Transcript Section 2.4
MTH 231 Section 2.4 Multiplication and Division of Whole Numbers Multiplication • Some of the conceptual models mentioned in the section: 1. Multiplication as repeated addition 2. Array model 3. Rectangular area model 4. Skip-count model Repeated Addition Array Rectangular Area Skip-Count Multiple Models Properties of Whole-Number Multiplication • Like addition, multiplication is: 1. Closed 2. Associative 3. Commutative • However, there are three new properties we need to discuss. 4. Multiplicative Identity Property • There is a “special” element in the whole numbers. This element has the property that any whole number multiplied by it gives back the number you started with: a x 1 = a and 1 x a = a for all whole numbers a 5. Multiplication-by-Zero Property • Any whole number multiplied by 0 gives a result of 0 b x 0 = 0 and 0 x b = 0 for all whole numbers b 6. Distributive Property • If a, b, and c are any three whole numbers: a x (b + c) = (a x b) + (a x c) and (a + b) x c = (a x c) + (b x c) • The official title of the property, “distributive property of multiplication over addition”, is reflected in the fact that both operations are present. Images More Images Division of Whole Numbers • Division is inherently more difficult to model than multiplication, yet there are fewer models: 1. Repeated-subtraction 2. Partition 3. Missing-factor Repeated-Subtraction • In this model, elements in a set are subtracted away in groups of a specified size. • This model is also called division by grouping. Partition • In this model, elements in a set are separated into groups of a specified size. Missing Factor • In this model, division is recognized as the inverse of multiplication. Division By Zero • Consider the following questions: 1. John has 12 pieces of candy. He wants to give each of his friends 0 pieces. How many friends will receive 0 pieces of candy? (repeated-subtraction) 2. John has 12 pieces of candy. He wants to divide them in groups of 0 pieces. How many groups of 0 pieces can John make? (partition) 3. Find a whole number c such that 0 x c = 12. (missing-factor) Division With Remainders • Sticking with the missing-factor model, we now consider those situations where a whole number c cannot be found: Find a whole number c such that 5 x c = 7. • The other models further support the idea that, in some cases, a remainder is needed to extend the division operation. The Division Algorithm • Let a and b be whole numbers with b not equal to zero (Why?). Then there exist whole numbers q and r such that a = q x b + r, with 0 < r < b. a is called the dividend. b is called the divisor. q is called the quotient. r is called the remainder. 7 Divided By 5, 3 Ways 1 5 7 5 2 7 1 5 2 7 5 1R 2