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The Real Number System

Each real number is a member of one or more of the following sets.
The sets of numbers described in the following table should look familiar to you.  It is sometimes handy to have names for these sets of numbers, so knowing their names can simplify, for example, describing domains of functions or comprehending theorems such as the rational zeros theorem.

Set

Description

Natural numbers

{1, 2, 3, 4, …. }

Whole numbers

{0, 1, 2, 3, 4, …}

Integers

{ …, -3, -2, -1, 0, 1, 2, 3, …. }

Rational numbers

All numbers that can be written as , where a and b are both integers, and b is not equal to 0.

Irrational numbers

Numbers such as

Real numbers

The union of the sets of rational numbers and irrational numbers

Things to notice:

 

 

 

Example

Classify the following numbers.  Remember that a number may belong to more than one category.
            0,   4,   -9, 0.23,

Solution 

Number                     Member of these sets

0                                  Whole, Integer, Rational (can be written as ), Real

4                                  Natural, Whole, Integer, Rational (can be written as ), Real

-9                                 Integer, Rational, Real

                                Rational, Real

                              Natural (), Whole, Integer, Rational, Real

                            Integer ( ), Rational, Real

                            Irrational  (≈ 3.31662479036…  This is not a terminating
                                    decimal and it does not repeat), Real

                                Whole (), Integer, Rational, Real

                       Rational ( =  ) , Real

                                Irrational, Real

                             Rational, Real

0.23                            Rational (terminating decimal equal to ), Real

                             Irrational (It is a fraction, but not a quotient of two integers), Real