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!. 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

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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!


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

  • 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

  • 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. 255.

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


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.


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)


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.


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.


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.


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