2.3 Multiplication and Division of Whole Numbers

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 Multiplicationof 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.