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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:

• 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