Integers

• Why integers were invented — to describe losses, depths, cold, and debt, not just gains and counts.
• To place, compare, and order positive and negative numbers on a number line.
• To add and subtract integers using sign rules and the key rewrite: a − b = a + (−b).
• To multiply integers using sign patterns — the bridge to algebra in Class 7.
• To model temperature, elevation, bank balances, and score changes with signed numbers.

Level 1 — What integers are

Verbal: Integers are whole numbers together with their negatives: …, −3, −2, −1, 0, 1, 2, 3, …

Symbolic: ℤ = {…, −2, −1, 0, 1, 2, …} — no fractions, no decimals.

Visual (number line): Draw a horizontal line with 0 at the centre. Right = positive, left = negative. Equal gaps between consecutive integers.

    −5   −4   −3   −2   −1    0    1    2    3    4    5
    ← colder / lower          |          warmer / higher →

Set relationship: Whole numbers W = {0, 1, 2, …} ⊂ ℤ. Every whole number is an integer, but −4 ∈ ℤ and −4 ∉ W.

Absolute value |a|: Distance from 0 on the number line, always ≥ 0.
|−9| = 9, |7| = 7, |0| = 0.
Think: "How far?" not "which direction?"

Additive inverse: For every integer a, there exists −a such that a + (−a) = 0.
Example: 5 + (−5) = 0. The additive inverse of −3 is +3.

Level 2 — Comparing integers

Number-line rule (master this): Further right = greater. −1 is greater than −8.

Between two integers: Integers strictly between −3 and 2 are −2, −1, 0, 1 (four values). Endpoints matter — read "between" vs "from … to … inclusive" carefully.

Level 3 — Addition and subtraction

Memory hook for "minus a negative": Two negatives in a row → the signs "become positive."
6 − (−2) → 6 + 2.

Closure: Integers are closed under +, −, × (results stay in ℤ) but not under ÷ (7 ÷ 2 = 3.5 ∉ ℤ).

Level 4 — Multiplication and powers (Class 6 extension)

Counting negatives: An even number of negative factors → positive product; odd → negative.

Powers:
(−3)² = (−3) × (−3) = 9 (even power → positive)
(−2)³ = (−2) × (−2) × (−2) = −8 (odd power keeps sign)
Critical bracket rule: (−3)² = 9 but −3² = −(3²) = −9 — the base is 3, not −3.

Evaluate: (−8) + 13 + (−5)

Step 1 — Group same-sign terms: (−8) + (−5) = −13
Step 2 — Combine with remaining term: −13 + 13 = 0
Step 3 — Number-line check: start at −8, move +13 → land on 5, then −5 → 0 ✓
Answer: 0

Temperature in Leh: Morning = −4 °C. By noon it rises 9 °C. By evening it drops 6 °C from noon. Final temperature?

Step 1 — Noon: −4 + 9 = 5 °C
Step 2 — Evening: 5 + (−6) = 5 − 6 = −1 °C
Answer: −1 °C (still below freezing — pack a jacket!)

Bank balance: Priya has ₹200. She spends ₹350 on books (overdraft allowed). Then her grandmother deposits ₹500. Final balance?

Step 1 — After spending: 200 + (−350) = −150 (₹150 overdraft)
Step 2 — After deposit: −150 + 500 = 350
Answer: ₹350

Compare and order: 0, −7, 3, −2, 5 from smallest to largest

Step 1 — Separate: negatives {−7, −2}, zero {0}, positives {3, 5}
Step 2 — Order negatives (smaller absolute value first when both negative): −7 < −2
Step 3 — Full order: −7 < −2 < 0 < 3 < 5
Answer: −7, −2, 0, 3, 5

• Order from smallest to largest: 0, −7, 3, −2, 5. (−7, −2, 0, 3, 5)
• Evaluate: 4 − (−6) and (−4) − 6. (10 and −10)
• A lift starts at floor −2 and goes up 5 floors. Which floor? (Floor 3)
• Which is greater: −11 or −9? Explain using the number line.
• Evaluate: (−2) × 5 and (−3) × (−4). (−10 and 12)

Revision tip: Draw one number line and mark real-life points: 0 °C (freezing), Mumbai sea level (0 m), −2 floor (basement), +10 runs (lead in cricket).

Open the Practice tab for graded questions on Integers.