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MathematicsClass 8Rational Numbers

Rational Numbers | Class 8 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 3 min read

Rational Numbers – this guide gives you a concise, exam-ready overview of Rational Numbers from Class 8 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

Properties of Rational Numbers

Rational numbers follow several important properties that help in performing arithmetic operations and understanding their behavior. These properties include closure, commutativity, associativity, distributivity, existence of identity elements, and existence of inverse elements. Closure property states that the sum, difference, or product of any two rational numbers is also a rational number. For example, adding 1/2 and 3/4 results in 5/4, which is rational. Commutativity means that changing the order of numbers in addition or multiplication does not change the result; for example, 1/3 + 2/5 = 2/5 + 1/3. Associativity means that when adding or multiplying three or more rational numbers, the way in which numbers are grouped does not affect the result; for example, (1/2 + 1/3) + 1/4 = 1/2 + (1/3 + 1/4). The distributive property connects multiplication and addition: a × (b + c) = a × b + a × c. Identity elements are special numbers that do not change other numbers when used in addition or multiplication. For addition, the identity is 0 because a + 0 = a; for multiplication, the identity is 1 because a × 1 = a. Inverse elements are numbers that when added or multiplied with a given number result in the identity element. The additive inverse of a rational number a/b is -a/b, because a/b + (-a/b) = 0. The multiplicative inverse of a rational number a/b (where a ≠ 0) is b/a, because (a/b) × (b/a) = 1. Understanding these properties is essential for simplifying expressions and solving equations involving rational numbers.

📊 Diagram: The textbook includes diagrams showing number lines and arithmetic operations illustrating the properties, such as two rational numbers being added in different orders to show commutativity, and grouping of numbers to show associativity.

🧪 Activity: An activity where students verify the properties of rational numbers by performing addition and multiplication with given rational numbers and checking if the properties hold.

🔗 Connection: Understanding these properties prepares students for performing arithmetic operations on rational numbers, which is covered in the next section.

Frequently asked questions

1. Name the property under multiplication used in each of the following. (i) \(\frac{-4}{5} \times 1 = 1 \times \frac{-4}{5} = -\frac{4}{5}\) (ii) \(-\frac{13}{17} \times \frac{-2}{7} = \frac{-2}{7} \times \frac{-13}{17}\) (iii) \(\frac{-19}{29} \times \frac{29}{-19} = 1\)

Solution: (i) The property used here is the Multiplicative Identity Property, which states that any number multiplied by 1 remains unchanged.

(ii) The property used here is the Commutative Property of Multiplication, which states that changing the order of factors does not change the product.

(iii) The property used here is the Multiplicative Inverse Property, which states that a number multiplied by its reciprocal equals 1.

2. Tell what property allows you to compute \(\frac{1}{3} \times \left(6 \times \frac{4}{3}\right)\) as \(\left(\frac{1}{3} \times 6\right) \times \frac{4}{3}\) .

Solution: The property used here is the Associative Property of Multiplication. It states that when three or more numbers are multiplied, the product is the same regardless of how the numbers are grouped.

So, \(\frac{1}{3} \times \left(6 \times \frac{4}{3}\right) = \left(\frac{1}{3} \times 6\right) \times \frac{4}{3}\).

3. The product of two rational numbers is always a

Solution: The product of two rational numbers is always a rational number.

Explanation: Rational numbers are closed under multiplication, meaning the product of any two rational numbers is also a rational number.

Which of the following numbers is a rational number?

\frac{3}{4}

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