This document contains notes from Chapter 2 on rational numbers and probability. Key concepts covered include: adding, subtracting, multiplying and dividing rational numbers; properties of numbers like commutative, associative, identity, and inverse; theoretical and experimental probability; and probability of compound events being dependent or independent. Examples are provided to illustrate concepts like finding probabilities of drawing different colored marbles from a bag without and with replacement.

1. Chapter 2 Notes
Lesson 2.1
Lesson 2.4
Lesson 2.2
Lesson 2.5
Lesson 2.3
Lesson 2.6
Lesson 2.7

2. 2.1 Adding Rational Numbers
identity property of addition
n+0=n
additive inverse
a when the opposite is added to a number the sum equals zero
additive identity property
a number plus its opposite equals zero
x + (-x) = 0
adding numbers with same signs
add numbers together the answer will have the same sign
adding numbers with different signs
subtract numbers; the answer will have the same sign as the larger number (not looking at sign)

3. Examples
-2 + (-6)
-8
4
0
4 + (-8)
16
-9 + 9
9+7
The temperature falls 15 degrees and then rises 18 degrees.
Use addition to find the change in temperature.
Evaluate the expression
n = -4 m=5
m+(-4) -n + 5
5 + (-4) = 1 -(-4)+5 = 9

4. Matrix
[ ]
4 6 Row
5
7 1 8
2 rows by 3 columns
Column
] [ ]
[
-2 5 4 -1
+ 3 0 =
7 1
[ ] [ ]
-2 + 4 5+(-1) = 2 4
7+3 1+0 10 1
End

5. 2.2 Subtracting Rational numbers
Subtracting numbers
change the second number to its opposite and follow the addition rules
Absolute value
do all the operations inside the absolute value then take the positive of the answer

6. Examples
2 - (-6)
4 - (-8)
-4 - 5
9- 2
Evaluate:
for n = -3 and m = 2
n-6 -m - 8

7. Examples
| -8 - 6| |6-2|
] [ ]
[
-2 5 4 -1
7 1 - 3 0
[ ] [ ]
-2 - 4 5-(-1) -6 6
=
7-3 1-0 4 1

8. 2.3 Multiplying and Dividing Rational Numbers
Identity Property of Multiplication
n*1=n
Multiplication Property of Zero
n*0=0
Multiplication Property of -1
n * (-1) = -n
Multiplying or Dividing Numbers with same sign
Multiply or Divide numbers together; answer is POSITIVE
Multiplying or Dividing Numbers with different signs
Multiply or Dividing numbers together; answer is NEGATIVE

9. Examples
-5 * (-10) 7*8
-8 * 5 -50
2
-30 75
-3 25
Evaluate:
for n = -2 and m = 8
8m -n2

10. Matrix scalar multiplication
] [ ]
[
5(4) 5(-5) 5(-1)
5 4 -5 -1 = 5(3) 5(0) 5(-9)
3 0 -9
]
[
Pull
20 -25 -5
15 0 -45

11. 2.4 The Distributive Property
Distributive property
a(b + c) = a(b) + a(c)

12. Example
2(3 + 7)
2(3) + 2(7)
6 + 14
20
3(4x -9)
3(4x) - 3(9)
12x - 27
-(6n + 8)
-1(6n + 8)
-6n - 8

13. Example
Like Terms
5a2 - 9ab - 18
Constant
coefficient
8x + 3(x + 4)
8x + 3x +12
11x + 12
End

14. 2.5 Properties of numbers
Commutative Property
a+b=b+a
a*b=b*a
Associative Property
(a + b) + c = a + (b + c)
(a * b) * c = a * (b * c)
Identity Property
a+0=a
a*1=a
Inverse Property
a + -a = 0
a * (1/a) = 1
Symmetric Property
if a = b then b = a

15. Distributive property
a(b + c) = ab + ac
a(b - c) = ab - ac
Multiplication Property of Zero
n*0=0
Mutliplication Property of -1
n * -1 = -n

16. End

17. 2.6 Theorectical and Experimental Probability
Probability Favorable outcome
total number of outcomes
Outcome
result of a single trial
Sample Space
all possible outcomes
Event
an outcome or group of outcomes
Theoretical Probability
how an event should turn out
Compliment of an event
the probability an event will not occur
Experimental Probability
how an event did turn out from a trial(s)

18. Example
P (green) = 1
5 = 20%
P (Red) = 2 = 40%
5
P ( not red) = 3 = 60%
5
Red Pink Orange Blue

19. 2.7 Probability of Compound events
Independant Events
events that do not effect each other
P ( A and B) = P(A) * P(B)
Dependant Event
events that do effect each other
P ( A and B) = P(A) * P(B after A)
End

20. You have a bag of marbles with 8 red marbles, 10 blue marbles,
7 yellow, and 5 black marbles.
What are the following probabilities WITHOUT replacement?
P( yellow and black)
P(2 reds)
P(purple)
What are the following probabilities with replacement?
P( yellow and black)
P(2 reds)
P(purple)