
Skills to Master
·
Determine the
real number sets to which a given number belongs
·
Name and use the
properties of real numbers
·
Graph subsets of
the real numbers
·
Use inequality
symbols
·
Use interval
notation and convert between interval notation, inequalities, and graphs for a
given subset of the real numbers
·
Write an
expression containing natural number exponents as one without exponents
·
Know and use the
rules of exponents
·
Know and use the
order of operations
·
Evaluate
expressions
·
Write numbers in
scientific notation and convert between decimal notation and scientific
notation
Number Sets
Natural Numbers {1, 2, 3, …}
Whole Numbers {0, 1, 2, …}
Integers {…, -2, -1, 0, 1, 2, …}
Rational Numbers ![]()
Irrational Numbers
is not rational}
Real Numbers {x:
x is rational or x is irrational}
PRACTICE
Classify the following numbers:
4_________________________
-7_________________________
3/5________________________
1.4________________________
0__________________________
π_________________________
Number Properties
The following properties are assumed to be true for
all real numbers:
·
Addition properties:
o
Closure:
adding any two real numbers results in a real number
o
Associativity:
for any real numbers a, b, and c, ![]()
o
Commutativity:
for any real numbers a and b, ![]()
o
Identity
property of zero: for any real number a, ![]()
o
Additive
inverse property: for any real number
,
there exists a unique
such that ![]()
·
Multiplication properties:
o
Closure:
multiplying any two real numbers results in a real number
o
Associativity:
for any real numbers a, b, and c, ![]()
o
Commutativity:
for any real numbers a and b, ![]()
o
Identity
property of one: for any real number a, ![]()
o
Multiplicative
inverse property: for any real number
,
there exists a unique
such that ![]()
·
Distributive property:
o
For any real numbers a, b,
and c,
and ![]()
Examples:
Name the number property:
![]()
_____________________
![]()
_____________________
![]()
_________________________
![]()
_________________________
Inequalities
|
Symbol |
Read as |
Examples |
|
|
|
“not equal to” |
|
|
|
|
“is less than” |
|
|
|
|
“is greater than” |
|
|
|
|
“is less than or equal to |
|
|
|
|
“is greater than or equal to” |
|
|
|
|
“is approximately equal to” |
|
|
Intervals and Interval Notation
For any real numbers a and b:
is equivalent to ![]()
is
equivalent to ![]()
is
equivalent to ![]()
is
equivalent to ![]()
is
equivalent to ![]()
is
equivalent to ![]()
is
equivalent to ![]()
is
equivalent to ![]()
Natural Number Exponents
Addition of two or more of
the same number can be written as a multiplication. For example,
. With variables:
. Exponents serve a
similar purpose, but for multiplication of several of the same number. For example,
. With variables,
. MAKE SURE YOU KNOW THE DIFFERENCE BETWEEN THESE FORMS OF
NOTATION! Exponents can only be affected
by multiplication.
In exponential notation, the b in the previous example is called the base and the 5 is the power to which the base is raised. If no exponent is given, the base is
understood to have the exponent 1. That
is,
. We will later see that
for all real numbers
other than zero.
Rules for Exponents
For any real number x and for natural numbers m and n:
![]()
![]()
![]()
![]()
For any real number y:
![]()
![]()
![]()
Examples:
Simplify.
![]()
![]()
![]()
Order of Operations and Evaluating Expressions
When
an expression includes several different operations, the operations must be
performed in the following order:
1.
Parentheses (or
other grouping symbols)—evaluate all expressions within parentheses
2.
Exponents—evaluate
all exponent operations
3.
Multiplication/Division—perform
multiplication and division operations as they occur from left to right.
4.
Addition/Subtraction—perform
addition and subtraction operations as they occur from left to right.
Example:
Evaluate ![]()
An
the variables in an expression are assigned values, the values can be
substituted in the expression. The
expression can then be evaluated.
Example:
Evaluate
the given expression for x = -2, y = 3, and z = 4
![]()
![]()
Scientific Notation
Format:
, where
and n is an integer.
Convert
from decimal to scientific notation:
1. Go to the decimal and move it either left or right
until there is a single digit to the
left of the decimal point.
2. Write a
after the last digit and
the exponent is the same as the number of places you moved the decimal. If the movement was to the left, the exponent is positive; if to
the right, the exponent is negative.
3. Examples:
![]()
![]()
![]()
![]()
![]()
Convert
from scientific notation to decimal:
1. Go to the decimal point and move it either:
a.
The same number
of spaces to the right as the
exponent on the 10 if the exponent is positive
b.
The same number
of spaces to the left as the
exponent on the 10 if the exponent is negative
2. Write your answer without the ![]()
3. Examples
![]()
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