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CHAPTER 1
SECTION 1-1
LANGUAGE OF
MATHEMATICS
SET– a collection or group
of, things, objects,
numbers, etc.
INFINITE SET – a set
whose members cannot
be counted.
If A= {1, 2, 3, 4, 5,…}
then A is infinite
FINITE SET – a set
whose members can be
counted.
If A= {e, f, g, h, i, j}
then A is finite and
contains six elements
SUBSET – all members
of a set are members of
another set
If A= {e, f, g, h, i, j}
and B = {e, i} , then
BA
EMPTY SET or NULL
SET – a set having no
elements.
A= { } or B = { } are
empty sets or null sets
written as 
The EMPTY SET is a
subset of every set
Every SET is a subset
of itself
VARIABLE – represents an
unknown number or
quantity and is usually
denoted by a letter such as
a, n, x, y, z
VARIABLE EXPRESSION
a statement containing a
number and/or variables
2x + 4
-10x – 22y – 33z
EQUATION – a statement
that two numbers or
expressions are equal.
-6 + 10 = 6 – 2 or
4x + 3 = 19
TRUE/FALSE
•9  { -3, 0, 3, 6,…}
•{a, b}  {a, b, e, i}
•The subsets of {b, c}
are {b}, {c}, {b, c}, 
SECTION 1-2
REAL NUMBERS
NATURAL NUMBERS
set of counting numbers
{1, 2, 3, 4, 5, 6, 7, 8…}
WHOLE NUMBERS - set
of counting numbers plus
zero
{0, 1, 2, 3, 4, 5, 6, 7, 8…}
INTEGERS - set of the
whole numbers plus their
opposites
{…, -3, -2, -1, 0, 1, 2, 3, …}
RATIONAL NUMBERS
numbers that can be
expressed as a ratio of
two integers a and b and
includes fractions,
repeating decimals, and
terminating decimals
EXAMPLES OF
RATIONAL
NUMBERS
½, ¾, ¼, - ½, -¾, -¼, .05
.76, .333…, .666…, etc
.
IRRATIONAL
NUMBERS
numbers that cannot be
expressed as a ratio of
two integers a and b and
can still be designated
on a number line
EXAMPLES OF
IRRATIONAL
NUMBERS
, 6, 29,
8.11211121114…, etc
.
The real number paired
with each point is the
coordinate of that point.
The distance between
any two points on the line
is equal to the absolute
value of the difference of
their coordinates.
SECTION 1-3
UNION AND
INTERSECTION OF
SETS
UNION OF SETS
Two or more sets
joining together to
form a new set
If A = {1,2,3} and
B = {-1, -2, -3} then
A  B = {-3, -2, -1, 1, 2, 3}
.
INTERSECTION OF
SETS
Two sets containing
elements common to
both sets
If A = {1, 2, 3} and
B = {-1, 0, 1} then
A  B = {1}
.
SYMBOLS
• - union of sets
• - intersection of
sets
• - complement of a
set (not)
VENN DIAGRAM
Diagram using circles
inside a rectangle to
represent the union
and intersection of
sets
VENN DIAGRAM
A = {4,5,6,7, 8, 9}
B = {8, 9, 12, 15, 16}
C = {18,20}
• Find A  B
• Find A  C
• Find B  C
A = {4,5,6,7, 8, 9}
B = {8, 9, 12, 15, 16}
C = {18,20}
• Find A  B
• Find A  C
• Find B  C
A = {4,5,6,7, 8, 9}
B = {8, 9, 12, 15,16}
C = {18,20}
• Find A 
• Find (A  B) 
• Find (A  C) 
SECTION 1-4
ADDITION,
SUBTRACTION AND
ESTIMATION
CLOSURE
PROPERTY
a + b is unique
7 + 5 = 12
COMMUTATIVE
PROPERTY
a+b =b+a
25 + 60 = 60 + 25
ASSOCIATIVE
PROPERTY
(a + b) + c = a + (b +c)
(5 + 15) + 20 = 5 + (15 +20)
IDENTITY
PROPERTY
a+0=0+a=a
-3 + 0 = 0 + -3 = -3
INVERSE
PROPERTY
a +(-a) = 0
-2 + (2) = 0
For real numbers a
and b
1.If a and b are negative
numbers, then a + b is
negative.
• - 16 + (-4) = -20
• (-14) -6 = -20
For real numbers a
and b
2.If a is a positive number, b
is a negative number, then
the sign of the sum of a + b
will be the sign of the
largest number when the
signs are ignored.
-9 + 5 = - 4
SECTION 1-5
MULTIPLICATION
AND DIVISION
CLOSURE
PROPERTY
ab is unique
7 • 5 = 35
COMMUTATIVE
PROPERTY
ab = ba
25 • 60 = 60 • 25
ASSOCIATIVE
PROPERTY
(ab)c = a(bc)
(5•15) • 20 = 5(15• 20)
IDENTITY
PROPERTY
a•1=1•a=a
-3 • 1 = 1 • -3 = -3
INVERSE
PROPERTY
a •1/a= 1/a •1 = 1
-2 • (-1/2)= 1
ZERO
PROPERTY
a •0= 0 •a = 0
5 • 0 = 0 •5 = 0
SECTION 1-7
DISTRIBUTIVE
PROPERTY AND
PROPERTIES OF
EXPONENTS
DISTRIBUTIVE
PROPERTY
a(b + c) = ab + ac
5(12 + 3) = 5•12 + 5 •3 = 75
EXPONENTIAL FORM –
number written such that it
has a base and an
exponent
3
4
= 4 •4 •4
BASE – tells what
factor is being
multiplied
EXPONENT – Tells
how many equal
factors there are
PROPERTY OF
EXPONENTS FOR
MULTIPLICATION
m
n
m+n
a •a = a
2
4
5 •5
=
2+4
5
=
6
5
PROPERTY OF
EXPONENTS FOR
MULTIPLICATION
m)n
mn
(a = a
2)4
(5
=
2·4
5 =
8
5
PROPERTY OF
EXPONENTS FOR
MULTIPLICATION
m
m
m
(ab) = a b
3
(5·3)
=
3
5
3
·3
PROPERTY OF
EXPONENTS FOR
DIVISION
m
n
m-n
a ÷a = a
5
2
5 5
=
5-2
5
=
3
5
PROPERTY OF
NEGATIVE
EXPONENTS
-m
m
a = 1/a
-2
5 =
2
1/5
= 1/25
PROPERTY OF
ZERO EXPONENT
0
a =
1
0
5 =
1
SECTION 1-8
EXPONENTS AND
SCIENTIFIC
NOTATION
SCIENTIFIC NOTATION –
A number having two
factors. The first factor
is greater than or equal
to 1 and less than 10.
The second is a power of
10
STANDARD FORM
– the customary
way numbers are
written
5,283.45
.00789
SCIENTIFIC FORM
496,000 = 4.96
.00059 = 5.9
5
•10
-4
•10
YEAH! THE END