Rational Numbers Class 7 Chapter Notes

What are rational numbers?

A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0. The word rational comes from ratio, so every rational number is a ratio of two integers. For example, 3/5, -7/9, 0/4, 8/1 and -12/5 are rational numbers. Every integer is also a rational number because it can be written with denominator 1, such as -6 = -6/1. Every fraction with whole number numerator and non-zero denominator is a rational number, but rational numbers also include negative fractions. The denominator cannot be zero because division by zero is not defined. In exams, always check three things: numerator and denominator are integers, denominator is not zero, and signs are handled correctly.

Key points

• A rational number has the form p/q, where p and q are integers and q is not equal to 0.
• All integers are rational numbers because a = a/1 for every integer a.
• 0 is a rational number because 0 can be written as 0/1, 0/2, 0/5 etc.
• The denominator of a rational number can never be 0; 5/0 is not a rational number.
• A rational number is positive when numerator and denominator have the same sign.
• A rational number is negative when numerator and denominator have opposite signs.
• Standard form means numerator and denominator have no common factor except 1, and the denominator is positive.
• Equivalent rational numbers are made by multiplying or dividing numerator and denominator by the same non-zero integer.
• On a number line, negative rational numbers lie to the left of 0 and positive rational numbers lie to the right of 0.
• For addition or subtraction, first make denominators equal using LCM, then add or subtract numerators.

Important definitions

Rational number

A number that can be expressed as p/q, where p and q are integers and q is not equal to 0.

Numerator

The integer written above the fraction bar in p/q; it shows the number of parts taken.

Denominator

The integer written below the fraction bar in p/q; it shows the number of equal parts and must not be 0.

Positive rational number

A rational number whose numerator and denominator have the same sign, such as 3/7 or -4/-9.

Negative rational number

A rational number whose numerator and denominator have opposite signs, such as -5/8 or 6/-11.

Standard form

A rational number is in standard form when numerator and denominator are co-prime and denominator is positive.

Equivalent rational numbers

Rational numbers that have the same value, such as 2/3, 4/6 and -6/-9.

Reciprocal

The reciprocal of a non-zero rational number p/q is q/p; zero has no reciprocal.

Signs and standard form rules

How to write a rational number in standard form

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Quick comparison: fractions, integers and rational numbers

Representing rational numbers on the number line

To place a rational number on a number line, first decide whether it is positive or negative. Positive numbers are placed to the right of zero, while negative numbers are placed to the left of zero. Then divide the required unit interval into equal parts according to the denominator. For example, to show 3/4, divide the interval from 0 to 1 into 4 equal parts and mark the third part from 0. To show -3/4, divide the interval from 0 to -1 into 4 equal parts and mark the third part to the left of 0. For improper rational numbers like 7/3, first notice that 7/3 = 2 1/3, so the point lies between 2 and 3. Number line questions test your understanding of order, equal divisions, and sign direction.

Operations on rational numbers: exam method

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Worked mini-examples

• {"title":"Example 1: Put -18/24 in standard form","bodyMarkdown":"HCF of 18 and 24 is 6. Divide numerator and denominator by 6: -18/24 = -3/4. Denominator is positive and 3 and 4 are co-prime, so standard form is -3/4."}
• {"title":"Example 2: Add -2/5 and 3/10","bodyMarkdown":"LCM of 5 and 10 is 10. Convert -2/5 = -4/10. Now -4/10 + 3/10 = -1/10. Answer: -1/10."}
• {"title":"Example 3: Divide -7/9 by 14/15","bodyMarkdown":"Change division to multiplication by reciprocal: -7/9 ÷ 14/15 = -7/9 × 15/14. Cancel 7 with 14 and 15 with 9: result = -5/6. Answer: -5/6."}

Must remember formulas and rules

• Equivalent form: p/q = (p × m)/(q × m), where m is any non-zero integer.
• Equivalent form by division: p/q = (p ÷ n)/(q ÷ n), where n is a common non-zero factor of p and q.
• Addition: a/b + c/d = (ad + bc)/bd, but using LCM is usually easier in Class 7.
• Subtraction: a/b - c/d = (ad - bc)/bd.
• Multiplication: a/b × c/d = ac/bd.
• Division: a/b ÷ c/d = a/b × d/c, where c/d is not zero.
• Additive inverse of a/b is -a/b because a/b + (-a/b) = 0.
• Reciprocal of a/b is b/a, only when a is not 0.
• Between any two rational numbers, there are infinitely many rational numbers.

Comparing rational numbers

Exam tips and common traps

Do not cancel terms across addition or subtraction; cancellation is safe mainly in multiplication after factors are visible. Always make the denominator positive in the final answer. In division, many students forget to take the reciprocal of the second rational number only. While comparing negative rational numbers, remember that -1/3 is greater than -2/3 because it lies closer to zero. For 2-mark questions, show at least one working step such as LCM conversion, HCF reduction, or reciprocal step; only the final answer may not get full marks if the method is required.

Quick revision checks

• Q: Is -8 a rational number? A: Yes. -8 = -8/1, so it is of the form p/q with q not equal to 0.
• Q: Write 36/-48 in standard form. A: 36/-48 = -36/48 = -3/4 after dividing by HCF 12.
• Q: Which is greater: -3/5 or -1/5? A: -1/5 is greater because it is closer to 0 on the number line.
• Q: Find 4/7 × -21/8. A: Cancel 4 with 8 and 21 with 7. Result = -3/2.

Frequently Asked Questions