Chapter 1 Number Systems Ex 1.2
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 (∵ 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.)
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.
No, the square roots of all positive integers are not irrational.
e.g., √l6 = 4
Here, ‘4’ is a rational number.
Show how √5 can be represented on the number line.
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.
Classroom activity (constructing the ‘square root spiral’).
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 OP1, of unit lengths Draw a line segment P1, P2 perpendicular to OP1 of unit length (see figure).
Now, draw a line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3. Continuing in this manner, you can get the line segment Pn-1 Pn by drawing a line segment of unit length perpendicular to
OPn-1. In this manner, you will have created the points P2, P3,…… Pn,….. and
joined them to create a beautiful spiral depicting √2,√3,√4,……