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)True.

All rational and irrational numbers together make up the collection of real numbers R.

(ii)False

Because there infinitely many numbers between √4 and √5

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:

False. Square roots of all positive integers are not irrational numbers.  For example 4, 9, 16, etc. are is a positive integers and their square roots are 2, 9 and 4 are rational numbers.

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

Solution:

To represent √5 on the number line

• Take a length of two units from 0 on the number in positive direction and one unit perpendicular to it. The hypotenuse of the triangle formed is of the length √5.
• With the help of the divider draw an arc which cuts the number line at √5 .
• Join the points to form the triangle.
• The length equal to the hypotenuse of √5 units can be cut on the number line.