Irrational Numbers – Concept & Its Uses

Key Concepts

• Fraction

• Rational number

• Decimal number

• Repeating decimal

• Non repeating/terminating decimal

•Non repeating/Non terminating decimal

• Irrational number

• Square roots

• Perfect square

Irrational Numbers: 

An irrational number is a type of real number that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio. When irrational numbers are expressed in the decimal form, they go on forever, even after the decimal point without repeating numbers. Thus, they are also known as non-terminating, non-repeating numbers. 

If N is irrational, then N is not equal to p/q where p and q are integers and q is not equal to 0. 

Let us see an example: 

If we find the value of , √5

the answer will be 2.23606797749979 

If we observe the numbers after decimal point, the numbers are non-terminating, non-repeating. 

Can we convert this decimal value into a fraction? 

It is not possible. 

Hence, we can say 2.23606797749979 is an irrational number. 

Irrational numbers in daily life: 

Do you remember the formula to calculate area of a circle? 

Area= πr² 

The most common irrational number is: 

Pi (π) = 22/7= 3.14159265358979… 

1.2.2 Identifying the Square roots as Irrational Numbers 

Perfect squares are numbers which are obtained by squaring a whole number. 

If you look at the above picture, you can observe that we can form a square with 4 marbles but not with 6 marbles. 

Here, 4 = 2 × 2 = 2² 

If n is a natural number, then √n is either a natural number or an irrational number. 

For any whole number b that is not a perfect square, √b is irrational. 

Let us look at some examples: 

The number 4 is a perfect square.  So √4 is 2 which is a rational number. 

The number 5 is not a perfect square.  So √5 is 2.23606797749979 which is an irrational number. 

1.2.3 Classification of Numbers into Rational and Irrational Numbers 

Any integer number that can be expressed in the form of x/y where both the numerator and the denominator are integers is a rational number. 

Any integer number that cannot be expressed in the form of x/y where both the numerator and the denominator are integers is an irrational number. In simple words, if the decimal form of a number does not stop and does not repeat, the number is irrational. 

Some of the examples of rational numbers. 

• Number 4 can be written in the form of 4/1 where 4 and 1 both are integers. 
• 0.25 can also be written as 1/4, or 25/100 and all terminating decimals are rational numbers. 

• √64 is a rational number, as it can be simplified further to 8, which is also the quotient of 8/1. 
• 0.888888 is a rational number because it is recurring in nature. 

Some of the examples of irrational numbers. 

• 3/0 is an irrational number, with the denominator equals to zero. 
• π is an irrational number that has value 3.142, and it is non-recurring and non-terminating in nature. 
• √3 is an irrational number, as it cannot be able to simplify further. 
• 0.21211211 is an irrational number as it is non-recurring and non-terminating in nature. 

Exercise:

1. Identify each of the following as rational or irrational:
   1. 0.58
   2. 0.475
   3. 3.605551275…
2. Identify each of the following as rational or irrational:
   1. √36
   2. √44
   3. √81
   4. √17
3. An elementary school has a square playground with an area of 3000 square feet. What is the width of the playground? Is the width a rational or irrational number?
4. Which of the following numbers are rational and which are irrational?
   1.  √24
   2. √25
   3. √36
   4. √37
5. The area of a square is 50 square feet. What are the lengths of its sides?
6. Find √125
7. Find √8
8. Find √27
9. Solve the following:
   11√8  + 15√21
10. Prove that  is 2√3/5 irrational number

What we have learnt:

• 1.2.1 About fractions, rational numbers, decimal numbers, irrational numbers, application of irrational numbers in everyday life and how to identify irrational numbers.

• 1.2.2 How to identify square roots as irrational numbers

• 1.2.3 How to classify a number as rational or irrational.

Concept Map:

Frequently Asked Questions (FAQ’s): 

1. Every real number is an irrational number?  
   All numbers are real numbers and all real numbers that are non-terminating are irrational numbers. 2,3,4 etc. Are examples of real numbers that are not irrational numbers. 

2. Why integers are not irrational numbers?  
   Integers that are positive, negative and zero are not irrational because they can be represented in the form of p/q (where Q ≠0) 

3. What are some commonly used irrational numbers? 
   √2, √3, π(pi), e (Euler’s Number) are some common irrational numbers. 

4.  How can you identify an irrational number?  
   Every number that is not a rational number is an irrational number. Irrational number can be written in the form of decimals but not in the form of fractions.  

5. What is the difference between a rational number and an irrational number? 
   Any integer number that can be expressed in the form of x/y where both the numerator and the denominator are integers is a rational number.  
   Any integer number that cannot be expressed in the form of x/y where both the numerator and the denominator are integers is an irrational number.