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Rational numbers look easy, because they are "just fractions", until a test asks whether subtraction is associative or how to place −7/3 on a number line. This guide covers the definition, the properties that matter, three worked examples and the mistakes that cost the most marks.
Where are rational numbers in the 2026-27 NCERT books?
The older Class 8 Mathematics book opens with "Rational Numbers". The new Class 8 books, Ganita Prakash Part I and Part II, do not have a chapter by that name. In the 2026-27 books the formal treatment is in Class 9 Ganita Manjari, Chapter 3, "The World of Numbers", section 3.4, "Filling the Spaces: Fractions and Rational Numbers". If you are in Class 8 now, the skills still matter: every fraction, percentage and equation chapter uses them.
What is a rational number?
Ganita Manjari defines it as any number that can be written as p/q, where p and q are integers and q ≠ 0. Three things follow:
Every integer is rational: 5 = 5/1 and −10 = −10/1.
A rational number has many equal forms: −1/3 = −2/6 = −3/9. We usually use the standard form, where p and q share no common factor other than 1 and q is positive.
The minus sign can sit in front, on top or below: −1/5 = (−1)/5 = 1/(−5).
Why must q not be zero? Because division by zero has no answer. Ganita Manjari asks you to explain this yourself, which is a good test of whether you understand it.
Which properties hold, and which fail?
The older Class 8 chapter checks each property for each operation. For rational numbers a, b and c:
Closure: adding, subtracting or multiplying two rationals always gives a rational. Division does too, except division by zero.
Commutative: a + b = b + a and a × b = b × a. Subtraction and division are not commutative: 1/2 − 1/3 is not 1/3 − 1/2.
Associative: (a + b) + c = a + (b + c), and the same for ×. Not for subtraction or division.
Identities: 0 for addition (a + 0 = a) and 1 for multiplication (a × 1 = a).
Inverses: the additive inverse of a is −a. The reciprocal of a/b is b/a, for any non-zero a/b. Zero has no reciprocal.
Distributive: a × (b + c) = a × b + a × c.
Three worked examples
Example 1: add with unlike denominators. Find −3/4 + 5/6.
The lowest common multiple of 4 and 6 is 12. −3/4 = −9/12 and 5/6 = 10/12. Sum = (−9 + 10)/12 = 1/12.
Example 2: check distributivity. Find −2/5 × (3/4 + 1/6) two ways.
Brackets first: 3/4 + 1/6 = 9/12 + 2/12 = 11/12. Then −2/5 × 11/12 = −22/60 = −11/30. Distributing: −2/5 × 3/4 = −6/20 = −3/10, and −2/5 × 1/6 = −2/30 = −1/15. Then −3/10 + (−1/15) = −9/30 − 2/30 = −11/30.
Both ways give −11/30, as the distributive property says.
Example 3: find rationals between two numbers. Find three rational numbers between 1/3 and 1/2.
Write both with denominator 24: 1/3 = 8/24 and 1/2 = 12/24. Three numbers between them: 9/24, 10/24 and 11/24, which simplify to 3/8, 5/12 and 11/24.
Need more? Use denominator 48 and you get seven. This is why the older chapter says there are "countless" rational numbers between any two rational numbers.
How do you place a rational number on the number line?
Turn it into a mixed number first. −7/3 = −2 1/3. So it lies between −3 and −2. Divide the gap from −2 to −3 into three equal parts, and mark one part to the left of −2.
The usual mistake is to count from −3 instead of from −2, which puts the point at −2 2/3 = −8/3.
What mistakes should you avoid?
Adding tops and bottoms: 1/2 + 1/3 is not 2/5. Make the denominators equal first; the answer is 5/6.
Losing the sign: −(−3/7) = +3/7.
Inverting the wrong number: to divide by 2/3, multiply by 3/2; do not flip the first fraction.
Comparing negatives by size of numerator: −3/4 is less than −2/3, because −9/12 is to the left of −8/12.
Claiming zero has a reciprocal. It does not.
A practice routine
Each day, do one sum for each operation with negative fractions, writing every step.
Pick one property and test it with your own numbers, including one case where it fails.
Mark four rational numbers on one number line, two positive and two negative.
Mix in two questions from an older chapter. In one classroom study of 126 seventh-graders, Rohrer, Dedrick and Stershic (2014) found that mixing problem types in practice gave higher test scores than practising one type at a time, both one day and 30 days later.
When a sign error keeps appearing and you cannot see where, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer. Next, use the same skills in linear equations in Class 8, or look ahead to number systems in Class 9.
Today, find five rational numbers between −1/2 and −1/3 and mark them on a number line.
