5.1 Addition, Subtraction, and Order Properties of Integers

1. 5.1 Addition, Subtraction, and Order Properties of Integers Remember to silence your cell phone and put it in your bag!

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

3. 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, …}

4. 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.

5. 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.

6. Modeling Integer Addition • 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.

7. Modeling Integer Addition • 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.

8. Procedures for Adding Integers • Review the procedures for adding two integers on p. 255. • Note: The procedures are not the emphasis for this class.

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

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

11. Modeling Integer Subtraction • 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.

12. Modeling Integer Subtraction • 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.

13. 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).

14. 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.