1. State whether the following statements are true or false. Justify your answers.
(i)Every irrational number is a real number.
According to the definition:
Irrational Numbers – A number is said to be irrational, if it cannot be written in the p/q, where p and q are integers and q ≠ 0.
i.e., Irrational numbers = π, e, √5, 6+√3, 7.241, 0.0110….
Real numbers – The collection of both rational and irrational numbers are known as real numbers.
i.e., Real numbers = √4, √7, , 0.102…
Hence, every irrational number is real number.
(ii) Every point on the number line is of the form √m , where m is a natural number.
Because negative numbers cannot be the square root of any natural number.
(iii) Every real number is an irrational number.
Because the real numbers include both irrational and rational numbers. Therefore, every real number cannot be an irrational number.
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.
No, the square roots of all positive integers are not irrational.
Let us take
√4 = 2 is rational.
Hence, the square roots of positive integer 4 is not irrational.
3. Show how √5 can be represented on the number line.
Step 1: Draw a line OB be of 2 unit on a number line.
Step 2: Draw a perpendicular line AB of length 1 unit at B.
Step 3: Now, join OA
Step 4: we observe that AOB is a right angled triangle. So, applying Pythagoras theorem,
AB2+OB2 = OA2
12+22 = OA2 = 5
Hence, OA = √5 .
Thus, OA is a line of length √5 unit.
Step 4: With center O, taking OA as a radius, draw an arc touching the number line. The point at which number line get intersected by arc is at √5 distance from 0.
4. 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 P1P2 perpendicular to OP1 of unit length (see Fig. 1.9). Now draw a line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3. Continuing in Fig. 1.9 :
Solution: Do it yourself.