Q.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.
(i) True statement, because all rational numbers and all irrational numbers form the group (collection) of real numbers.
(ii) False statement, because no negative number can be the square root of any natural number.
(iii) False statement, because rational numbers are also a part of real numbers.
Q.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.
Ans. No, if we take a positive integer, say 4 its square root is 2, which is a rational number.
According to the Pythagoras theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In the figure: OB2 =OA2 + AB2
If OB2 =12 + 12 = 2 ⇒ OB = √2
Q.3. Show how √5 can be represented on the number line.
Ans. Let us take the horizontal line XOX" as the x-axis. Mark O as its origin such that it represents 0.
Cut off OA = 1 unit, AB = 1 unit.
∴ OB = 2 units Draw a perpendicular BC ⊥ OX.
Cut off BC = 1 unit.
Since OBC is a right triangle.
∴ OB2 + OC2 = OC2
or 22 + 12 = OC2 or 4 + 1 = OC2
or OC2 = 5
⇒ OC = √5
With O as centre and OC as radius, draw an arc intersecting OX at D.
∴ OD represents √5 on XOX".