1. 2.3 Multiplication and Division of Whole Numbers Remember to Silence Your Cell Phone and Put It In Your Bag!

2. What is Multiplication? Review – Addition is the joining together of two sets or two lengths. Multiplication is the joining together of equal-sized sets (equivalent sets) or equal-sized lengths.

3. Interpretations of Multiplication Repeated Addition Rectangular Array Area Cartesian Product

4. Additional Set Operation The Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (x, y) such that x is an element of A and y is an element of B.

5. Definition of Multiplication of Whole Numbers For any whole numbers m and n, m  0, m  n = n + n + n +... + n where n occurs m times If m = 0, 0  n = 0. Factors Product Note – I am not using the definition the book gives on p. 91. Consider it as an alternate definition.

6. Properties of Multiplication Let a, b, c  W Closure Property a  b is a unique whole number Identity Property There exists a unique whole number, 1, such that a  1 = a and 1  a = a.

7. Properties of Multiplication (cont.) Commutative Property a  b = b  a Associative property (a  b)  c = a  (b  c) Zero Property a  0 = 0 and 0  a = 0

8. Properties of Multiplication (cont.) Distributive property of multiplication over addition a  (b + c) = (a  b) + (a  c) (b + c)  a = (b  a) + (c  a)

9. What is Division? Division is separating a quantity into groups of the same size Division is separating a set of objects into equivalent subsets Note – Division is the inverse operation of multiplication

10. Interpretations of Division Finding how many in each subset Sharing Finding how many subsets Repeated subtraction Missing Factor

11. Definition of Division For a, b  W, b  0, a  b = c iff c is a unique whole number such that c  b = a. Dividend Divisor Quotient

12. In other words... Definition of Division Dividend  Divisor = Quotient iff Quotient  Divisor = Dividend

13. The Division Algorithm For a, b  W, b  0, a division process for a  b can be used to find unique whole numbers q and r such that a = b  q + r and 0  r < b. a is the dividend, b is the divisor, q is the quotient, and r is the remainder

14. Division Involving Zero If a  0, then 0  a = 0 b/c 0  a = 0. If a  0, then a  0 is undefined b/c there is no number q such that q  0 = a. q  0 always equals 0. 0  0 is undefined b/c there is no unique number q such that q  0 = 0. For any number q  0 = 0.