Download Natural Numbers: The counting numbers starting at 1: {1, 2, 3,

1.1
Natural Numbers,
Whole Numbers,
Integers,
: The counting numbers starting at 1: {1, 2, 3, ... }.
:
The counting numbers again, but starting at 0: {0, 1, 2, 3, ... }.
: The whole numbers and their opposites: {…, -3, -2, -1, 0, 1, 2, 3, ... }.
Rational numbers,
: Rational numbers are all the possible ratios of two integers,
provided that 0 is not in the denominator. It is important to note that as decimals, rational
numbers either stop or repeat. For example, ½ = 0.5 (stops), 1/3 = 0.33333… (repeats), 1/4 =
0.25 (stops), 1/5 = 0.2 (stops), 1/6 = 0.166666… (repeats), 6/2 = 3 (stops), 60/4 = 15 (stops),
etc.
Irrational numbers,
: Irrational numbers are all the numbers that can’t be written as a
ratio of two integers. It is important to note that as decimals, irrational numbers neither stop
nor repeat. For example,  is an irrational number that is approximately 3.14159. When we
write 3.14159, we are actually writing a rational number approximation for the irrational
number . Another example of an irrational number is the square root of 2. The square root
of 2 does not stop and it does not have a repeating part. It is approximately 1.4142, but again
1.4142 is a rational number approximation.
Real numbers,
numbers.
: The real numbers are all the rational numbers and all the irrational
Prime number: A prime is a natural number, greater than 1, that has only 2 factors. The
factors of a prime number are 1 and itself. The only way to get a prime number by
multiplication is taking 1 times the prime number. {2, 3, 5, 7, 11, 13, 17, 19, …}
Composite number: A composite number is a natural number, greater than 1, that is not
prime. Composite numbers have more than 2 factors. {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, …}
Which are prime and which are composite?
39?
48?
57?
59?
The Intersection of two sets, denoted
, is only those things that are in the first set and in
the second set, just as the intersection of two streets is only the part that is in the first street
and in the second street, the part that they share in common. If we look at a recipe for cake
and a recipe for quiche lorraine, the things that the two recipes have in common would be the
intersection, namely flour, eggs, and salt.
The Union of two sets, denoted
, is all the things that are in the first set or in the second
set, we just put the two streets together, not worrying about what they share in common. If
we look at a recipe for cake and a recipe for quiche lorraine, the union of the recipes would
not be appetizing—it would be a cake flavored with pie crust, bacon, onion, and swiss cheese.
A  k | k is prime, k  , k is between 1 and 14 and
B  k | k is odd, k  , k is between -1 and 16
If
then that means that A={2, 3, 5, 7, 11, 13} and B={1, 3, 5, 7, 9, 11, 13, 15}.
What is
A
B?
What is
A
B?
We might use a factor tree:
… or repeated division by prime factors:
When we write a number as a product of primes, we call it the prime factorization of the
number.
Subset: One set A is a subset of another set B if everything in A is in B. We write this
A  B . We could say that the set of all Saturn automobiles is a subset of all GM vehicles,
every Saturn is a GM car.
Element of: An item V is an element of a set B if it comes from or is contained in the set B.
We write this V  B . We could say that I am an element of the faculty at SCC and write
Steve Willott  SCC Faculty . Some people read the symbol as “is a member of”,
“comes from”, “is a type of”, “is an element of”, or “is one of”.

True of false:


3 

3 
4
5


2 

9
To recap how to graph inequalities on a number line:
 greater than (open circle or parentheses)
 greater than or equal to (closed circle or brackets)
< less than (open circle or parentheses)
 less than or equal to (closed circle or brackets)
For example:
graph
x2
graph
x2
graph
x2
graph
x  3
graph
3  x
graph
3 x
Algebraic properties are like law; studying law can be very dry and studying properties of
algebra can also be very dry. We may not know which part of legal code says we can’t drive
150 mph on Interstate 70, but we know that there is such a law (and we can research it). We
often don’t know the particular property that lets us change the direction that we multiply two
numbers, but we know that changing the order we multiply is allowed (and now we are going
to research it).
Properties are on pages 6-9. Which properties (mathematical legal code) are used here:
(2  x)5  10  5 x
(2  x)5  5(2  x)
(2  x)  5  2  ( x  5)
7 25  95
1.2
Order of Operations
Parentheses and Exponents
Multiplication and Division
Addition and Subtraction
(9-6)(2-5)
-5(6-2[3-2(-2-2)])
PE
MD
AS
Please Excuse
My Dear
Aunt Sally
-13-(2-5)
-2
23 +5[32 -2(5-2(3)+2)+1]+7
1.3
Like terms have the same combination of variables and exponents on those variables.
Like terms can be added, subtracted, multiplied, or divided.
Unlike terms either have different combinations of variables and/or have different
combinations of exponents on the variables. Unlike terms can’t be added or subtracted; they
can only be multiplied or divided,
5a+4a+3ab-2ab
5a(-3a)
3(x+2y)+4(3x-2y)
x(x 3 )(x 5 )
(x 3 )5
-2x 4 (x 7 )
(-2x)4 (x 7 )
22
 2 2
y2 + y[y+y(y-4)]
x-5[x-5(x-5)]
1.4 Translating into Algebraic Form
a. Nine more than a number
b. Two less than three times a number
c. There are two numbers, the smaller of which is 9 less than 4 times the larger.
d. Represent four consecutive odd integers.
e. Represent the cube of the sum of two consecutive integers.
page 39, #28.
The length of a rectangular plot of land is 20 meters and its width is 12 meters. Heavy- duty
fence for the length costs $5 per meter, and regular fence for the width costs $2 per meter.
(a) How much regular fence is needed?
(b) How much will the regular fence cost?
(c) How much heavy-duty fence is needed?
(d) How much will the heavy-duty fence cost?
(e) What will the total cost for fencing the plot of land be?
1.5 First-degree equations and inequalitites
#6 Which of these satisfy the equation?
#24 Solve 4-3b = 5b +7
#44 Solve
7 x
7x- =
8 4
#58 Solve 2(x-y) =3x+4 for y
x= -4, 3, 6 6x-x 2 =0
#34 Solve 4(x-3)-2x=4(x-1)
#56 Solve 2x-9y=11 for x