NCERT Solutions for Class 7 Maths Chapter 8 Exercise 8.1

All the solutions and study material are updated according to latest NCERT Books 2024-25. In class 7 math exercise 8.1, we will learn to compare the rational numbers and plotting a rational number on number line. Finding more rational numbers between two given rational number is also important for next classes.

The counting number are called natural numbers. Thus, 1, 2, 3, 4, 5, 6, …, etc., are all natural numbers.

All natural numbers together with 0 (zero) are called whole numbers. Thus, 0, 1, 2, 3, 4, 5, 6, …., etc., are all whole numbers. Clearly, every natural number is a whole number but 0 is a whole number which is not a natural number.

All natural numbers, 0 and negatives of counting numbers are called integers.
Thus, …., – 5, – 4, – 3, -2, – 1, 0, 1, 2, 3, 4, 5, …, etc., are all integers.
1, 2, 3, 4, 5, 6, …., etc., are all positive integers.
– 1, – 2, – 3, – 4, – 5, – 6, …, etc., are all negative integers.
Zero is an integer which is neither positive nor negative.
Clearly, a positive integer is the same as a natural number

The numbers of the form, where a and b are natural numbers, are called fractions.
Thus 2/3, 5/8, 3/11, 9/23, etc., are all fractions.

The numbers of the form p/q, where p and q are integers and q ≠ 0. are called rational numbers.
Examples of rational numbers
1. Each of the number 6/8, 9/12, 6/15, is a rational number.
2. Zero is a rational number, since we can write 0 = 0/1, which is the quotient of two integers with a nonzero denominator.
3. Every natural number is a rational number. We can write.
1 = 1/1, 2 = 2/1, 3 = 3/1, ………
In general, if n is a natural number, then we can write it as, which is a rational number.

4. Every integer is a rational number.
If m is an integer then we can write it as m/1, which is clearly a rational number. Thus, every integer is a rational number.
5. Every integer is a rational number. Let a/b be a fraction. Then, a and b are whole numbers and b ≠ 0. But, every whole number is an integer.
Thus, is the quotient of two integers such that b ≠ 0.
So, a/b is a rational number.
Hence, every fraction is a rational number.

A rational number is said to be positive if its numerator and denominator are either both positive or both negative.
Example: Each of the number 5/7, 13/8, 17/9 etc.

A rational number is said to be negative if its numerator and denominator are such that one of them is a positive integer and the other is a negative integer.
Example: Each of the number -3/5, 5/-7, -9/11, is a negative rational number.

Theorem-1 of Rational Numbers:
If p/q is a rational number and m is a nonzero integer. Then = (p/q) x (m/m).
Thus, a rational number remains unchanged. if its numerator and denominator are multiplied by the same nonzero integer.

Theorem-2 of Rational Numbers:
If p/q is a rational number and m is a common divisor of p and q, then p/q = (p/m)/ (q/m).
Thus, on dividing the numerator and denominator of a rational number by a common divisor, it remains unchanged.

On multiplying the numerator and denominator of a given rational number by the same nonzero number, we get a rational number equivalent to the given rational number.
Similarly, on dividing the numerator and denominator of a given rational number by a common divisor, we get a rational number equivalent to the given rational number.

A rational number p/q
METHOD: In order to express a given rational number in standard form, we first convert it into a rational number whose denominator is positive and then we divide its numerator and denominator by their HCF. is said to be in standard form, if q is positive, and p and q have no common divisor other than 1.

Theorem-3 of Rational Numbers
For any two rational numbers a/b and c/d we have (a x d) = (b x c).