## Chapter 1 Number Systems 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 √m , where m is a natural number.**
**(iii) Every real number is an irrational number.**
**Solution:**

(i) True (∵ Real numbers = Rational numbers + Irrational numbers.)

(ii) False (∵ no negative number can be the square root of any natural number.)

(iii) False (∵ rational numbers are also present in the set of real numbers.)

**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:**

No, the square roots of all positive integers are not irrational.

e.g., √l6 = 4

Here, ‘4’ is a rational number.

**Question 3.**
**Show how √5 can be represented on the number line.**
**Solution:**

Now, take O as centre OP = √5 as radius, draw an arc, which intersects the line at point R. .

Hence, the point R represents √5.

**Question 4.** **Classroom activity (constructing the ‘square root spiral’).** **Solution:**

Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP_{1}, of unit lengths Draw a line segment P_{1}, P_{2} perpendicular to OP_{1} of unit length (see figure).

Now, draw a line segment P_{2}P_{3} perpendicular to OP_{2}. Then draw a line segment P_{3}P_{4} perpendicular to OP_{3}. Continuing in this manner, you can get the line segment P_{n-1} P_{n} by drawing a line segment of unit length perpendicular to

OP_{n-1}. In this manner, you will have created the points P_{2}, P_{3},…… P_{n},….. and

joined them to create a beautiful spiral depicting √2,√3,√4,……