Vardaan
Class 7 Maths • Chapter 01

Integers

Vardaan Learning Institute • Detailed Notes with Practice Questions

🔢 1. The Number System — Where Integers Fit

Before diving into integers, let's understand how they fit in the entire number system:

NUMBERS Real Numbers Imaginary (√−1) Rational (p/q) Irrational (√2, π) Integers ← YOU ARE HERE Fractions/Decimals Negative (…−2,−1) Zero (0) Positive (1,2,3…) Key Points: • Integers = { …−3, −2, −1, 0, 1, 2, 3… } • Whole Numbers = { 0, 1, 2, 3… } ⊂ Integers • Natural Numbers = { 1, 2, 3… } ⊂ Integers • Every natural number is an integer • 0 is neither positive nor negative

📏 2. The Number Line

Every integer has a position on the number line. Numbers to the right are greater; numbers to the left are smaller.

−5 −4 −3 −2 −1 0 1 2 ← Smaller Larger →

➕ 3. Operations on Integers

A. Addition of Integers — Rules

Situation Rule Example
Both positive Add and keep (+) sign (+5) + (+3) = +8
Both negative Add magnitudes, keep (−) sign (−5) + (−3) = −8
Different signs Subtract smaller from larger; take sign of the larger magnitude (−8) + (+3) = −5 ; (+8) + (−3) = +5

B. Subtraction of Integers

a − b = a + (−b) → Change sign of the number to be subtracted, then add
Examples:
• 7 − (−3) = 7 + 3 = 10
• −5 − (+4) = −5 + (−4) = −9
• −6 − (−2) = −6 + 2 = −4

C. Multiplication of Integers — Sign Rules

Multiplication Result Sign Example
(+) × (+) Positive (+) 5 × 3 = 15
(−) × (−) Positive (+) (−5) × (−3) = 15
(+) × (−) Negative (−) 5 × (−3) = −15
(−) × (+) Negative (−) (−5) × 3 = −15
Any number × 0 Zero (0) (−99) × 0 = 0
🧠 Memory Trick — Sign Rules Same signs → Positive. Different signs → Negative.
Think of it like: Good × Good = Good; Bad × Bad = Good; Good × Bad = Bad

D. Division of Integers — Same Sign Rules Apply

Division Result Sign Example
(+) ÷ (+) Positive 12 ÷ 4 = 3
(−) ÷ (−) Positive (−12) ÷ (−4) = 3
(+) ÷ (−) Negative 12 ÷ (−4) = −3
(−) ÷ (+) Negative (−12) ÷ 4 = −3

📋 4. Properties of Integers

Property Addition Multiplication
Closure a + b is always an integer ✅ a × b is always an integer ✅
Commutative a + b = b + a ✅ a × b = b × a ✅
Associative (a+b)+c = a+(b+c) ✅ (a×b)×c = a×(b×c) ✅
Identity a + 0 = a (0 is additive identity) a × 1 = a (1 is multiplicative identity)
Distributive a × (b + c) = a × b + a × c ✅ (multiplication over addition)
Subtraction NOT Commutative: a−b ≠ b−a ❌ NOT Commutative: a÷b ≠ b÷a ❌

💡 5. Solved Examples

Example 1: Find (−15) + (+8) + (−3)
= −15 + 8 − 3 = −7 − 3 = −10

Example 2: Verify: (−5) × [(3) + (−2)] = (−5)×3 + (−5)×(−2) [Distributive law]
LHS = (−5) × [1] = −5
RHS = −15 + 10 = −5 ✓

Example 3: Temperature at noon = 8°C. It drops 12°C by night. Temperature at night = 8 + (−12) = −4°C

Example 4: A diver is 30 m below sea level (−30). She rises 18 m. New position = −30 + 18 = −12 m (still below)
✏️ Practice Questions — Integers (30 Questions)

Section A — Fill in the Blanks / MCQ Easy

Q1. (−7) + (−8) = ____
Q2. 15 + (−20) = ____
Q3. (−6) − (−4) = ____
Q4. The additive inverse of −13 is ____
Q5. (−1) × (−1) × (−1) = ____
Q6. 0 ÷ (−5) = ____
Q7. The successor of −99 is ____
Q8. Which is greater: −8 or −2? ____

Section B — Short Answer Medium

Q9. Compute: (−3) × (−4) × (−2)
Q10. Find: [(−36) ÷ (−6)] + 5
Q11. Verify: (−8) + (−5) = (−5) + (−8)
Q12. Simplify: (−25) × 4 + (−25) × 6 [use distributive law]
Q13. If a = −3, b = 4, find: a × b − b ÷ 2
Q14. Find 5 integers between −3 and 3
Q15. Arrange in ascending order: −5, 3, −9, 0, 7, −2
Q16. Compute: (−17) − [(−7) + (−3)]
Q17. What is (−1)²⁰ + (−1)¹⁵?
Q18. Find the sum of all integers from −5 to +5

Section C — Word Problems Medium

  1. Q19. The temperature of a city was −6°C on Monday. It rose by 9°C on Tuesday and fell by 4°C on Wednesday. Find Wednesday's temperature.
  2. Q20. A submarine is at a depth of 1200 m. It ascends 400 m, then descends 300 m. What is its final depth?
  3. Q21. A shop earns ₹500 on Mon, loses ₹200 on Tue, earns ₹750 on Wed, loses ₹125 on Thu. Find net profit/loss.
  4. Q22. A mountain peak is at 3726 m above sea level. A valley nearby is at 186 m below sea level. What is the vertical distance between them?

Section D — Higher Order Thinking Challenge

  1. Q23. If a × b = −18 and a + b = −3, find all possible integer pairs (a, b).
  2. Q24. Product of two integers is −72. One of them is −8. List all possible values of the other integer.
  3. Q25. A = {−3, −2, −1, 0, 1, 2, 3}. Find all pairs from A such that their sum is 0.
  4. Q26. Simplify: (−2) × 3 × (−4) × 5 × (−6) ÷ [(−3) × (−2)]
  5. Q27. Is it true that (a − b) is always less than (a + b) when b > 0? Justify with two examples.
  6. Q28. Fill: If p × q > 0, then both p, q have ____ signs. If p × q < 0, they have ____ signs. If p × q = 0 then ____.
  7. Q29. Write all pairs of integers whose product is −36 and both lie between −10 and +10.
  8. Q30. A lift starts at floor 0. It goes up 5 floors, down 8, up 3, down 2, and up 7. Which floor is it on now?
✅ Answers (Section A): Q1: −15  |  Q2: −5  |  Q3: −2  |  Q4: +13  |  Q5: −1  |  Q6: 0  |  Q7: −98  |  Q8: −2

📝 Quick Revision

  1. Integers: …−3, −2, −1, 0, 1, 2, 3… (includes negative, zero, positive)
  2. On number line: right = greater. Every positive integer > 0 > every negative integer
  3. Additive inverse of a = −a. Their sum = 0
  4. Same sign × or ÷ = Positive. Different sign = Negative
  5. Properties: Closure ✓, Commutative ✓, Associative ✓, Distributive ✓ for multiplication over addition
  6. a − b = a + (−b): always change the sign and add
  7. Subtraction and division are NOT commutative or associative for integers