5.1 Addition, Subtraction, and Order Properties of Integers Remember to silence your cell phone and put it in your bag!

Opposite For every natural number n, there is a unique number the opposite of n, denoted by –n, such that n + -n = 0.

The Set of Integers The set of integers, I, is the union of the set of natural numbers, the set of the opposites of the natural numbers, and the set that contains zero. I = {1, 2, 3, …}  {-1, -2, -3...}  {0} I = { …, -3, -2, -1, 0, 1, 2, 3, …}

Opposite (revisited) For every integer n, there is a unique integer, the opposite of n, denoted by –n, such that n + -n = 0. Note: The opposite of 0 is 0.

Definition of Absolute Value The absolute value of an integer n, denoted by |n|, is the number of units the integer is from 0 on the number line. Note: |n|  0 for all integers.

Modeling Integer Addition 1. Chips (counters) Model 1. Black Chips (or yellow) represent a positive integer. 2. Red Chips represent a negative integer. 3. A black chip (or yellow) and a red chip together represent 0.

Modeling Integer Addition 2. Number Line Model (this is different than the model in the book) 1. A person (or car) starts at 0, facing in the positive direction, and walks (or moves) on the number line. 2. Walk forward to add a positive integer. 3. Walk backward to add a negative integer.

Procedures for Adding Integers Review the procedures for adding two integers on p Note: The procedures are not the emphasis for this class.

Properties of Integer Addition For a, b, c  I 1. Inverse property For each integer a, there is a unique integer, -a, such that a + (-a) = 0 and (-a) + a = Closure Property a + b is a unique integer.

Properties of Integer Addition (cont.) 3. Identity Property 0 is the unique integer such that a + 0 = a and 0 + a = a. 4. Commutative Property a + b = b + a 5. Associative property (a + b) + c = a + (b + c)

Modeling Integer Subtraction 1. Chips (counters) Model 1. Use the take-away interpretation of subtraction. 2. Because a black-red pair is a “zero pair,” you can include as many black-red pairs as you want when representing an integer, without changing its value.

Modeling Integer Subtraction 2. Number Line Model (this is different than the model in the book) 1. A person (or car) starts at 0, facing in the positive direction, and walks (or moves) on the number line. 2. Walk forward for a positive integer. 3. Walk backward for a negative integer. 4. To subtract, you must change the direction of the walker.

Integer Subtraction (Cont) Definition of Integer Subtraction For a, b, c  I, a – b = c iff c + b = a. The missing addend interpretation of subtraction may be used for integers. Theorem: To subtract an integer, you may add its opposite. a, b  I, a – b = a + (-b).

Definition of Greater Than and Less Than for Integers a < b iff there is a positive integer p such that a + p = b. b > a iff a < b.