12. Find two rational numbers between  and .

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  where m is a natural number.
(iii)  Every real number is an irrational number.

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 length. Draw a line segment P₁ P₂ perpendicular to OP₁ of unit length (see in given Fig.). Now draw a line segment P₂ P₃ perpendicular to OP₂ . Then draw a line segment P₃ P₄ perpendicular to OP₃ . Continuing in this manner, you can get the line segment P_n-1P_n by drawing a line segment of unit length perpendicular to OP_n-1. In this manner, you will have created the points P₂ , P₃ ,...., P_n ,... ., and joined them to create a beautiful spiral depicting 2, 3, 4, ...



Constructing square root spiral