Summary & Quick Quiz പുനരവലോകനം
Every rule of the chapter in one place, the sign tables, the idea behind all the rules, the mistakes to avoid, flashcards, a scored quiz and your study checklist.
1. All the rules
For positive x, y:
−x + y = y − x
−x − y = −(x + y)
3 − 7 = −4; −6 + 8 = 2; −6 − 8 = −14.
x ഏതു സംഖ്യയായാലും, y ഏതു അധിസംഖ്യയായാലും
5 + (−3) = 2; (−5) + (−3) = −8. Final position z = x + y for every move.
For any x and positive y:
4 − (−7) = 11: how much to add to −7 to reach 4. Change of position = y − x.
For positive x:
−(−8) = 8.
x, y ഏതു സംഖ്യകളായാലും
10 − 3 = 7 = −(3 − 10) = −(−7).
For positive x, y:
(−x)(−y) = xy
(−10) × 2 = −20; (−10)(−2) = 20. Position s = vt in every direction.
For positive x, y:
(−x)/(−y) = x/y
(−6) ÷ 2 = −3; 12 ÷ (−4) = −3; (−6) ÷ (−3) = 2.
Even power 1, odd power −1. ½(1 + (−1)ⁿ) is 1 for even n, 0 for odd n.
2. The sign tables
× multiplication
| × | + | − |
|---|---|---|
| + | + | − |
| − | − | + |
Same signs ⇒ positive; different signs ⇒ negative.
÷ division
| ÷ | + | − |
|---|---|---|
| + | + | − |
| − | − | + |
Exactly the same pattern, because division is undoing multiplication.
None of these rules is a guess. Each new operation was defined (നിർവചനം) so that an old rule keeps working:
- 5 + (−3) = 2 keeps "order of adding doesn't matter" and makes final = start + change true for every move.
- 4 − (−7) = 11 keeps "subtraction asks how much to add" and makes change = target − start true.
- (−10)(−2) = 20 keeps the distributive law (x + y)z = xz + yz and makes s = vt true before and after, left and right.
3. Common mistakes
- −3 − 5 is not −2. Both moves go left: −3 − 5 = −(3 + 5) = −8.
- −(−4) is not −4. The negative of a negative is positive: −(−4) = 4.
- (−2)² is not −2². (−2)² = (−2)(−2) = 4, but −2² means −(2²) = −4. Use brackets when you square a negative.
- Forgetting brackets: write 6 − (−4), not 6 − −4; write 3 × (−7), not 3 × −7.
- Sign of a long product: count the negative factors – odd number ⇒ negative, even ⇒ positive.
- Substituting negatives: x² + 3x with x = −3 is 9 + (−9) = 0, not 9 + 9.
4. Flashcards
5. Scored quiz
6. Your study checklist
- Right is positive, left is negative; positions and changes become numbers with signs.
- Adding a negative is subtracting; subtracting a negative is adding; −(−x) = x.
- Same signs multiply or divide to positive, different signs to negative.
- Every rule was chosen so that one equation – z = x + y, y − x, s = vt – works in all cases.
Next: Model test paper ▶