Exercise 1.5

Question 1:

1Classify the following numbers as rational or irrational:

(i)  (ii)  (iii) 

(iv)  (v) 2π

Ans.

(i) = 2 − 2.2360679…

= − 0.2360679…

As the decimal expansion of this expression is non-terminating non-recurring, therefore, it is an irrational number.

(ii) 

As it can be represented in form, therefore, it is a rational number.

(iii) 

As it can be represented in form, therefore, it is a rational number.

(iv) 

As the decimal expansion of this expression is non-terminating non-recurring, therefore, it is an irrational number.

(v) 2π = 2(3.1415 …)

= 6.2830 …

As the decimal expansion of this expression is non-terminating non-recurring, therefore, it is an irrational number.

Question 2:

Simplify each of the following expressions:

(i)  (ii) 

(iii)  (iv) 

Ans.

(i) 

(ii)  

= 9 − 3 = 6

(iii) 

(iv) 

= 5 − 2 = 3

Question 3:

Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, . This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

Ans.There is no contradiction. When we measure a length with scale or any other instrument, we only obtain an approximate rational value. We never obtain an exact value. For this reason, we may not realise that either c or d is irrational. Therefore, the fraction  is irrational. Hence, π is irrational.

Question 4:

Represent  on the number line

Ans.

Mark a line segment OB = 9.3 on number line. Further, take BC of 1 unit. Find the mid-point D of OC and draw a semi-circle on OC while taking D as its centre. Draw a perpendicular to line OC passing through point B. Let it intersect the semi-circle at E. Taking B as centre and BE as radius, draw an arc intersecting number line at F. BF is.