What are Facts about the real number system?

Answer 1

The set of real numbers can be intuitively defined as the group of numbers that can be written in some decimal expression (whether finite or infinite). For example, the following numbers all can be expressed in some decimal expansion:

42

3.75

3/4 = 0.75

1/3 = 0.3333333...

pi = 3.141592653589...

sqrt(2) = 1.4142135...

8.24386820395768403...

Alternatively, we can define the set of real numbers as the group of numbers that can be expressed as points on an infinite number line.

These definitions are not very rigorous, and more rigorous definitions will be given below.

In general, real numbers can be rational or irrational as well as algebraic or transcendental. All real numbers are positive, negative, or zero. The real numbers are not algebraically closed, as all odd polynomials have at least one imaginary root. Finally, the real numbers are uncountable.

To define the rational numbers more rigorously, we can construct them as all numbers that can be created through the process of a convergence of a series similar to the one below:

{3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, ...}

All numbers in this set are real, as well as the number which is the result of the convergence of the set (pi).