Ex - 1.2
Question 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
, where m is a natural number.
(iii) Every real number is an irrational number.
Solution
(i) True, since real numbers consists of rational and irrational numbers.
(ii) False, Since negative integers cannot be expressed as the square root of any natural number.
(iii) False, real number includes both rational and irrational numbers. So every real number can not be an irrational number.
Concept Insight: Mentioning the reasons is important in this problem. Real Numbers consists of rational and irrational numbers and not vice versa. Every real number corresponds to a point on number line and vice versa.
Recall real number includes negative numbers also. Square root of negative numbers is not defined.
Question 2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Solution
Thus the square roots of all positive integers are not irrational
Concept Insight: In general only the square root of a prime number is irrational.
Question 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
, where m is a natural number.(iii) Every real number is an irrational number.
Solution
(i) True, since real numbers consists of rational and irrational numbers.
(ii) False, Since negative integers cannot be expressed as the square root of any natural number.
(iii) False, real number includes both rational and irrational numbers. So every real number can not be an irrational number.
Concept Insight: Mentioning the reasons is important in this problem. Real Numbers consists of rational and irrational numbers and not vice versa. Every real number corresponds to a point on number line and vice versa.
Recall real number includes negative numbers also. Square root of negative numbers is not defined.
Question 2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Solution
Square roots of all square numbers are rational.
For example
For example
Thus the square roots of all positive integers are not irrational
Concept Insight: In general only the square root of a prime number is irrational.
Therefore square root of perfect square numbers are rational.
Question 3. Show how 
can be represented on the number line.
can be represented on the number line.
Solution
Using Pythagoras Theorem: 5=22+12
Taking positive square root we get
Taking positive square root we get
1. Mark a point 'A' representing 2 units on number line.
2. Now construct AB of unit length perpendicular to OA. Join OB
3. Now taking O as centre and OB as radius draw an arc, intersecting number line at point C.
4. Point C represents on number line. [length (OB) = length (OC)]
2. Now construct AB of unit length perpendicular to OA. Join OB
3. Now taking O as centre and OB as radius draw an arc, intersecting number line at point C.
4. Point C represents
on number line. [length (OB) = length (OC)]
Concept Insight: For a positive integer n, 
can be located on number line , if 
is located using Pythagoras Theorem . If is a perfect square then this method is useful.
can be located on number line , if is located using Pythagoras Theorem . If is a perfect square then this method is useful.
To represent the irrational number key idea is to use Pythagoras theorem and create a length of units by constructing a right triangle of base and perpendicular of length 2 and 1 units.
key idea is to use Pythagoras theorem and create a length of units by constructing a right triangle of base and perpendicular of length 2 and 1 units. 
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