Copyright © 2016, 2013, and 2010, Pearson Education, Inc. 5 Chapter Integers Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

5-1 Addition and Subtraction of Integers The meaning of integers and their representation on a number line. Models for addition and subtraction of integers. Properties of addition and subtraction of integers.

Representations of Integers The set of integers is denoted by I: The negative integers are opposites of the positive integers. –4 is the opposite of positive 4 3 is the opposite of –3

Example For each of the following, find the opposite of x. a. x = 3 b. x = −5 −x = 5 c. x = 0 −x = 0

Integer Addition Chip Model Black chips represent positive integers and red chips represent negative integers. Each pair of black/red chips neutralize each other.

Integer Addition Charged-Field Model Similar to the chip model. Positive integers are represented by +’s and negative integers by –’s. Positive charges neutralize negative charges.

Integer Addition Number-Line Model Positive integers are represented by moving forward (right) on the number line; negative integers are represented by moving backward (left).

Example The temperature was −4°C. In an hour, it rose 10°C. What is the new temperature? −4 + 10 = 6 The new temperature is 6°C.

Integer Addition Pattern Model Beginning with whole number facts, a table of computations is created by following a pattern. 4 + 3 = 7 4 + 2 = 6 4 + 1 = 5 4 + 0 = 4 Basic facts 4 + −1 = 3 4 + −2 = 2 4 + −3 = 1 4 + −4 = 0 4 + −5 = −1 4 + −6 = −2

Absolute Value The absolute value of a number a, written |a|, is the distance on the number line from 0 to a. |4| = 4 and |−4| = 4 Absolute value is always positive or zero.

Definition For any integer x,

Example Evaluate each of the following expressions. a. |20| |20| = 20 b. |−5| |−5| = 5 c. |0| |0| = 0 d. −|−3 | −|−3| = −3 e. |2 + −5| |2 + −5| = |−3| = 3

Properties of Integer Addition Integer addition has all the properties of whole- number addition. Given integers a, b, and c. Closure property of addition of integers a + b is a unique integer. Commutative property of addition of integers a + b = b + a. Associative property of addition of integers (a + b) + c = a + (b + c).

Properties of Integer Addition Identity property of addition of integers 0 is the unique integer such that, for all integers a, 0 + a = a = a + 0. Additive Inverse Property of Integers For every integer a, there exists a unique integer −a, the additive inverse of a, such that a + −a = 0 = −a + a.

Properties of the Additive Inverse By definition, the additive inverse, −a, is the solution of the equation x + a = 0. For any integers a and b, the equation x + a = b has a unique solution, b + −a. For any integers a and b −(−a) = a and −a + −b = −(a + b).

Example Find the additive inverse of each of the following. a. −(3 + x) 3 + x b. a + −4 −(a + −4) = −a + 4 c. −3 + −x −(−3 + −x) = 3 + x

Integer Subtraction Chip Model for Subtraction To find 3 − −2, add 0 in the form 2 + −2 (two black chips and two red chips) to the three black chips, then take away the two red chips.

Integer Subtraction Charged-Field Model for Subtraction To find −3 − −5, represent −3 so that at least five negative charges are present. Then take away the five negative charges.

Integer Subtraction Number-Line Model While integer addition is modeled by maintaining the same direction and moving forward or backward depending on whether a positive or negative integer is added, subtraction is modeled by turning around.

Integer Subtraction Pattern Model for Subtraction Using inductive reasoning and starting with known subtraction facts, find the difference of two integers by following a pattern. 3 − 2 = 1 3 − 3 = 0 3 − 4 = 3 − 5 = 3 − 2 = 1 3 − 1 = 2 3 − 0 = 3 3 − −1 = −1 −2 4

Integer Subtraction Subtraction Using the Missing Addend Approach Subtraction of integers, like subtraction of whole numbers, can be defined in terms of addition. We compute 3 – 7 as follows: 3 – 7 = n if and only if 3 = 7 + n. Because 7 + –4 = 3, then n = –4.

Definition Subtraction For integers a and b, a − b is the unique integer n such that a = b + n.

Example Use the definition of subtraction to compute the following: Let 3 − 10 = n. Then 10 + n = 3, so n = −7. b. −2 − 10 Let −2 − 10 = n. Then 10 + n = −2, so n = −12.

Integer Subtraction Subtraction Using Adding the Opposite Approach Subtracting an integer is the same as adding its opposite. For all integers a and b a − b = a + −b.

Example Using the fact that a − b = a + −b, compute each of the following: a. 2 − 8 2 − 8 = 2 + −8 = −6 b. 2 − −8 2 − −8 = 2 + −(−8) = 2 + 8 = 10 c. −12 − −5 −12 − −5 = −12 + −(−5) = −12 + 5 = −7 d. −12 − 5 −12 − 5 = −12 + −5 = −17

Example Rewrite each expression without parentheses. a. −(b − c) −(b − c) = −(b + −c) = −b + −(−c) = −b + c b. a − (b + c) a − (b + c) = a + −(b + c) = a + (−b + −c) = (a + −b) + −c = a + −b + −c

Example Simplify each of the following expressions. a. 2 − (5 − x) 2 − (5 − x) = 2 + −(5 + −x) = 2 + −5 + −(−x) = 2 + −5 + x = −3 + x or x − 3 b. 5 − (x − 3) 5 − (x − 3) = 5 + −(x + −3) = 5 + −x + −(−3) = 5 + −x + 3 = 8 + −x = 8 − x

Example (continued) Simplify each of the following expressions. c. −(x − y) − y −(x − y) − y = −(x + −y) + −y = [−x + −(−y)] + −y = (−x + y) + −y = −x + (y + −y) = −x

Order of Operations Recall that subtraction is neither commutative nor associative. An expression such as 3 − 15 − 8 is ambiguous unless we know in which order to perform the subtractions. Mathematicians agree that 3 − 15 − 8 means (3 − 15) − 8. Subtractions are performed in order from left to right.

Example Compute each of the following. a. 2 − 5 − 5 2 − 5 − 5 = −3 − 5 = −8 b. 3 − 7 + 3 3 − 7 + 3 = −4 + 3 = −1 c. 3 − (7 − 3) 3 − (7 − 3) = 3 − 4 = −1