5.2 The Integers; Order of Operations

Define the Integers 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 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.

Graphing Integers on a Number Line 3 4 Solution: Place a dot at the correct location for each integer.

Use the Symbols < and > Numbers 4 and 1 are graphed below. Note that 4 is to the left of 1 on the number line. This means that -4 is less than -1. Also note that 1 is to the right of 4 on the number line. This means that 1 is greater then 4.

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.

Your Turn Insert either < or > in the shaded area between the integers to make each statement true: 4 3 -4 < 3 1 5 -1 > -5 5 2 -5 < -2 0 3 0 > -3

Use the Symbols < and > The symbols < and > may be combined with an equal sign, as shown in the following table:

Use the Symbols < and > The symbols < and > may be combined with an equal sign, as shown in the following table:

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: |3| b. |5| c. |0| : |-3| = 3 |5| = 5 |0| = 0

Addition of Integers Find: 5 + 5 5 + 5 = 10 Find: -5+ (-3) -5 + (-3) = -8 Find: 3 + (-5) 3 + (-5) = -2 Find: -3 + (-7) -3 + (-7) = -10 -8 -5 -3 -2 3 -5 -10 -3 -7

Addition of Integers Rule If the integers have the same sign, Add their absolute values. The sign of the sum is the same sign of the two numbers. If the integers have different signs, Subtract the smaller absolute value from the larger absolute value. The sign of the sum is the same as the sign of the number with the larger absolute value.

Your Turn Find the sums 12 + (-7) 12 + (-7) = 5 -8 + (-5) -8 + (-5) = -13 13 + (-15) 13 + (-15) = -2 -5 + (-7) -5 + (-7) = -12

Study Tip 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,

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 + (18) = 0 ( 7) + 7 = 0 In general, the sum of any integer and its additive inverse is 0: a + (a) = 0

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.

Subtracting of Integers a. 17 – (–11) b. –18 – (–5) c. –18 – 5

Your Turn 5 – 8 5 – 8 = 5 + (-8) = -3 5 – (-8) 5 – (-8) = 5 + (8) = 13 Find the difference in each case. 5 – 8 5 – 8 = 5 + (-8) = -3 5 – (-8) 5 – (-8) = 5 + (8) = 13 -5 – 8 -5 – 8 = -5 + (-8) = -13 -5 – (-8) -5 – (-8) = -5 + (8) = 3

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,

Multiplication of Integers: Rules The product of two integers with different signs is found by multiplying their absolute values. The product is negative. The product of two integers with the same signs is found by multiplying their absolute values. The product is positive. The product of 0 and any integer is 0: Examples 7(5) = 35 (6)(11) = 66 17(0) = 0

Multiplication of Integers: Rules If no number is 0, a product with an odd number of negative factors is found by multiplying absolute values. The product is negative. 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

Exponential Notation Because exponents indicate repeated multiplication, rules for multiplying can be used to evaluate exponential expressions. E.g., (-2)3 = (-2)(-2)(-2) = -8 (-3)4 = (-3)(-3)(-3)(-3) = 81

Evaluating Exponential Notation Evaluate: a. (6)2 b. 62 c. (5)3 d. (2)4 Solution:

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. 12 4 = ? ? ∙ 4 = 12; So, ? = 3 This means that 4(3) = 12.

Division of Integers Rules 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. Examples

Order of Operations Perform all operations within grouping symbols (Parentheses). Exponential expressions. Multiplications and divisions, from left to right. Additions and subtractions, from left to right.

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 = 18 + 1 = 19

Your Turn Simplify 5 ∙ 3 + 6/2 + 32 5 ∙ 3 + 6/2 + 32 = 5 ∙ 3 + 6/2 + 9 = 15 + 3 + 9 = 27 5 ∙ 3 + (-6/2) – 32 = 5 ∙ 3 + (-3) – 32 = 5 ∙ 3 + (-3) – 9 = 15 + (-3) – 9 = 12 – 9 = 3