CHAPTER 1: NUMBER SYSTEMS

CHAPTER 1: NUMBER SYSTEMS

Rational number: A number ‘r’ is called a rational number, if it can be written in the form of  where p and q are integers and q ≠ 0.

Example 1: Are the following statements true or false? Give reasons for your answers.
(i)    Every whole number is a natural number.
(ii)    Every integer is a rational number.
(iii)    Every rational number is an integer.

Solution:
(i) False, because zero is a whole number but not a natural number.
(ii) True, because every integer m can be expressed in the form , and so it is a rational number.
(iii) False, because is not an integer.

Exercise 1.1

1.    Is zero a rational number? Can you write it in the form , where p and q are integers and q ≠ 0 ?
Answer: Zero is a rational number.
Zero can be written as  , or , or  etc.

2.    Find six rational numbers between 3 and 4.
Solution:

Now,

3.    Find five rational numbers between  and
Solution:


Now,



4.    State whether the following statements are true or false. Give reasons for your answers.
(i)    Every natural number is a whole number.
Answer: True, because whole number contains zero and all the natural numbers.

(ii)    Every integer is a whole number.
Answer: False, because -1 is an integer but not a whole number.

(iii)    Every rational number is a whole number.
Answer: False, because is a rational number but not a whole number.

Additional questions
Q. Look at several examples of rational numbers in the form of  (q ≠ 0), where p and q are integers with no common factor other than 1 and having terminating decimal  representations. Can you guess what property q must satisfy?
Solution: A rational number  is a terminating decimal only, when prime factorization of q must have only powers of 2 or 5 or both.