5.1 Addition, Subtraction, and Order Properties of Integers

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5.1 Addition, Subtraction, and Order Properties of Integers

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5.1 Addition, Subtraction, and Order Properties 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, …}

- 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.

- For each integer a, there is a unique integer, -a, such that 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.

- 0 is the unique integer such that
- 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.

- 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.