axiomatic definition of the real numbers
Axiomatic definition of the real numbers
is an Abelian group:
For , we have
there exists an element such that for all ,
every has an inverse such that .
The operations and are compatible with the order :
If , , and , then .
If , , with and , then .
has the least upper bound property: If , then an element is an for if
If is non-empty, we then say that is bounded from above. That has the least upper bound property means that if is bounded from above, it has a least upper bound . That is, has an upper bound such that if is any upper bound from , then .
Here it should be emphasized that from the above we can not deduce that a set 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).
In condensed form, the above conditions state that is an ordered field with the least upper bound property. In particular is a ring, and is a group, and we have the following basic properties:
The additive inverse is unique (proof) (http://planetmath.org/UniquenessOfAdditiveIdentityInARing).
The additive identity is unique (proof) (http://planetmath.org/UniquenessOfAdditiveIdentityInARing2).
The multiplicative inverse is unique (proof) (http://planetmath.org/UniquenessOfInverseForGroups).
If are non-zero, then (proof) (http://planetmath.org/InverseOfAProduct).
In view of property 2, we can write simply instead of and .
Because of the additive inverse of a real number is unique (by property 1 above), and , we see that the additive inverse of is , or that . Similarly, if , then (or we’ll end up with ), and therefore by Property 6 above, has a unique multiplicative inverse. Since , we see that is the multiplicative inverse of . In other words, .
For let us also define , which is called the difference of and . By commutativity, . It is also common to leave out the multiplication symbol and simply write . Suppose and is non-zero. Then divided (http://planetmath.org/Division) by is defined as
In consequence, if and are non-zero, then
|Title||axiomatic definition of the real numbers|
|Date of creation||2013-03-22 15:39:29|
|Last modified on||2013-03-22 15:39:29|
|Last modified by||matte (1858)|