CHAPTER 5 Number Theory and the Real Number
CHAPTER 5 Number Theory and the Real Number System Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 1
5. 2 The Integers; Order of Operations Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 2
1. 2. 3. 4. 5. 6. Objectives Define the integers. Graph integers on a number line. Use symbols < and >. Find the absolute value of an integer. Perform operations with integers. Use the order of operations agreement. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 3
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”. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 4
The Number Line 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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 5
Example: Graphing Integers on a Number Line Graph: a. 3 b. 4 c. 0 Solution: Place a dot at the correct location for each integer. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 6
Use the Symbols < and > 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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 7
Use the Symbols < and > The symbols < and > are called inequality symbols. These symbols always point to the lesser of the two real numbers when the inequality statement is true. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 8
Example: Using the Symbols < and > Insert either < or > in the shaded area between the integers to make each statement true: a. 4 3 b. 1 5 c. 5 2 d. 0 3 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 9
Example: Using the Symbols < and > continued 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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 10
Use the Symbols < and > The symbols < and > may be combined with an equal sign, as shown in the following table: Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 11
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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 12
Example: Finding Absolute Value Find the absolute value: a. | 3| b. |5| Solution: c. |0| 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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 13
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. Examples 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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 14
Study Tip A good analogy for adding integers is temperatures above and below zero on thermometer. Think of a thermometer as a number line standing straight up. For example, Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 15
Additive Inverses Additive inverses have the same absolute value, but lie on opposite sides of zero on the number line. When we additive inverses, the sum is equal to zero. For example: 1. 18 + ( 18) = 0 2. ( 7) + 7 = 0 In general, the sum of any integer and its additive inverse is 0: a + ( a) = 0 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 16
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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 17
Example: Subtracting of Integers Subtract: a. 17 – (– 11) b. – 18 – (– 5) Copyright © 2015, 2011, 2007 Pearson Education, Inc. c. – 18 – 5 Section 5. 2, Slide 18
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, Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 19
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: Copyright © 2015, 2011, 2007 Pearson Education, Inc. Examples • 7( 5) = 35 • ( 6)( 11) = 66 • 17(0) = 0 Section 5. 2, Slide 20
Multiplication of Integers: Rules Rule Examples 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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. • • Section 5. 2, Slide 21
Exponential Notation Because exponents indicate repeated multiplication, rules for multiplying can be used to evaluate exponential expressions. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 22
Example: Evaluating Exponential Notation Evaluate: a. ( 6)2 Solution: b. 62 Copyright © 2015, 2011, 2007 Pearson Education, Inc. c. ( 5)3 d. ( 2)4 Section 5. 2, Slide 23
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. This means that 4( 3) = 12. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 24
Division of Integers Rule 1. The quotient of two integers with different signs is found by dividing their absolute values. The quotient is negative. 2. The quotient of two integers with the same sign is found by dividing their absolute values. The quotient is positive. 3. Zero divided by any nonzero integer is zero. 4. Division by 0 is undefined. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Examples • • Section 5. 2, Slide 25
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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 26
Example: Using the Order of Operations Simplify 62 – 24 ÷ 22 · 3 + 1. Solution: There are no grouping symbols. Thus, we begin by evaluating exponential expressions. 62 – 24 ÷ 22 · 3 + 1 = 36 – 24 ÷ 4 · 3 + 1 = 36 – 6 · 3 + 1 = 36 – 18 + 1 = 19 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 2, Slide 27
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