Lecture Notes: Multiplication and Division

Introduction to Multiplication and Division

Developed from human necessity due to complex interactions in civilizations.
Multiplication and division are essential arithmetic operations following addition and subtraction.

Definition: Multiplication combines two numbers to yield a product.
Can be viewed as repeated addition.
- Example: Collecting apples in groups of two.
  - Adding two apples five times: 2 + 2 + 2 + 2 + 2 = 10
  - Alternative: 2 (the number of apples) x 5 (the number of times) = 10
Factors and Product:
- In the example, 2 and 5 are factors; 10 is the product.
Symbols for Multiplication:
- Early on: X (times)
- Later: dot (•), asterisk (*), or adjacency (no symbol).

Definition: Division is the inverse operation of multiplication; it breaks larger numbers into smaller ones.
Example: To find out how many baskets of 5 apples are needed for 15 apples:
- 15 (dividend) ÷ 5 (divisor) = 3 (quotient).
Symbols for Division:
- Early on: ÷
- Later: line notation (either angled or horizontal).

Commutative Property:
- Multiplication is commutative: 2 x 5 = 5 x 2.
- Division is not commutative: 10 ÷ 2 ≠ 2 ÷ 10.
Associative Property:
- Multiplication is associative: (2 x 3) x 4 = 2 x (3 x 4).
- Division is not associative: (10 ÷ 5) ÷ 2 ≠ 10 ÷ (5 ÷ 2).

The four basic arithmetic operations: addition, subtraction, multiplication, and division.
Importance of practice and memorization of multiplication tables.
Caution against over-reliance on calculators for basic operations.
Future lessons will include tips on mental math.