Multiplication with Whole Numbers

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Lizbeth Osborne
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1 Math "Multiplication and Division with Whole Numbers" Objectives * Be able to multiply and understand the terms factor and product. * Properties of multiplication commutative, associative, distributive, multiplicative identity, and zero factor law. * Understand the concept of area. * Be able to divide and understand the terms divisor, dividend, quotient, and remainder. * Learn how to estimate products and quotients with rounded numbers. Multiplication with Whole Numbers To process of repeated addition with the same number can be shortened considerably by learning to multiply and memorizing the basic facts of multiplication with single-digit numbers. Multiplication is generally much easier than repeated addition, particularly if the numbers are large. The result of multiplication is called a ; and the numbers being multiplied are called of the product. Symbols for Multiplication Symbol Example 1 Raised dot 2 Numbers inside or next to parentheses 3 Cross sign Properties of Multiplication As with addition, the operation of multiplication has several properties. Multiplication is commutative and associative, and the number 1 is the multiplicative identity. Also, multiplication by 0 always gives a product of 0, and this fact is called the zero factor law. Commutative Property of Multiplication For any whole numbers a and b, (The order of multiplication can be reversed.) Associative Property of Multiplication For any whole numbers a, b and c, (The grouping of the numbers can be changed.) Page 1

2 Multiplicative Identity Property For any whole number a, (The product of any number and 1 is that same number.) The number 1 is called the Zero Factor Law For any whole number a, (The product of a number and 0 is always 0) Example 1 (Using the properties of multiplication) Find the product and identify which property of multiplication is being illustrated. a) 8 12 = 12 8 b) 0 57 c) 35 1 d) 7 (3 2) = (7 3) 2 Example 2 (Using the properties of multiplication) Find the values of the variable that will make each statement true and name the property of multiplication that is used. a) 25 1 = x b) 14 5 = 5 n c) (9 2) y = 9 (2 1) d) t 17 = 0 Page 2

3 The following example illustrates how the commutative and associative properties of multiplication are related to the method of multiplication of numbers that end with 0 s. Example 3 (Using the properties of multiplication) Find the following products. a) b) The Distributive Property To understand the technique for multiplying two whole numbers, we use expanded notation, the method of multiplication by powers of 10; and the following property, called the distributive property of multiplication over addition (or sometimes simply the distributive property). Distributive Property of Multiplication Over Addition If a; b; and c are whole numbers, then For example, But we can also multiply rst and then add, in the following manner Or vertically, Example 4 (Using the distributive property) Use the distributive property to nd each of the following products. a) 5 (2 + 6) b) 6 (4 + 8) Example 5 (Multiplication of whole numbers) Find the following products. 67 a) 52 b) Page 3

4 The Concept of Area Area id the measure of the interior, or enclosed region, of a plane surface and is measured in square units. Some of the units of area in the metric system are square meters, square decimeters, square centimeters, and square millimeters. Example 6 (Finding the area) The area of a rectangle (measured in square units) is found by multiplying its length by its width. rectangular plot of land with dimensions as shown here. Find the area of a Division with Whole Numbers In division, we want to nd how many times one number is contained in another. Division can be thought of as the reverse of multiplication. The number being divided is called the ; the number dividing the dividend is called the, and the result of division is called the. Division does not always involve factors (or exact divisors), and we need a more general idea and method for performing division. The more general method is a process known as the division algorithm (or long division). The remainder must be less than the divisor. If the remainder is 0, then both the divisor and quotient are factors (or divisors) of the dividend. Moreover, division is not commutative nor associative. The symbols we use for division are Example 7 (Dividing with whole numbers) Find the following quotients. a) 48 6 b) 72 8 Example 8 (Dividing with whole numbers) Find the quotient and remainder for a) 31)7982 b) Page 4

5 Example 9 (Dividing with whole numbers) Show that 22 and 32 are both factors of 704 by using long division. [Check the answer] Example 10 (Dividing with whole numbers) A plumber purchased 32 pipe ttings. What was the price of one tting if the bill was $512 before taxes? Division with 0 Case 1 If a is any nonzero whole number, then For example Case 2 If a is any whole number, For example Estimating Products and Quotients Products and quotients can be estimated just as sums and di erences can be estimated. Example 11 (Estimating products and quotients) Estimate the following products and quotients. Then nd the respective product or quotient. 39 a) b) c) 13)260 d) 39; Page 5

How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left. The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics