Transcript 1.1 Algebraic Expression and Real Numbers

Chapter 1
Algebra, Mathematical Models,
and Problem Solving
§ 1.1
Algebraic Expressions and Real Numbers
Variables in Algebra
Algebra uses letters such as x and y to
represent numbers. If a letter is used to
represent various numbers, it is called a
variable. For example, the variable x might
represent the number of minutes you can lie
in the sun without burning when you are not
wearing sunscreen.
Blitzer, Algebra for College Students, 6e – Slide #3 Section 1.1
Suppose you are wearing number 6
sunscreen. If you can normally lie in the sun
x minutes without burning, with the number
6 sunscreen, you can lie in the sun 6 times as
long without burning - that is, 6 times x or
6x would represent your exposure time
without burning.
Blitzer, Algebra for College Students, 6e – Slide #4 Section 1.1
Algebraic Expressions
A combination of variables and numbers
using the operations of addition, subtraction,
multiplication, or division, as well as powers
or roots, is called an algebraic expression.
Blitzer, Algebra for College Students, 6e – Slide #5 Section 1.1
Translating Phrases into Expressions
English Phrase
sum
plus
increased by
more than
difference
minus
decreased by
less than
product
times
of (used with fractions)
twice
quotient
divide
per
ratio
Mathematical Operation
Addition
Subtraction
Multiplication
Division
Blitzer, Algebra for College Students, 6e – Slide #6 Section 1.1
Translating Phrases into Expressions
EXAMPLE
Write the English phrase as an algebraic expression. Let x represent
the number.
Four more than five times a number
SOLUTION
5x  4
Blitzer, Algebra for College Students, 6e – Slide #7 Section 1.1
Evaluating an Algebraic Expression
EXAMPLE
The formula C  F  32  expresses the relationship between
9
Fahrenheit temperature, F, and Celsius temperature, C. Use the
formula to convert the given Fahrenheit temperature to its
equivalent temperature on the Celsius scale.
5
50F
SOLUTION
5
C  F  32 
9
C
5
50  32
9
Replace F with 50
Blitzer, Algebra for College Students, 6e – Slide #8 Section 1.1
Evaluating an Algebraic Expression
CONTINUED
5
C  18
9
Subtract
C  10
Multiply
Therefore 50F  10C .
Blitzer, Algebra for College Students, 6e – Slide #9 Section 1.1
In evaluating expressions, what comes first?
• #1 Start with the
parentheses. Parentheses
say “Me First!”
• #3 Multiplications and
divisions are equal in the
order of operations –
Perform them next.
• #2 Then evaluate the
exponential expressions.
• #4 Additions and
subtractions are also equal
to each other in order – and
they come last.
Remember by “PEMDAS” parentheses, exponents, multiplication, division, addition, subtraction
Blitzer, Algebra for College Students, 6e – Slide #10 Section 1.1
Order of Operations - PEMDAS
Order of Operations
1) First, perform all operations within
grouping symbols
2) Next, Evaluate all exponential
expressions.
3) Next, do all multiplications and
divisions in the order in which they
occur working from left to right.
4) Finally, do all additions and
subtractions in the order in which they
occur, working from left to right.
Blitzer, Algebra for College Students, 6e – Slide #11 Section 1.1
Order of Operations - PEMDAS
EXAMPLE
Evaluate R 3  26  R 4 for R  3 .
SOLUTION
R  26  R 
3
3  26  3
3
3  23
3
4
27  23
4
4
4
Replace R with 3
Evaluate inside parentheses first
3
Evaluate 3 – first exponent
Blitzer, Algebra for College Students, 6e – Slide #12 Section 1.1
Order of Operations - PEMDAS
CONTINUED
4
Evaluate 3 – second exponent
27-2(81)
27-162
Multiply
-135
Subtract
Blitzer, Algebra for College Students, 6e – Slide #13 Section 1.1
Number Sets
Sets of Numbers
Definition
Natural Numbers
All numbers in the set {1,2,3,4,…}
Whole Numbers
All numbers in the set {0,1,2,3,4,…}
Integers
All numbers in the set {…-3,-2,-1,0,1,2,3,…}
Rational Numbers
All numbers a/b such that a and b are integers
Irrational Numbers
All numbers whose decimal representation
neither terminate nor repeat
Real Numbers
All numbers that are rational or irrational
NOTE: “…” means continue without end
Blitzer, Algebra for College Students, 6e – Slide #14 Section 1.1
Three Common Number Sets
Note that…
The natural numbers are the numbers we use for counting.
The set of whole numbers includes the natural numbers and
0. Zero is a whole number, but is not a natural number.
The set of integers includes all the whole numbers and
their negatives. Every whole number is an integer, and
every natural number is an integer.
These sets are just getting bigger and bigger…
Blitzer, Algebra for College Students, 6e – Slide #15 Section 1.1
Set-Builder Notation
EXAMPLE
Express x > 10 using set-builder notation
SOLUTION
{x | x is a real number and greater than 10}
Blitzer, Algebra for College Students, 6e – Slide #16 Section 1.1
Rational Numbers
Definition
The set of rational numbers is the set of all numbers that can
be expressed as the quotient of two integers with the
denominator not zero.
That is, a rational number is any number that can be written in
the form a/b where a and b are integers and b is not zero.
Rational numbers can be expressed either in fraction or in
decimal notation. Every integer is rational because it can be
written in terms of division by one.
Blitzer, Algebra for College Students, 6e – Slide #17 Section 1.1
Symbols
 
and

The symbol
is used to indicate that a number or object is
in a particular set. Here is an example:
7  {1,2,5,7,9}

The symbol
is used to indicate that a number or object is
not in a particular set. For example:
3
 {4,6}
Blitzer, Algebra for College Students, 6e – Slide #18 Section 1.1
Inequalities
Inequalities
<
>

Meanings
Examples
is less than
10 < 32
-5 < 3
-7 < -2
is greater than
is less than or is equal to

is greater than or is equal to
Blitzer, Algebra for College Students, 6e – Slide #19 Section 1.1
6 > -4
11 > 8
-6 > -12
3.4
-2


4.5
-2
5 5
0  -3