Transcript 5.1 Addition, Subtraction, and Order Properties of Integers

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. 255.
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 = 0.
2. 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.