CAT Irrational Number: Know Tips and Tricks to Solve Irrational Numbers Questions

By BYJU'S Exam Prep

Updated on: September 13th, 2023

In the CAT exam, questions on Irrational numbers are often asked. Aspirants preparing for the CAT 2022 exam should know the concept of irrational numbers. Below mentioned are experts’ tips and tricks to solve CAT questions on irrational numbers. Understand the basic concept of irrational numbers, their properties, along with sample questions.

Before preparing for the irrational numbers for the CAT exam, the candidate must know the definition of irrational numbers. So, what is an irrational number?  Irrational numbers are real numbers that cannot be written as a simple fraction. It means these numbers cannot be expressed in a ratio form, like p/q, in which both p and q are the integers, and q is not equal to 0.


The candidates will find that the calculation of these numbers is a little complicated therefore, to ace CAT questions on irrational numbers, aspirants are required to practice consistently. For example, √5, √11, √13, √17 are irrational numbers. If these numbers are used in any mathematical calculations, the candidates must find the values under root. It should be kept in mind that sometimes the CAT irrational number values will be recurring also.

What are Irrational Numbers?

It is important to know that irrational numbers are always expressed in the form of fractions, while rational numbers can never be expressed in the form of fractions. Any numbers such as √2 and √5 etc., are irrational numbers and always expressed in the form of P/Q, in which the P and Q are two integers. Another important point is that Q should always not be equal to 0; otherwise, the number will be irrational.

The most important irrational number is Pi, denoted as the symbol of (π). It values 22/7 that is approximately 3.14.

Important Properties Of Irrational Numbers

The irrational numbers will always obey the following properties in the real number system.

  • The sum of a rational and an irrational number will always be a rational number.
  • If an irrational number is multiplied by a non-zero rational number, the result will always be the irrational number.
  • If two irrational numbers are added or multiplied ever, the result will always be the rational number.

How to solve CAT Questions On Irrational Numbers?

Below mentioned are two kinds of CAT questions from the Irrational numbers section. 

CAT Question Type 1: Irrational numbers on the number line

The real number is either rational or irrational. So, you can see that each real number can be represented uniquely on the number line. If you consider an irrational number in the form of √n, in which the n is a positive integer on the number line, then the following steps will be followed:

  • Step 1: Write the provided number as the sum of the squares of two natural integers (without the root).
  • Step 2: Find the distance equal to these two natural integers on the number line.
  • Step 3: To calculate the distance, use Pythagoras’ theorem.

CAT Question Type 2:Find the irrational number between the rational numbers.

Suppose you have to find the irrational number between 2 and 3.

  • Then find the square root of 4, that is 2; √4=2
  • And the square root of 9, that is 3; √9=3

Then the irrational number will be √5, √6, √7, and √8. Because these are not the perfect squares, so can not be calculated further

CAT Irrational Numbers Questions

Following are some questions with solutions related to irrational numbers, and the candidate might find it beneficial for their CAT preparation.

Question 1: Find the rational number between √3 and √5.

  1. ½ (√3-√5)
  2. ½ (√3+√5)
  3. 2.1
  4. 3.1

Answer: 2.1

Solution: √3=1.73, √5=2.23. So, the number 2.1 lies between these two numbers.

Question 2: Find the value of (√5+√7)^2

  1. 12
  2. √35
  3. √5+√7
  4. 12+2√35

Answer: 12+2√35

Solution:  (√5+√7)^2 =( (√5)^2 +(√7)^2+2( (√5)(√7) =5+7+2√35 =12+2√35


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