📌 Key Points
- Integers are {..., -3, -2, -1, 0, 1, 2, 3, ...}. They include positive numbers, negative numbers, and zero.
- Positive integers are natural numbers: 1, 2, 3, 4, ...
- Negative integers are less than zero: -1, -2, -3, -4, ...
- Number line: Larger numbers are to the right, smaller to the left.
- Comparing integers: Every positive > every negative. For negatives: -1 > -5 (closer to zero).
- Additive inverse of a is -a. Example: Inverse of 5 is -5. Inverse of -3 is 3.
- Absolute value |a| is distance from zero. Always non-negative: |-5| = 5, |5| = 5.
- Addition rule: Positive + Positive = Positive. Negative + Negative = Negative.
- Mixed signs addition: Subtract smaller absolute value from larger, use sign of larger.
- Subtraction: a - b = a + (-b). Convert subtraction to addition of opposite.
- Multiplication signs: + × + = +, - × - = +, + × - = -, - × + = -
- Count negative signs for multiple multiplications: Even count = positive, odd count = negative.
- Division signs follow same rules as multiplication for sign determination.
- Division by zero is undefined. We cannot divide any number by 0.
- Closure property: Sum, difference, product of integers are always integers.
- Commutative property: a + b = b + a and a × b = b × a. NOT true for subtraction/division.
- Associative property: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
- Distributive property: a × (b + c) = (a × b) + (a × c)
- Additive identity: 0. Any integer + 0 = itself. a + 0 = a
- Multiplicative identity: 1. Any integer × 1 = itself. a × 1 = a
📘 Important Definitions
🔢 Formulas & Laws
Additive Inverse
Inverse of a = -a
a + (-a) = 0
Absolute Value
|a| = a if a ≥ 0; |a| = -a if a < 0
Absolute value is always non-negative
Successor Formula
Successor of n = n + 1
Example: Successor of 5 is 6; Successor of -3 is -2
Predecessor Formula
Predecessor of n = n - 1
Example: Predecessor of 5 is 4; Predecessor of -3 is -4
Subtraction Rule
a - b = a + (-b)
Convert subtraction to addition of opposite
Distributive Property
a × (b + c) = (a × b) + (a × c)
Used to simplify expressions with parentheses
⚠️ Common Mistakes
✗ Wrong: Thinking -5 < -2 because 5 > 2
✓ Correct: Correct: -2 > -5 because -2 is closer to zero on the number line
✗ Wrong: 5 - (-3) = 5 - 3 = 2
✓ Correct: Correct: 5 - (-3) = 5 + 3 = 8 (subtract means add opposite)
✗ Wrong: (-3) × (-2) = -6
✓ Correct: Correct: (-3) × (-2) = 6 (negative × negative = positive)
✗ Wrong: (-4) ÷ (-2) = -2
✓ Correct: Correct: (-4) ÷ (-2) = 2 (negative ÷ negative = positive)
✗ Wrong: Subtraction is commutative: 5 - 3 = 3 - 5
✓ Correct: Correct: Subtraction is NOT commutative. 5 - 3 = 2, but 3 - 5 = -2
✗ Wrong: (-2) + (-3) + (-4) = -9 is unclear how we got it
✓ Correct: Correct: Add absolute values (2 + 3 + 4 = 9), then add negative sign = -9
📝 Exam Focus
These questions are frequently asked in CBSE exams:
🎯 Last-Minute Recall
Close your eyes and try to recall: Key definitions, formulas, and 3 common mistakes. If you can recall 80% without looking, you're exam-ready!