Introduction

Before this exercise, we are going to learn about Rational Number and Irrational Number. So far, all the numbers you have come across in the previous exercise, are of the form p/q, where p and q are integers and q ≠ 0 ( called Rational Numbers ). So, you may have question: are there numbers which are not of this form? There are indeed such numbers ( and this is the type of Irrational Numbers ).

Some Important Definition

1. Rational Numbers: The numbers that can be expressed in the form of a fraction p/q​, where p and q are integers, and q≠0. The decimal representation of rational numbers is either terminating or non-terminating but repeating. Example: 3/4, -(5/2), 0, 7, Decimal Numbers like: 0.25 (Terminating) or 0.333... (Non-Terminating or Repeating) etc
2. Irrational Numbers: The numbers that cannot be expressed in the form of a fraction p/q​, where p and q are integers and q≠0. These numbers have non-terminating and non-repeating decimal expansions. √2​, √3, √5 (Square roots of non-perfect squares), π, e (Euler's number) etc.

Exercise 1.2

1. State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is of the form m , where m is a natural number.
(iii) Every real number is an irrational number.

Sol: (i) True ( Justification: Irrational numbers are part of the set of real numbers. Real numbers consist of both rational and irrational numbers. For example, √2 and π are irrational numbers, and they exist on the real number line. )

Sol: (ii) False ( Justification: The number line represents all real numbers, including integers (negative and zero), fractions, decimals, and irrational numbers. Natural numbers are a subset of real numbers and start from 1,2,3,… Points like −2,0,1/2,√3 are on the number line but are not natural numbers. )

Sol: (iii) False ( Justification: Real numbers include both rational and irrational numbers. While numbers like π and √2 are irrational, numbers like 2,3/4,0.5 are rational and are also part of real numbers. Thus, not every real number is irrational. )

2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a umber that is a rational number.

Sol: No, the square roots of all positive integers are not irrational. Example: 
The square root of 4 is 2, which is a rational number. Similarly, the square root of 9 is 3, and the square root of 16 is 4, both of which are rational numbers.
( Explanation: The square root of a positive integer is rational if the number is a perfect square (like 1,4,9,16,25,…). However, if the number is not a perfect square (like 2,3,5,6,7,8,10,…), its square root will be irrational. For example, √2 and √3 are irrational. )

3. Show how √5 can be represented on the number line.

Sol: Steps to Construct √5 on the Number Line:

1. Draw a Number Line: Draw a horizontal line, mark a point as 0, and then mark 1,2,3,4,5 to the right of 0.

2. Represent √5 Using the Pythagorean Theorem:

   • Draw a line segment of 2 units starting from 0 to 2.
   • From the point 2, draw a perpendicular line of 1 unit in length.

3. Create a Right Triangle:

   • Connect the endpoint of the perpendicular line back to 0.
   • This forms a right triangle where:
     • One side is 2 units.
     • The other side is 1 unit.

4. Calculate the Hypotenuse: By the Pythagorean theorem:

   Hypotenuse = √(2²+1²) = √(4+1) = 5.

5. Mark √5 on the Number Line:

   • Use a compass to measure the hypotenuse.
   • Place the compass pointer at 0 and mark the arc where the hypotenuse intersects the number line.
   • This point represents 5
     
     Visual Representation:
     
     
     
     

Important Questions Based On Ex. 1.2
( Ex. 1.2 पर आधारित महत्वपूर्ण प्रश्न )

1. Locate √2 on the number line.
2. Locate √3 on the number line.
3. State whether the following statements are true or false. Justify your answers.
   1. Every irrational number is a real number.
   2. Every point on the number line is of the form √a , where m is a natural number.
   3. Every real number is an irrational number.