4.1 Integers and Operations of Addition and
Subtraction
Symbol "-" means Subtract, Opposite, Negative
**our
text uses a raised negative as -4 or -a to mean
opposite or negative)
Negative
Numbers -
useful - debt, temperature,
"sea-level"
Integers {...-3, -2, -1,0,1,2,3,...}
Positive
Integers NonPositive
Integers
Negative
Integers NonNegative
Integers
-3 where the -
means "the opposite of"
CSMP uses ^ above the number
Example
If x
= -4, or x =3,
find -x.
Integer
Addition Models
Chip
Model Black +, Red - or sometimes White is -
4 + 7 -8
+ -4 -5
+3
Charged
Field Model +
and - ** Zero is ±
-5 + 7 -6 + -8
Patterns Read on own in text pg. 173
Number-Line
Model Car faces right and + is forward, - is backward
CSMP Do some on transparencies
Absolute
Value "distance to zero"
Examples
|5| |-2| |4 - 7| -3 + |6 + -8|
-|-2| -|5|
Properties
of Integer Addition
Closure
on Integers, Commutative, Associative, Identity
Uniqueness
of Additive Inverse
For
every integer, a, there is a unique number -a such
that
a + -a = 0 = -a
+ a
Examples: Find Additive Inverse
7 -3 - (4 + x) m - -2 -5 + -x
Integer
Subtraction
***
Subtraction is CLOSED on the set of integers ***
Chip
Model "Take away" red or black
chips. Sometimes need to put in more
"red/black" zero sets to have enough to subtract
6 - 8 -4
- 5 6 - -3
Charged
Field Model
"take away" +'s or -s. Again may need to put in zeros as ±
5 - 2 4
- -6 3 - -6 3 - 6
Number
Line Model Subtraction "turn car left", but forward and
backward are still + and -
6 - 4 3
- 5 2 - -4 -5 - 3 -5 - -1
Definition
of Subtraction
For all integers a and b,
a - b = n, a unique number,
such that
b + n = a
Examples: Use the definition of subtraction to find:
4 -
7 -3
- -5
Property For all integers a - b = a + -b
This is the traditional method use for
subtracting integers
3 -
8 + 4 5
- (6 - 2)
7 -
(10 - 3x)
#7b,
c, e
#14
b, d
#28
Black
Assignment: 4.1
pg 180; 1, 2a, 3r, 4r, 5, 7r, 9,
11, 12, 13r, 14r, 15, 17, 18, *19, 23r, 25r, 26a,b, 28r, 30r, 31r