Class 9 - Operations on Real Numbers: Rules, Examples and Solved Problems

Performing operations on real numbers involves adding, subtracting, multiplying and dividing rational and irrational numbers. The real numbers set includes both rational and irrational numbers. A rational number is a number in the form p/q where p is the numerator and q is the denominator. An irrational number is a number that cannot be written in the form p/q, (for example: 2=1.4142…). These operations are either performed on two rational or two irrational numbers or between rational and irrational numbers. Understanding the basic rules of operations on real numbers will enable you to perform these operations easily.

In this topic, we will cover the various types of operations performed on rational numbers and irrational numbers along with the rules of performing operations on them with examples and solved problems.

Table of Contents

• Operations on Two Rational Numbers
• Operations on Two Irrational Numbers
• Operations on Rational and Irrational Numbers
• Solved Examples on Operation on Real Numbers

Operations on Two Rational Numbers

We can perform all four arithmetic operations (addition, subtraction, multiplication and division) between two rational numbers. When we add, subtract, multiply or divide (except by zero) two rational numbers, we get a rational number (that is, rational numbers are ‘closed’ with respect to addition, subtraction, multiplication and division). Rational numbers satisfy the commutative, associative and distributive laws for addition and multiplication. For example, result of addition of rational numbers is a rational number: 43+23=63=2

Operations on Two Irrational Numbers

We can perform all arithmetic operations on two irrational numbers. However, the sum, difference, quotients and products of irrational numbers are not always irrational. Irrational numbers also satisfy the commutative, associative and distributive laws for addition and multiplication. For Example, in the following operations the result of all four operations on irrational numbers is a rational number:

• Addition of (6)+(−6)=0   The value of 6≈2.449, so: 2.449+(−2.449)=0
• Subtraction of (2)−(2)=0
• Multiplication of (3)⋅(3)=3
• Division of (17)(17)=1
  
  

Read more: Real numbers

Operations on Rational and Irrational Numbers

Addition of Rational and Irrational Numbers

The sum of a rational number and an irrational number is irrational.

2+3=2+3

Rational numbers + Irrational Number = Irrational Number

The difference between a rational number and an irrational number is irrational. Examples: Rational − Irrational → Irrational

3−5=3−5

Multiplication of Rational and Irrational Numbers

The product of a non-zero rational number with an irrational number is irrational.

Examples: Rational × Irrational → Irrational (if rational ≠ 0)

4×2=42

Division of Rational and Irrational Numbers

The quotient of a non-zero rational number with an irrational number is irrational.

Keypoints:

• (i) The sum or difference of a rational number and an irrational number is irrational.
• (ii) The product or quotient of a non-zero rational number with an irrational number is irrational.
• (iii) If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.
  
  

Read more:

• Class 9 - Exponent Rules for Real Numbers
• Class 9 - Decimal Expansion of Real Numbers
  
  

Solved Problems on Operation on Real Numbers

Example 1: Add 2+53 and 2−33

Solution:

(22+53)+(2−33)

=(22+2)+(53−33)

=(2+1)2+(5−3)3=32+23

Example 2: Divide the following: 815÷23

Solution:

815÷23

=83×523

=45

Example 3: Multiply 65 by 25

Solution:

65×25=6×2×5×5=12×5=60