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 

Natural numbers 

Whole numbers 

Integers 

Rational numbers 

Irrational numbers 

Real numbers 

Things to notice:
 The set of Whole numbers is the same as the set of Natural numbers, except that it includes 0. To help remember this, think “o” is in “whole.”
 The set of Integers is the same as the set of whole numbers and the negatives of the whole numbers.
 We can think of Rational numbers as fractions. To remind us, notice that the word “ratio” is embedded in the word “rational.” A ratio is a fraction.
 The set of Rational numbers includes all decimals that have either a finite number of decimal places or that repeat in the same pattern of digits. For example, 0.333333… = 1/3 and .245245245…. = 245/999.
 The set of Natural numbers is a subset of the set of Whole numbers, which is contained in the set of Integers, which is inside of the set of Rational numbers.
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