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# 5.1 Addition, Subtraction, and Order Properties of Integers - PowerPoint PPT Presentation

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.

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### 5.1 Addition, Subtraction, and Order Properties of Integers

Remember to silence your cell phone and put it in your bag!

• 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, 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.

• 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.

• 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.

• 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.

• Review the procedures for adding two integers on p. 255.

• Note: The procedures are not the emphasis for this class.

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.

• 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)

• 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.

• 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).

• a < b iff there is a positive integer p such that a + p = b.

• b > a iff a < b.