Integers and Rational Numbers: Class 7 Study Guide
Rational numbers extend integers to include fractions and make division always possible (except by zero).
Integers are …, −3, −2, −1, 0, 1, 2, 3, … A rational number is any number that can be written as p/q, where p and q are integers and q ≠ 0. So 3/4, −5/2, 7 (= 7/1) and 0.25 (= 1/4) are all rational numbers.
A rational number is in standard form when its denominator is positive and p and q have no common factor except 1. For example, −12/18 in standard form is −2/3.
Rules for integers
| Operation | Rule | Example |
|---|---|---|
| (+) × (+) | positive | 3 × 4 = 12 |
| (−) × (−) | positive | (−3) × (−4) = 12 |
| (+) × (−) | negative | 3 × (−4) = −12 |
| (−) ÷ (+) | negative | (−12) ÷ 4 = −3 |
| Subtracting | add the opposite | 5 − (−2) = 7 |
Operations on rational numbers
To add or subtract, use a common denominator: −1/2 + 3/4 = −2/4 + 3/4 = 1/4. To multiply, multiply numerators and denominators: (−2/3) × (3/5) = −6/15 = −2/5. To divide, multiply by the reciprocal: (3/4) ÷ (−1/2) = (3/4) × (−2) = −3/2. Between any two rational numbers there are infinitely many more rational numbers.
Positive denominator.
Common denominator, or multiply.
Mind the signs.
Divide by the HCF.
Comparing and representing
To compare −3/4 and −2/3, convert to a common denominator: −9/12 and −8/12. Since −8 > −9, −2/3 > −3/4. On a number line, positive rationals are to the right of zero and negatives to the left, with larger values further right.
Worked examples and common mistakes
Example: The temperature falls from 5 °C to −7 °C: change = −7 − 5 = −12 °C. Common mistakes: forgetting that a negative times a negative is positive, comparing negative fractions as if they were positive and leaving a negative denominator. Check with decimals when unsure.
Check your understanding: model answers
1. −30. 2. −4/10 + 5/10 = 1/10. 3. −3/5 = −9/15, −2/3 = −10/15, so −3/5 is greater.
✨ Key Takeaways
- Rational numbers are p/q with q ≠ 0.
- Like signs multiply to positive; unlike signs to negative.
- Divide by multiplying by the reciprocal.
- Compare using a common denominator.
💬 Questions for Discussion & Reflection
- Find (−6) × 5.
- Find −2/5 + 1/2.
- Which is greater: −3/5 or −2/3?
Print graph paper for number lines →