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.
Solution.
(i) Every real number is either rational or irrational, so every irrational number is a real number. True.
(ii) Real numbers can be represented on the number line. True.
(iii) Rational and irrational numbers together make up real numbers. False.Ans.
Question 2. Are the square roots of all positive integers irrational? If not, give an example of a square root that is rational.
Solution. No — for example, √9 = 3, which is rational. So the statement is not correct.Ans.
Question 3. Show how √5 can be represented on the number line.
Solution. Draw a number line and mark point O (0). Let point A represent 2. Construct a right-angled triangle OAB, right-angled at A, with OA = 2 units and AB = 1 unit.
By the Pythagoras theorem: OB = √(OA² + AB²) = √(4 + 1) = √5
Draw an arc with centre O and radius OB, cutting the number line at C. Then OC = OB = √5, so point C represents √5.Ans.
Question 4. Classroom activity (Constructing the ‘square root spiral’).
Solution. Classroom activity – Do as directed.