we then say that

A

is bounded from above.
That

ℝ

has the least upper bound property means that
if

A

⊂

ℝ

is bounded from above, it has a least upper bound

m

∈

ℝ

. That is,

A

has an upper bound

m

such that if

M

is any upper bound from

M

,
then

m

≤

M

.

Here it should be emphasized that from the above we can not deduce that
a set

ℝ

with operations

+

,

⋅

,

<

exists. To settle this question such
a set has to be explicitly constructed. However, this can be done in various ways, as
discussed on this page (

http://planetmath.org/RealNumber

).
One can also show the above conditions uniquely determine the real numbers
(up to an

isomorphism

). The proof of this can be found on
this page (

http://planetmath.org/EveryOrderedFieldWithTheLeastUpperBoundPropertyIsIsomorphicToTheRealNumbers

).

Basic properties

In condensed form, the above conditions state that

ℝ

is an

ordered
field

with the least upper bound property. In particular

(

ℝ

,

+

,

⋅

)

is a ring, and

(

ℝ

∖

{

0

}

,

⋅

)

is a group, and we have the following basic properties:

Lemma 1

.

Suppose

a

,

b

∈

R

.

1.

The

additive inverse

(

-

a

)

is unique (proof) (

http://planetmath.org/UniquenessOfAdditiveIdentityInARing

).

2.

The

additive identity

0

is unique (proof) (

http://planetmath.org/UniquenessOfAdditiveIdentityInARing2

).

3.

(

-

1

)

⋅

a

=

(

-

a

)

(proof) (

http://planetmath.org/1cdotAA

).

4.

(

-

a

)

⋅

(

-

b

)

=

a

⋅

b

(proof) (

http://planetmath.org/XcdotYXcdotY

).

5.

0

⋅

a

=

0

(proof) (

http://planetmath.org/0cdotA0

)

6.

The

multiplicative inverse

a

-

1

is unique (proof) (

http://planetmath.org/UniquenessOfInverseForGroups

).

7.

If

a

,

b

are non-zero, then

(

a

⁢

b

)

-

1

=

b

-

1

⁢

a

-

1

(proof) (

http://planetmath.org/InverseOfAProduct

).

In view of property

2

, we can write simply

-

a

instead of

(

-

1

)

⋅

a

and

(

-

a

)

.

Because of the additive inverse of a real number is unique (by property 1 above), and

(

-

a

)

+

a

=

a

+

(

-

a

)

=

0

, we see that the additive inverse of

-

a

is

a

, or that

-

(

-

a

)

=

a

. Similarly, if

a

≠

0

, then

a

-

1

≠

0

(or we’ll end up with

1

=

a

⁢

a

-

1

=

a

⁢

0

=

0

), and therefore by Property 6 above,

a

-

1

has a unique multiplicative inverse. Since

a

⁢

a

-

1

=

a

-

1

⁢

a

=

1

, we see that

a

is the multiplicative inverse of

a

-

1

. In other words,

(

a

-

1

)

-

1

=

a

.

For

a

,

b

∈

ℝ

let us also define

a

-

b

=

a

+

(

-

b

)

, which is called
the

difference

of

a

and

b

.
By

commutativity

,

a

-

b

=

-

b

+

a

.  It is also common to leave out the

multiplication

symbol and simply write

a

⁢

b

=

a

⋅

b

.  Suppose

a

∈

ℝ

and

b

∈

ℝ

is non-zero.  Then

b

divided (

http://planetmath.org/Division

) by

a

is defined as

a

b

=

a

⁢

b

-

1

.

In

consequence

, if

a

,

b

,

c

,

d

∈

ℝ

and

b

,

c

,

d

are non-zero, then

•

a

b

c

d

=

b

⁢

d

a

⁢

c

,

•

a

⁢

b

b

=

a

.

For example,

a

b

c

d

=

a

⁢

b

-

1

c

⁢

d

-

1

=

a

⁢

b

-

1

⁢

(

c

⁢

d

-

1

)

-

1

=

a

⁢

b

-

1

⁢

d

⁢

c

-

1

=

a

⁢

d

b

⁢

c

.

Title

axiomatic definition of the real numbers

Canonical name

AxiomaticDefinitionOfTheRealNumbers

Date of creation

2013-03-22 15:39:29

Last modified on

2013-03-22 15:39:29

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

17

Author

matte (1858)

Entry type

Definition

Classification

msc 54C30

Classification

msc 26-00

Classification

msc 12D99

Related topic

RealNumber

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