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

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.

Interpretations of Multiplication Repeated Addition Rectangular Array Area Cartesian Product

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.

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.

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.

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

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

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

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

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

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

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

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.