A real number is a number that falls on the real number line. It can have any value. Some examples of real numbers are:, and so on. Numbers that are not real are , , , i.e. complex numbers, and quaternions.
The set of real numbers, denoted by , is a subset of complex numbers(). Commonly used subsets of the real numbers are the rational numbers (), integers (), natural numbers () and irrational numbers (sometimes, but not universally, denoted ). In addition means positive integers and means negative integers. The real numbers can also be divided between the algebraic numbers and transcendental numbers, although these two classes are best understood as subsets of the complex numbers.
The set of Real numbers is a complete, ordered field under addition and multiplication.
Dedekind developed a method to construct , the set of Real numbers from the set of rational numbers, using the very elegant idea of cuts.
Let be the set of rational numbers.
Let be non-empty
We say that is a cut if and only if
(1) and is bounded above
(2) If ,
(3) If is a rational and then there exists a rational such that
We define to be the set of all cuts
Using this definition, we can show that possesess all the properties mentioned above.
Let , be cuts
We say that iff
Let be an upper bound of
Let be the set of all rationals such that .
As is bounded above, is non empty.
Consider the set
We see that is a cut, say
Thus, the cut is the supremum of set