© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 5 Number Theory and the Real Number System

© 2010 Pearson Prentice Hall. All rights reserved The Integers; Order of Operations

© 2010 Pearson Prentice Hall. All rights reserved. 3 Objectives 1.Define the integers. 2.Graph integers on a number line. 3.Use symbols. 4.Find the absolute value of an integer. 5.Perform operations with integers. 6.Use the order of operations agreement.

© 2010 Pearson Prentice Hall. All rights reserved. 4 Define the Integers The set consisting of the natural numbers, 0, and the negatives of the natural numbers is called the set of integers. Notice the term positive integers is another name for the natural numbers. The positive integers can be written in two ways: 1.Use a “+” sign. For example, +4 is “positive four”. 2.Do not write any sign. For example, 4 is also “positive four”.

© 2010 Pearson Prentice Hall. All rights reserved. 5 The number line is a graph we use to visualize the set of integers, as well as sets of other numbers. Notice, zero is neither positive nor negative. The Number Line

© 2010 Pearson Prentice Hall. All rights reserved. 6 Graph: a.  3 b.4 c.0 Solution: Place a dot at the correct location for each integer. Example 1: Graphing Integers on a Number Line

© 2010 Pearson Prentice Hall. All rights reserved. 7 Use the Symbols Looking at the graph,  4 and  1 are graphed below. Observe that  4 is to the left of  1 on the number line. This means that -4 is less than -1. Also observe that  1 is to the right of  4 on the number line. This means that  1 is greater then  4.

© 2010 Pearson Prentice Hall. All rights reserved. 8 The symbols are called inequality symbols. These symbols always point to the lesser of the two real numbers when the inequality statement is true. Use the Symbols

© 2010 Pearson Prentice Hall. All rights reserved. 9 Insert either in the shaded area between the integers to make each statement true: a.  4 3 b.  1  5 c.  5  2 d. 0  3 Example 2: Using the Symbols

© 2010 Pearson Prentice Hall. All rights reserved. 10 a.  4 < 3 (negative 4 is less than 3) because  4 is to the left of 3 on the number line. b.  1 >  5 (negative 1 is greater than negative 5) because  1 is to the right of  5 on the number line. c.  5 <  2 ( negative 5 is less than negative 2) because  5 is to the left of  2 on the number line. d.0 >  3 (zero is greater than negative 3) because 0 is to the right of  3 on the number line. Example 2: Using the Symbols continued

© 2010 Pearson Prentice Hall. All rights reserved. 11 The symbols may be combined with an equal sign, as shown in the following table: Use the Symbols

© 2010 Pearson Prentice Hall. All rights reserved. 12 Absolute Value The absolute value of an integer a, denoted by |a|, is the distance from 0 to a on the number line. Because absolute value describes a distance, it is never negative. Example: Find the absolute value: a.|  3|b. |5|c. |0| Solution:

© 2010 Pearson Prentice Hall. All rights reserved. 13 Example 3: Finding Absolute Value Find the absolute value: a.|  3|b. |5|c. |0| Solution: a.|  3| = 3 because  3 is 3 units away from 0. b.|5| = 5 because 5 is 5 units away from 0. c.|0| = 0 because 0 is 0 units away from itself.

© 2010 Pearson Prentice Hall. All rights reserved. 14 Addition of Integers Rule If the integers have the same sign, 1.Add their absolute values. 2.The sign of the sum is the same sign of the two numbers. If the integers have different signs, 1.Subtract the smaller absolute value from the larger absolute value. 2.The sign of the sum is the same as the sign of the number with the larger absolute value. Examples

© 2010 Pearson Prentice Hall. All rights reserved. 15 A good analogy for adding integers is temperatures above and below zero on the thermometer. Think of a thermometer as a number line standing straight up. For example, Study Tip

© 2010 Pearson Prentice Hall. All rights reserved. 16 Additive Inverses Additive inverses have the same absolute value, but lie on opposite sides of zero on the number line. When we add additive inverses, the sum is equal to zero. For example: (  18) = 0 2.(  7) + 7 = 0 In general, the sum of any integer and its additive inverse is 0:a + (  a) = 0

© 2010 Pearson Prentice Hall. All rights reserved. 17 Subtraction of Integers For all integers a and b, a – b = a + (  b). In words, to subtract b from a, add the additive inverse of b to a. The result of subtraction is called the difference.

© 2010 Pearson Prentice Hall. All rights reserved. 18 Example 4: Subtracting of Integers Subtract: a. 17 – (–11)b. –18 – (–5)c. –18 – 5

© 2010 Pearson Prentice Hall. All rights reserved. 19 Multiplication of Integers The result of multiplying two or more numbers is called the product of the numbers. Think of multiplication as repeated addition or subtraction that starts at 0. For example,

© 2010 Pearson Prentice Hall. All rights reserved. 20 Multiplication of Integers: Rules Rule 1.The product of two integers with different signs is found by multiplying their absolute values. The product is negative. 2.The product of two integers with the same signs is found by multiplying their absolute values. The product is positive. 3.The product of 0 and any integer is 0: Examples 7(  5) =  35 (  6)(  11) = 66  17(0) = 0

© 2010 Pearson Prentice Hall. All rights reserved. 21 Multiplication of Integers: Rules Rule 4.If no number is 0, a product with an odd number of negative factors is found by multiplying absolute values. The product is negative. 5.If no number is 0, a product with an even number of negative factors is found by multiplying absolute values. The product is positive. Examples

© 2010 Pearson Prentice Hall. All rights reserved. 22 Exponential Notation Because exponents indicate repeated multiplication, rules for multiplying can be used to evaluate exponential expressions.

© 2010 Pearson Prentice Hall. All rights reserved. 23 Example 6: Evaluating Exponential Notation Evaluate: a. (  6) 2 b.  6 2 c. (  5) 3 d. (  2) 4 Solution:

© 2010 Pearson Prentice Hall. All rights reserved. 24 Division of Integers The result of dividing the integer a by the nonzero integer b is called the quotient of numbers. We write this quotient as or a / b. Example: This means that 4(  3) =  12.

© 2010 Pearson Prentice Hall. All rights reserved. 25 Rule The quotient of two integers with different signs is found by dividing their absolute values. The quotient is negative. The quotient of two integers with the same sign is found by dividing their absolute values. The quotient is positive. Zero divided by any nonzero integer is zero. Division by 0 is undefined. Division of Integers Rules Examples

© 2010 Pearson Prentice Hall. All rights reserved. 26 Order of Operations 1.Perform all operations within grouping symbols. 2.Evaluate all exponential expressions. 3.Do all the multiplications and divisions in the order in which they occur, working from left to right. 4.Finally, do all additions and subtractions in the order in which they occur, working from left to right.

© 2010 Pearson Prentice Hall. All rights reserved. 27 Simplify 6 2 – 24 ÷ 2 2 · Solution: There are no grouping symbols. Thus, we begin by evaluating exponential expressions. 6 2 – 24 ÷ 2 2 · = 36 – 24 ÷ 4 · = 36 – 6 · = 36 – = = 19 Example 7: Using the Order of Operations