In your study of numbers, you started by counting objects around you. The numbers you use for counting are called counting numbers or natural numbers. They are 1, 2, 3, 4, and so on. When we include 0 with natural numbers, we get the whole numbers, which are 0, 1, 2, 3, and so on.
Then we combined the negatives of natural numbers with whole numbers to create integers. Integers are numbers like 3, 2, 1, 0, 1, 2, 3, and so on. So, we expanded the number system from natural numbers to whole numbers and from whole numbers to integers.
You also learned about fractions, which are numbers written in the form of numerator/denominator, where the numerator can be 0 or a positive whole number, and the denominator is a positive whole number.
Showing 1 part of a pizza in the form of a fraction
In this chapter, we will introduce the concept of rational numbers, along with exploring how to add, subtract, multiply, and divide them.
We already learned how integers can be used to represent opposite situations involving numbers. For example, we used positive integers to denote distances to the right of a place and negative integers to represent distances to the left. Similarly, we used positive integers to represent profits and negative integers for losses.
Now, some situations involve fractional numbers, similar to the examples mentioned earlier. For instance, we can represent a distance of 750m above sea level as 3/4 km. But can we represent 750m below sea level in km? Can we denote the distance of 3/4 km below sea level as 3/4? We observe that 3/4 is neither an integer nor a fractional number. Therefore, we need to expand our number system to include such numbers.
Pic depicting above and below sea level
A rational number is a number that can be written in the form p/q, where p and q are integers and q ≠ 0.
To reduce a rational number to its standard form, divide its numerator and denominator by its Highest Common Factor (HCF). To find the standard form of a rational number with a negative integer as the denominator, divide its numerator and denominator by their HCF with a minus sign.
(ii) Integers:
(iii) Rational Numbers:
Note: Rational numbers exclude zero in the denominator for division closure.
Division by zero is undefined in the set of rational numbers.
(ii) Integers:
(iii) Rational Numbers:
(ii) Integers:
(iii) Rational Numbers:
 1+(−1)=(−1)+1=0, so the negative of 1 is 1.
 2+(−2)=(−2)+2=0, making 2 the negative or additive inverse of 2, and vice versa.
 In general, for an integer a, a+(−a)=(−a)+a=0.
=>For rational numbers, similar principles apply.
In general, for a rational number
For example, Zero has no reciprocal because there is no rational number that, when multiplied by 0, results in 1.
Zero has no reciprocal because there is no rational number that, when multiplied by 0, results in 1.
In general, ifis the reciprocal of , then
While comparing positive rational numbers with the same denominator, the number with the greatest numerator is the largest. It is easy to compare these numbers if their denominators are the same.
Eg:
A positive rational number is always greater than a negative rational number. While comparing negative rational numbers with the same denominator, compare their numerators ignoring the minus sign. The number with the greatest numerator is the smallest.
Positive rational numbers lie to the right of 0, while negative rational numbers lie to the left of 0 on the number line.
To compare rational numbers with different denominators, convert them into equivalent rational numbers with the same denominator, which is equal to the LCM of their denominators. You can find infinite rational numbers between any two given rational numbers.
The following points are to be noted while finding the sum of rational numbers:
Thus, the sum is .
To divide one rational number by another, we actually multiply the first number with the reciprocal of the second number.
Example 1: Simplify the rational number 16/(24)
Ans:
Step 1: Identify the common factors of both numerator and denominator. In this case, the common factors are 2, 4, and 8.
Step 2: Divide both numerator and denominator by the greatest common factor, which is 8.
(16) ÷ 8 = 2
(24) ÷ 8 = 3
So, 16/(24) simplifies to 2/3.
Example 2: Add the rational numbers 2/5 and 3/10
Ans:
Step 1: Find the least common denominator (LCD) of both denominators. In this case, the LCD is 10.
Step 2: Convert both fractions to equivalent fractions with the LCD as the new denominator.
2/5 = 4/10 (multiply both numerator and denominator by 2)
3/10 = 3/10 (no change needed)
Step 3: Add the equivalent fractions.
4/10 + 3/10 = (4 + 3) / 10 = 7/10
So, 2/5 + 3/10 = 7/10.
1. Positive Rational Numbers are the numbers whose both the numerator and denominator are positive.
Example: 3/4, 12/24 etc.
2. Negative Rational Numbers are the numbers whose one of the numerator or denominator is negative.
Example: (2)/6, 36/(3) etc.
Remark: The number 0 is neither a positive nor a negative rational number.
Representation of whole numbers, natural numbers and integers on a number line is done as follows
Rational Numbers can also be represented on a number line like integers i.e. positive rational numbers are on the right to 0 and negative rational numbers are on the left of 0.
Representation of rational numbers can be done on a number line as follows:
Between any two rational numbers, there are infinitely many rational numbers.
To find the rational numbers between two rational numbers, we have to make their denominator the same then we can find the rational numbers.
Example: Find three rational numbers between 1/4 and 1/2.
Ans:
(i) First, we find the mean of given numbers.
Mean is
(ii) Again we find another rational number between 1/4 and 3/8 .For this, again we calculate mean of 1/4 and 3/8.
Mean is
(iii) For the third rational number, we again find mean of 3/8 and 1/2.
Mean is
Hence, 5/16, 3/8, and 7/16 are 3 rational numbers between 1/4 and 1/2.
Example: Find rational numbers between 3/5 and 3/7.
Ans: To find the rational numbers between 3/5 and 3/7, we have to make their denominator same.
LCM of 5 and 7 is 35.
Hence the rational numbers between 3/5 and 3/7 are
These are not the only rational numbers between 3/5 and 3/7.
If we find the equivalent rational numbers of both 3/5 and 3/7 then we can find more rational numbers between them.
Hence we can find more rational numbers between 3/5 and 3/7.
Remark: Between any two given rational numbers, we need not necessarily get an integer but there are countless rational numbers between them.
76 videos345 docs39 tests

1. What is the need for understanding Rational Numbers in mathematics? 
2. How are Rational Numbers defined and what sets them apart from other types of numbers? 
3. What are some properties of Rational Numbers that make them unique? 
4. How do you perform operations on Rational Numbers, such as addition, subtraction, multiplication, and division? 
5. Can Rational Numbers be both positive and negative, and how do we differentiate between the two? 

Explore Courses for Class 7 exam
