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 √m, where m is a natural number.
(iii) Every real number is an irrational number.
Reason: Real numbers are any number which can we think about. Thus, every irrational number is a real number.
Reason: A number line may have negative or positive number. Since, no negative can be the square root of a natural number, thus every point on the number line cannot be in the form of √m, where m is a natural number.
Reason: All numbers are real number and non terminating numbers are irrational number. For example 2, 3, 4, etc. are some example of real numbers and these are not irrational.
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.
Answer: No. Square roots of all positive integers are not irrational. Example 4, 9, 16, etc. are positive integers and their square roots are 2, 9 and 4 which are rational numbers.
Question 3: Show how √5 can be represented on the number line.
Steps to show √5 on a number line.
Step: 1 – Draw a number line mm’
Step: 2 – Take OA equal to one inch, i.e. one unit.
Step: 3 – Draw a perpendicular AB equal to one inch (1 inch) on point A.
Step: 4 – Join OB. This OB will be equal to √2
Step: 5 – Draw a line BC perpendicular to OB on point B equal to OA i.e. one inch.
Step: 6 – Join OC. This OC will be equal to √3
Step: 7 – Draw a line CD equal to OA and perpendicular to OC.
Step: 8 – Join OD. This will be √4 i.e. equal to 2.
Step: 9 – Draw ED equal to 1 inch and perpendicular to OD.
Step: 10 – Join OE. This will be equal to √5
Step: 11 – Cut a line segment OF equal to OE on number line. This line segment OF will be equal to √5
See how, OE is equal to √5
Here, OD = 2, DE = 1 and angle ODE = 90º
Thus, according to Pythagoras theorem.