The Real Number System

All numbers that we will be mentioned in this lesson belong to the set of the real numbers. The set of the real numbers is denoted by the symbol . There are five subsets within the set of real numbers. Let’s go over each one of them.


Five (5) Subsets of Real Numbers

1) The Set of Natural or Counting Numbers

 The set of the natural numbers (also known as counting numbers) contains the elements,

the symbol of the set of natural numbers is N. moreover, the elements of the set of natural numbers can be expressed as {1,2,3,4,5,6,...}.

The ellipsis “…” signifies that the numbers go on forever in that pattern.


2) The Set of Whole Numbers

 The set of whole numbers includes all the elements of the natural numbers plus the number zero (0).

the symbol W indicates the set of whole numbers. on the other hand, the elements of the set of whole numbers can be expressed as {0, 1, 2, 3, 4, 5, 6, ...}.

The slight addition of the element zero to the set of natural numbers generates the new set of whole numbers. Simple as that!


3) The Set of Integers

The set of integers includes all the elements of the set of whole numbers and the opposites or “negatives” of all the elements of the set of counting numbers.

the symbol for the set of integers is Z while the elements of the set of integers can be expressed as {-4, -3, -2, -1, 0, 1, 2, 3, 4, ...}.

4) The Set of Rational Numbers

 The set of rational numbers includes all numbers that can be written as a fraction or as a ratio of integers. However, the denominator cannot be equal to zero.

the symbol Q indicates the set of rational numbers. meanwhile, the elements of the set of rational numbers are expressed as a/b where a and b are integers but b≠0.

A rational number may also appear in the form of a decimal. If a decimal number is repeating or terminating, it can be written as a fraction, therefore, it must be a rational number.

Examples of terminating decimals:

0.12 = 12/100 = 3/25
0.456 = 456/1,000 = 57/125

Examples of repeating decimals:

0.3333... = 1/3
0.636363... = 7/11
0.142857142857... = 1/7

5) The Set of Irrational Numbers

The set of the irrational numbers can be described in many ways. These are the common ones.

a) Irrational numbers are numbers that cannot be written as a ratio of two integers. This description is exactly the opposite that of the rational numbers.

b) Irrational numbers are the leftover numbers after all rational numbers are removed from the set of the real numbers. You may think of it as,

irrational numbers = real numbers “minus” rational numbers

c) Irrational numbers if written in decimal forms don’t terminate and don’t repeat.

There’s really no standard symbol to represent the set of irrational numbers. But you may encounter the one below.

the symbol for the set of irrational numbers is RQ while the elements of the set of rational numbers can be expressed as real numbers "minus" rational numbers.

Examples:

a) Pi

π = 3.141592653589793238462643...

b) Euler’s number

e=2.7182818284590452353602874...

c) Square root of 2

sqrt2=1.41421356237309504880168...

Here’s a quick diagram that can help you classify real numbers.

a diagram of the real number system showing real numbers classified into two sets, namely, the sets of rational and irrational numbers. within the set of rational numbers are the set of natural or counting numbers, the set of whole numbers, and the set of integers.

Practice Problems on How to Classify Real Numbers

Example 1: Tell if the statement is true or false.  Every whole number is a natural number.

Solution: The set of whole numbers include all natural or counting numbers and the number zero (0). Since zero is a whole number that is NOT a natural number, therefore the statement is FALSE.


Example 2: Tell if the statement is true or false.  All integers are whole numbers.

Solution: The number -1 is an integer that is NOT a whole number. This makes the statement FALSE.


Example 3: Tell if the statement is true or false. The number zero (0) is a rational number.

Solution: The number zero can be written as a ratio of two integers, thus it is indeed a rational number. This statement is TRUE.

0=0/2 → 0=0/13 → 0=0/1,000

Example 4: Name the set or sets of numbers to which each real number belongs.

1) 7

It belongs to the sets of natural numbers, {1, 2, 3, 4, 5, …}. It is a whole number because the set of whole numbers includes the natural numbers plus zero. It is an integer since it is both a natural and whole number. Finally, since 7 can be written as a fraction with a denominator of 1, 7/1, then it is also a rational number.

2) 0

This is not a natural number because it cannot be found in the set {1, 2, 3, 4, 5, …}. This is definitely a whole number, an integer, and a rational number. It is rational since 0 can be expressed as fractions such as 0/3, 0/16, and 0/45.

3) 0.318318...

This number obviously doesn’t belong to the set of natural numbers, set of whole numbers and set of integers. Observe that 18 is repeating, and so this is a rational number. In fact, we can write it a ratio of two integers.

0.318318... = 7/22

4) \sqrt 5

This is not a rational number because it is not possible to write it as a fraction. If we evaluate it, the square root of 5 will have a decimal value that is non-terminating and non-repeating. This makes it an irrational number.