Real Numbers and the Dedekind Cut
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Numbers as solutions to equations
Equations are omitted for technical reasons - download the original pdf
You should be familiar with the following sets of numbers. [Equation goes here - download the original to see it.] (Note, there is some debate as to whether 0 is included in the set of natural numbers or not.) Integers are used to solve certain types of equation. Given integers a and b, there exist positive integers c, d such that [Equation goes here - download the original to see it.] However, sometimes equations of the form [Equation] where a and b are integers may be solved to give integer solution x; but in general an equation of this form does not always have integer solutions. To solve it we must extend the number system to include numbers of the form [Equation] where a and b are integers. Such numbers are called rational numbers. The set of all rational numbers is denoted by [Equation]. [Equation] All integers are rational numbers, but not all rational numbers are integers. Equations of the form [Equation] where a is a rational number may sometimes be solved with rational numbers. For example, [Equation] has solution [Equation]. We denote the positive solution to the equation [Equation], where a is an integer, is called the square root of a and is denoted by [Equation]. For example, the positive square root of 2 is denoted [Equation]. We shall show below that [Equation] is not a rational number. Because it is not rational it is called irrational. Hence, in order to solve equations of the form [Equation], for integers n and a, we need to extend the number system yet further, to include the irrational numbers. The set of all rational and irrational numbers combined is called the set of real numbers, and is denoted by [Equation]. [Equation]
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Contents of
Real Numbers and the Dedekind Cut
1 Numbers as solutions to equations
2 The irrationality of root 2
3 The set of all rational numbers is dense
4 The Dedekind cut
5 The set of real numbers as an ordered field
6 The ordered field axioms
7 The sets of rational and real numbers are ordered fields.
8 Axiom of completeness
9 Bounded sets of numbers
10 Maximum and minimum
11 Bounded set
12 Least upper bound
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