Learning Objectives
- Understand properties of addition and subtraction of integers
- Learn multiplication and division of integers
- Apply properties of commutativity, associativity, and distributivity
- Solve word problems involving integers
Key Concepts
Properties of Integer Addition
Closure Property: The sum of any two integers is always an integer. If a and b are integers, then a + b is also an integer.
Commutative Property: a + b = b + a. The order of addition does not matter.
Associative Property: (a + b) + c = a + (b + c). Grouping does not matter.
Additive Identity: a + 0 = 0 + a = a. Zero is the additive identity.
Multiplication of Integers
Rules for multiplication:
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
- Negative × Positive = Negative
Distributive Property: a × (b + c) = a × b + a × c
Division of Integers
Division rules follow the same sign rules as multiplication:
- Positive ÷ Positive = Positive
- Negative ÷ Negative = Positive
- Positive ÷ Negative = Negative
- Negative ÷ Positive = Negative
Note: Division by zero is not defined.
Summary
Integers include all whole numbers and their negatives. They follow closure, commutative, associative, and distributive properties for addition and multiplication. Multiplication and division of integers follow specific sign rules. Zero is the additive identity and 1 is the multiplicative identity.
Important Terms
- Integers
- The set of whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
- Additive Identity
- Zero (0), because adding 0 to any integer gives the same integer
- Multiplicative Identity
- One (1), because multiplying any integer by 1 gives the same integer
- Additive Inverse
- For any integer a, the additive inverse is -a, because a + (-a) = 0
Quick Revision
- Integers are closed under addition, subtraction, and multiplication but NOT under division
- Product of two negative integers is always positive
- Product of a positive and negative integer is always negative
- a × 0 = 0 for any integer a
- Distributive property: a × (b + c) = ab + ac
Practice Tips
- Practice sign rules by creating multiplication tables with negative numbers
- Use the number line to visualize integer operations
- Solve word problems involving temperature changes, bank transactions, and elevation