Chapter 3 - Rational Numbers Revision Notes - Class 7 Mathematics

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📌 Key Points

Rational numbers are of form p/q where p, q are integers and q ≠ 0.
Every integer is a rational number (e.g., 5 = 5/1).
Standard form: denominator must be positive and GCD(p,q) = 1.
To reduce fraction: divide numerator and denominator by their GCD.
Comparing rationals: use same denominator (LCM) or cross-multiplication.
For positive rationals: larger numerator (same denominator) = larger number.
For negative rationals: closer to zero = greater (e.g., -1/3 > -1/2).
Addition/subtraction: convert to same denominator, then operate on numerators.
Multiplication: multiply numerators and denominators separately, then simplify.
Division: multiply by reciprocal (flip the divisor).
Additive inverse of a/b is -a/b (sum equals zero).
Multiplicative inverse (reciprocal) of a/b is b/a (product equals 1).
Closure: sum, difference, product of rationals is always rational.
Commutative for addition and multiplication but NOT for subtraction/division.
Additive identity is 0; multiplicative identity is 1.

📘 Important Definitions

Rational Number

A number that can be expressed as p/q where p and q are integers and q ≠ 0.

Standard Form

Form where denominator is positive and GCD of numerator and denominator is 1.

Additive Inverse

The opposite of a rational number. For a/b, it is -a/b. Sum equals 0.

Multiplicative Inverse (Reciprocal)

For rational a/b (a ≠ 0), the multiplicative inverse is b/a. Product equals 1.

Equivalent Rationals

Two rationals representing the same value but with different numerators and denominators.

GCD (Greatest Common Divisor)

The largest positive integer that divides both numerator and denominator.

LCM (Least Common Multiple)

The smallest positive integer that is a multiple of both denominators.

Closure Property

A set is closed under operation if result of operation on two elements is in the set.

🔢 Formulas & Laws

Addition of Rationals

a/b + c/d = (ad + bc)/(bd) or use LCM method

Convert to same denominator first

Subtraction of Rationals

a/b - c/d = (ad - bc)/(bd)

Same as addition but subtract numerators

Multiplication of Rationals

(a/b) × (c/d) = (a×c)/(b×d)

Simplify before multiplying if possible

Division of Rationals

(a/b) ÷ (c/d) = (a/b) × (d/c)

Multiply by reciprocal of divisor

Comparison using Cross-Multiplication

For a/b and c/d: if a×d > b×c, then a/b > c/d

Quick method for comparing two rationals

⚠️ Common Mistakes

✗ Wrong: Writing 3/-4 in standard form as is

✓ Correct: Convert to -3/4. Negative must be in numerator, denominator must be positive.

✗ Wrong: Adding fractions by adding numerators and denominators: 1/2 + 1/3 = 2/5

✓ Correct: Convert to common denominator: 1/2 + 1/3 = 3/6 + 2/6 = 5/6

✗ Wrong: Dividing by flipping the numerator: (1/2) ÷ (1/3) = (1/2) × (1/3)

✓ Correct: Multiply by reciprocal: (1/2) ÷ (1/3) = (1/2) × (3/1) = 3/2

✗ Wrong: Thinking -1/3 < -1/4 because 3 > 4

✓ Correct: -1/3 ≈ -0.33, -1/4 = -0.25. So -1/4 > -1/3 (closer to zero)

✗ Wrong: Not simplifying final answer after operations

✓ Correct: Always reduce to standard form (lowest terms) at the end.

📝 Exam Focus

These questions are frequently asked in CBSE exams:

Reducing to standard form and identifying equivalent rationals

1m★★★

Comparing and ordering rational numbers

2m★★★

Operations on rationals (addition, subtraction, multiplication, division)

2m★★★

Properties of rational numbers (commutative, associative, distributive, closure)

2m★★

Word problems involving rationals (sharing, recipes, speed/distance)

3m★★

Verification of properties and identities

🎯 Last-Minute Recall