1.2 Rational Numbers [30m]
"Ration Cards" are issued to families to whom the government wants to provide food grains at subsidised prices. The names, gender, and age of all family members are written in the ration card. Based on this information, the amount of each food grain that the family can buy is calculated. The ration shop owner keeps a record of how much food grains are given to each family each month.
Do you know why these cards are called "Ration Cards" or why these shops are called "Ration Shops"? By the end of this lesson, you will discover the answer to this question.
DEFINITION: RATIONAL NUMBERS
Rational numbers are those numbers which can be written in the form where p and q are integers and q ≠ 0.
EXERCISE: Is zero a rational number? Can you write it in the form , where p and q are integers and q ≠ 0?
ALTERNATE DEFINITION OF RATIONAL NUMBERS
A number whose decimal representation is either terminating or non terminating recurring is called a rational number.
A BEAUTIFUL RESULT: There are INFINITE rational numbers between any two rational numbers.
This makes rational numbers different from integers in two ways:
(i) Every integer has a successor and a predecessor. But, no rational number has a successor or a predecessor.
(ii) Between any two integers there are always a finite number of integers. But, between any two rational numbers there are always infinite rational numbers.
In the next exercise, you will find 2 methods to discover rational numbers that lie between any two rational numbers.
EXERCISE: Find six rational numbers between 3 and 4.
We can also use the two methods discussed in the video above to discover rational numbers between two rational numbers that are expressed in the p/q form.
EXERCISE: Find five rational numbers between and .
SELF PRACTICE: Find five rational numbers between:
1. and .
2. and .
3. and .
EXERCISE: State whether the following statements are true or false. Give reasons for your answers.
1. Every natural number is a whole number.
2. Every integer is a whole number.
3. Every rational number is a whole number.
SELF PRACTICE: State whether the following statements are true or false. Give reasons for your answers.
1. Every natural number is an integer.
2. Every integer is a natural number.
3. Every rational number is an integer.
TAKEAWAYS OF THIS LESSON:
- All the numbers we knew so far (natural numbers, whole numbers, integers, and fractions) are rational numbers.
- All numbers that can be written as a ratio of an integer to a non-zero integer is a rational number.
- Any decimal number that is terminating or recurring is a rational number.
- There are infinite rational numbers between any two rational numbers.