Integers — The World of Positive and Negative Numbers
"Integers extend the number line below zero — letting us represent debts, temperatures, depths, and more."
1. What Are Integers?
Whole numbers (0, 1, 2, 3, ...) together with negative numbers (...−3, −2, −1) form the set of INTEGERS. The set is written as Z = {..., −3, −2, −1, 0, 1, 2, 3, ...}. Zero is neither positive nor negative. Every integer has an opposite: the opposite of +5 is −5; the opposite of −8 is +8.
Integer Number Line
←—|——|——|——|——|——|——|——|——|——|——|——|——|——|→
−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7
Numbers INCREASE as we move RIGHT. Numbers DECREASE as we move LEFT. Every point on the number line represents a unique integer.
2. Absolute Value
The absolute value of an integer is its distance from zero on the number line — ALWAYS non-negative. Notation: |a|. Examples: |5| = 5. |−5| = 5. |0| = 0. 'Absolute value tells us MAGNITUDE without sign.'
3. Addition of Integers
Same Sign — Add and Keep the Sign
- (+3) + (+4) = +7
- (−3) + (−4) = −7
Different Signs — Subtract the Smaller Absolute Value from the Larger. Keep the Sign of the Larger Absolute Value.
- (−7) + (+4) = −3 (7 − 4 = 3. Sign of larger absolute value 7 → negative)
- (+7) + (−4) = +3 (7 − 4 = 3. Sign of larger absolute value 7 → positive)
Using the Number Line for Addition
'Start at the first number. Move RIGHT for positive. Move LEFT for negative.'
| Example | Steps | Result |
|---|---|---|
| (−2) + 3 | Start at −2. Move 3 right. | +1 |
| 4 + (−6) | Start at 4. Move 6 left. | −2 |
| (−3) + (−2) | Start at −3. Move 2 left. | −5 |
4. Subtraction of Integers
KEY RULE: Subtracting an integer is the SAME as adding its opposite. a − b = a + (−b). 'This single rule makes subtraction just a special case of addition.'
| Problem | Rewrite as | Result |
|---|---|---|
| 8 − 15 | 8 + (−15) | −7 |
| 5 − (−3) | 5 + (+3) | 8 |
| −7 − (−4) | −7 + (+4) | −3 |
| −10 − 3 | −10 + (−3) | −13 |
Common Mistake
'Two negatives do NOT always make a positive.' −7 − 3 = −10 (NOT −4). Two negatives make a positive ONLY when we have SUBTRACTION of a NEGATIVE: 5 − (−3) = 5 + 3 = 8.
5. Multiplication of Integers
| Sign Pattern | Result | Example |
|---|---|---|
| (+) × (+) | + | 3 × 4 = 12 |
| (+) × (−) | − | 3 × (−4) = −12 |
| (−) × (+) | − | (−3) × 4 = −12 |
| (−) × (−) | + | (−3) × (−4) = 12 |
Rule: 'Product of EVEN number of negative integers is POSITIVE. Product of ODD number of negative integers is NEGATIVE.'
| Example | Number of Negatives | Result |
|---|---|---|
| (−2) × (−3) × (+4) | 2 (even) | +24 |
| (−2) × (−3) × (−4) | 3 (odd) | −24 |
| (−1) × (−1) × (−1) × (−1) | 4 (even) | +1 |
6. Division of Integers
Same sign rule as multiplication:
| Sign Pattern | Result | Example |
|---|---|---|
| (+) ÷ (+) | + | 12 ÷ 3 = 4 |
| (+) ÷ (−) | − | 12 ÷ (−3) = −4 |
| (−) ÷ (+) | − | (−12) ÷ 3 = −4 |
| (−) ÷ (−) | + | (−12) ÷ (−3) = 4 |
Important: Division by zero is NOT defined. Any integer divided by zero is undefined. Zero divided by any non-zero integer equals zero: 0 ÷ 5 = 0.
'Division is the INVERSE of multiplication.' Since (−4) × (−3) = 12, we have 12 ÷ (−3) = −4 and 12 ÷ (−4) = −3.
7. Properties of Integers — Complete Reference Table
| Property | Addition | Multiplication | Subtraction | Division |
|---|---|---|---|---|
| Closure | a + b is an integer ✓ | a × b is an integer ✓ | a − b is an integer ✓ | a ÷ b MAY NOT be an integer ✗ |
| Commutative | a + b = b + a ✓ | a × b = b × a ✓ | a − b ≠ b − a ✗ | a ÷ b ≠ b ÷ a ✗ |
| Associative | (a+b)+c = a+(b+c) ✓ | (a×b)×c = a×(b×c) ✓ | (a−b)−c ≠ a−(b−c) ✗ | (a÷b)÷c ≠ a÷(b÷c) ✗ |
| Identity | a + 0 = a (0 is identity) | a × 1 = a (1 is identity) | No identity | No identity |
| Inverse | a + (−a) = 0 | a × (1/a) — not always integer | — | — |
| Distributive | — | a(b+c) = ab + ac ✓ | a(b−c) = ab − ac ✓ | — |
Worked Example — Using Properties
Simplify (−37) × 15 + (−37) × 85 using the distributive property: = (−37) × (15 + 85) = (−37) × 100 = −3700. 'This is MUCH faster than multiplying separately and then adding.'
8. Word Problems with Integers
Temperature Problems
'The temperature in Hyderabad at noon is 35°C. It drops by 3°C per hour for the next 5 hours.' Temperature after 5 hours = 35 + 5 × (−3) = 35 − 15 = 20°C. At midnight it drops another 8°C: 20 + (−8) = 12°C. 'In Kullu-Manali, the temperature can drop to −8°C — negative temperatures are real!'
Financial Problems
Ravi has Rs 500. He buys goods worth Rs 650 on credit. His net balance = 500 − 650 = −150 (he owes Rs 150). 'Debt is a negative integer in real life. If he then deposits Rs 200: −150 + 200 = +50.'
Depth Problems
'A submarine is at −200 m (below sea level). It descends another 50 m: −200 − 50 = −250 m. It then rises 80 m: −250 + 80 = −170 m.'
9. Common Mistakes and Fixes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| −3 − 5 = −2 | Subtracting 5 from −3 means moving FURTHER left | −3 − 5 = −3 + (−5) = −8 |
| (−4)² = −8 | Misapplying multiplication | (−4) × (−4) = +16 |
| −2 + 5 = −7 | Confusing addition direction | Start at −2, move 5 right → +3 |
| (−3) × 4 = +12 | Wrong sign rule | (+) × (−) = −, so (−3) × 4 = −12 |
10. AP SSC Exam Focus
| Topic | Marks | Question Type |
|---|---|---|
| Properties of integers | 2-3 | True/False, Fill in blanks |
| Addition and subtraction | 2-3 | Direct computation |
| Multiplication and division | 2-3 | Direct computation |
| Word problems | 3-4 | Application (temperature, finance) |
| Absolute value | 1-2 | Definition and simple problems |
Quick Self-Test
Q1. Evaluate: (−25) × 8 + (−25) × 2. A1. (−25) × (8 + 2) = (−25) × 10 = −250.
Q2. What is the integer that is 6 less than −3? A2. −3 − 6 = −9.
Q3. Verify: Is subtraction commutative for integers? Give a counterexample. A3. No. Example: 15 − 10 = 5 but 10 − 15 = −5. Since 5 ≠ −5, subtraction is NOT commutative.
Q4. The temperature in Srinagar on Tuesday was −4°C. On Wednesday it dropped by 7°C. What was the temperature on Wednesday? A4. −4 − 7 = −11°C.
Q5. Find the product using properties: (−47) × 102. A5. (−47) × (100 + 2) = (−47) × 100 + (−47) × 2 = −4700 + (−94) = −4794.
Q6. Is (−8) ÷ 0 defined? A6. No. Division by zero is undefined.
Q7. What is the absolute value of −15? A7. |−15| = 15.
