Real Numbers: Definition, Properties, Sets & Examples Explained

Real Numbers

In mathematics, the number system is primarily composed of real numbers and imaginary numbers. Real numbers encompass both rational and irrational numbers, all of which can be plotted on a number line. Unlike real numbers, imaginary numbers do not exist on the standard number line. In this guide, we will explore the definition, classifications, sets, and properties of real numbers, supported by illustrative examples.

What are Real Numbers?

Real numbers are defined as the complete set of rational and irrational numbers. Symbolized by the letter ‘R′, this category includes positive and negative values, natural numbers, fractions, and decimals.

Set of Real Numbers

A real number is any value that can be represented as a specific point on the number line. The set of real numbers includes:

1. Natural Numbers
2. Whole Numbers
3. Integers
4. Rational Numbers
5. Irrational Numbers

Types of Real Numbers
Type	Definition	Examples
Natural Numbers	Numbers that begin from 1 and end at infinity.	All numbers such as 1, 2, 3, 4, 5, 6,…..…
Whole Numbers	Numbers including 0 and natural numbers.	All numbers such as 0, 1, 2, 3, 4, 5, 6, ………
Integers	Numbers that are whole numbers and negative of all natural numbers	Includes: -infinity (-∞),……..-4, -3, -2, -1, 0, 1, 2, 3, 4, ……+infinity (+∞)
Rational Numbers	Numbers that are represented in p/q form, where q ≠ 0	Examples of rational numbers are 6/4, 5/4, 12/6,3/9, 18/9, etc.
Irrational Numbers	Numbers that are not represented in p/q form.	Numbers that are non-terminating and non-repeating, such as, √,5–√,π, …. etc.

Symbols of Real Numbers

Real numbers, which include natural numbers, whole numbers, integers, rational numbers, and irrational numbers, are classified as follows:

Symbols of Real Numbers
Types of Real Numbers	Symbol
Natural Numbers	N
Whole Numbers	W
Integers	Z
Rational Numbers	Q
Irrational Numbers	Q’

Real Numbers Chart

Real numbers are categorized into several types: natural numbers (counting numbers starting from 1), whole numbers (natural numbers including 0), integers (whole numbers and their negative counterparts), rational numbers (numbers expressed as fractions p/q), and irrational numbers (numbers that cannot be expressed as p/q). These real number classifications will help clarify the hierarchy of the number system.

Properties of Real Numbers

The core algebraic properties of real numbers include:

Properties of Real Numbers		Examples
Commutative property	The Commutative Property: For any two real numbers m and n, the order of operation does not change the result: m + n = n + m for addition, and m × n = n × m for multiplication. Addition: m + n = n + m. Multiplication: m × n = n × m.	Example of Commutative Property: Addition: 2 + 3= 3+2 Multiplication: 2 ×3= 3 ×2
Associative property	The Associative Property: For any three real numbers m, n, and r, the grouping of numbers does not change the result: m + (n + r) = (m + n) + r for addition, and m(nr) = (mn)r for multiplication. Addition: The general form will be m + (n + r) = (m + n) + r. Multiplication: (mn) r = m (nr).	Example of Associative Property: Addition: 20 +( 10+10)= (20 + 10) + 10 Multiplication:  (5 ×2) 2= 5(2 ×2)
Distributive property	The Distributive Property: For any three real numbers m, n, and r, the distributive property states that multiplying a sum by a number is the same as multiplying each addend individually: m(n + r) = mn + mr and (m + n)r = mr + nr.	Example of Distributive Property: 5(2 + 3) = (5 × 2) + (5 × 3) = 10 + 15 = 25.
Identity property	Real numbers also adhere to the additive and multiplicative identity properties. For addition: m + 0 = m. (0 is the additive identity) For multiplication: m × 1 = 1 × m = m.	Example of Identity Properties: Addition: 2+ 0 = 2 Multiplication: 2 ×1= 1× 2= 2

Real Numbers on Number Line

The real number line is a horizontal axis representing all real values. The origin point is “0”, with positive numbers extending to the right and negative numbers extending to the left at uniform intervals.

Real Numbers Examples

Example 1: Determine if the following statements are true or false and provide reasoning.

i.) Every whole number is a natural number.

ii.) Every integer is a rational number.

iii.) Every rational number is an integer.

Solution: i.) False, because zero is a whole number but not a natural number.

ii.) True, because every integer m can be expressed as m/1, which fits the definition of a rational number.

iii.) False, because 3/5 is a rational number but not an integer.

Example 2: Find five rational numbers between 1 and 2.

Solution: We can find rational numbers between any two values r and s by calculating their average.

To find a rational number between r and s, use the formula (r + s) / 2.

Calculating for 1 and 2: (1 + 2) / 2 = 3/2.

Applying this formula, we find the first number is 3/2.

Thus, 3/2 is a rational number located between 1 and 2.

You can repeat this process to find as many rational numbers as needed.

Additional rational numbers between 1 and 2 include 5/4, 11/8, 13/8, and 7/4.

Example 3: Solve for 'b' using the associative property: (24 + 222) + 654 = 24 + (b + 654)

Solution: Based on the associative property, m + (n + r) = (m + n) + r.

246 + 654 = 24 + (b + 654)

900 = 678 + b

b = 900 - 678

b = 222