Real Numbers: Definition, Set, Chart, Properties & Examples

Real Numbers

Real Numbers: In Mathematics, the number system is composed of two types of numbers, that is, Real numbers and imaginary numbers. Real numbers are the sum of rational and irrational numbers which can be represented on a number line. Unlike real numbers, imaginary numbers are not represented on the number line. So this article, we will discuss in detail real numbers, its definition, types, sets of real numbers, and real number charts with some examples.

What are Real Numbers?

A combination of both rational numbers and irrational numbers is known as real numbers. Real numbers can be positive and negative, represented by ‘R′ natural numbers, fractions, and decimals all come under this category.

Set of Real Numbers

The real number is the number that is represented on the number line. Set of real numbers, that is,

Natural Numbers
Whole Numbers
Integers
Rational Numbers
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 such as natural numbers, whole numbers, Integers, rational numbers, and Irrational numbers are symbolized as-

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 of different types, namely, natural numbers ( starting from 1 to infinity), whole numbers( 0 and natural numbers collection), Integers(whole numbers and negative of all natural numbers), rational numbers( can be written in p/q form), irrational number(a number which cannot be represented in p/q form). These real number charts will help you to understand the concept well.

Properties of Real Numbers

The properties of real numbers are-

Properties of Real Numbers

Examples

Commutative property

If m and n are the numbers, then the general form will be m + n = n + m for addition and m× n = n×m for multiplication.

Addition: m + n = n + m.
Multiplication: m × n = n × m.

For Example-

Addition: 2 + 3= 3+2
Multiplication: 2 ×3= 3 ×2

Associative property

If m, n, and r are the numbers. The general form will be m + (n + r) = (m + n) + r for addition(mn) r = m(nr) for multiplication.

Addition: The general form will be m + (n + r) = (m + n) + r.
Multiplication: (mn) r = m (nr).

For Example-

Addition: 20 +( 10+10)= (20 + 10) + 10
Multiplication: (5 ×2) 2= 5(2 ×2)

Distributive property

For three numbers m, n, and r, which are real in nature, the distributive property is represented as:

m (n + r) = mn + mr and (m + n) r = mr + nr.

For Example-

5(2 + 3) = 5 × 2 + 5 × 3

Identity property

There are additive and multiplicative identities.

For addition: m + 0 = m. (0 is the additive identity)
For multiplication: m × 1 = 1 × m = m.

For Example-

Addition: 2+ 0 = 2
Multiplication: 2 ×1= 1× 2= 2

Real Numbers on Number Line

The real number line is a horizontal line with arrows at both ends. The “0” is the origin point. All the positive numbers lie on the right side of the origin and all the negative numbers lie on the left side of the origin, having a definite scale.

Real Numbers Examples

Example1: Are the following statements true or false? Give a reason for your answers.

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 in the form m/1, and so it is a rational number.

iii.) False, because 3/5 is not an integer.

Example2: Find five rational numbers between 1 and 2.

Solution: One of the methods to find the rational number between 1 and 2.

As we know in order to find a rational number between “r” and “s” you have to add “r” and “s” and then divide it by 2,

that is, r+s/2.

Apply this to the above question, we get 3/2

So, 3/2 is one of the rational numbers between 1 and 2.

You can proceed in this manner to find out four more rational numbers.

These four rational numbers are 5/4, 11/8, 13/8, and 7/4.

Example3: Find the value of 'q' using the associative property of real numbers: (24 + 222) + 654 = 24 + (b + 654)

Solution: According to the associative property, m + (n + r) = (m + n) + r.

246 + 654 = 24 + b + 654

900 = 678 + b

b = 900-678

b = 222

Real Numbers- FAQs

Ans. A combination of both rational numbers and irrational numbers is known as real numbers. Real numbers can be positive and negative, which is represented by ‘R′ natural numbers, fractions, and decimals all come under this category.

Ans. Zero is both real as well as imaginary number.

Ans. There are five subsets real numbers, that is, Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers.

Ans. The properties of real numbers are- Commutative property, Associative property, Distributive property and Identity property.

Ans. Yes, √2 is a real number. We know that a real number is a combination of both rational and irrational numbers. Since √3 is an irrational number, we can also say that √2 is a real number.