Some Basics of Algebra
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Transcript Some Basics of Algebra
1.1
Some Basics of Algebra
• Algebraic Expressions and Their
Use
• Translating to Algebraic
Expressions
• Evaluating Algebraic Expressions
• Sets of Numbers
Terminology
A letter that can be any one of various numbers
is called a variable. If a letter always
represents a particular number that never
changes, it is called a constant.
Algebraic Expressions
An algebraic expression consists of
variables, numbers, and operation signs.
Examples:
y
t,
4
2 l 2 w,
m x b.
When an equal sign is placed between two
expressions, an equation is formed.
Translating to Algebraic
Expressions
Key Words
Addition
add
sum of
plus
Subtraction
subtract
difference of
minus
increased by
decreased by
more than
less than
Multiplication
multiply
product of
times
Division
divide
divided by
quotient of
twice
ratio
of
per
Example
Translate to an algebraic expression:
Eight more than twice the product of 5
and a number.
Solution
8 2 5 n
Eight more than twice the product of 5 and a number.
Evaluating Algebraic
Expressions
When we replace a variable with a number,
we are substituting for the variable. The
calculation that follows is called evaluating
the expression.
Example
Evaluate the expression
8xz y for x 2, y 7, and z 3.
Solution
8xz – y = 8·2·3 – 7
= 48 – 7
= 41
Substituting
Multiplying
Subtracting
Example
The base of a triangle is 10 feet and the
height is 3.1 feet. Find the area of the
triangle.
Solution
h
1
1
b h 10·3.1
2
2
= 15.5 square feet
b
Exponential Notation
The expression an, in which n is a counting
number means
a a a a a
n factors
In an, a is called the base and n is called the
exponent, or power. When no exponent
appears, it is assumed to be 1. Thus a1 = a.
Rules for Order of Operations
1. Simplify within any grouping symbols.
2. Simplify all exponential expressions.
3. Perform all multiplication and division
working from left to right.
4. Perform all addition and subtraction
working from left to right.
Example
Evaluate the expression
2
2
2 x 3 12 x for x 2.
Solution
2(x + 3)2 – 12 x2
= 2(2 + 3)2 – 12 22
2
2
2 5 12 2
2 25 12 4
50 3
47
Substituting
Working within parentheses
Simplifying 52 and 22
Multiplying and Dividing
Subtracting
Example
Evaluate the expression
4 x 2 xy z for x 3, y 2, and z 8.
2
Solution
4x2 + 2xy – z = 4·32 + 2·3·2 – 8
= 4·9 + 2·3·2 – 8
= 36 + 12 – 8
= 40
Substituting
Simplifying 32
Multiplying
Adding and
Subtracting
Part 2 of 1.1
Sets of Numbers
Sets of Numbers
Natural Numbers (Counting Numbers)
Numbers used for counting: {1, 2, 3,…}
Whole Numbers
The set of natural numbers with 0 included:
{0, 1, 2, 3,…}
Integers
The set of all whole numbers and their
opposites: {…,-3, -2, -1, 0, 1, 2, 3,…}
Sets of Numbers
Rational Numbers
Numbers that can be expressed as an
integer divided by a nonzero integer are
called rational numbers:
p
p is an integer, q is an integer, and q 0 .
q
Converting Fractions to Decimals
Divide the numerator by the denominator
5
6
.8 3 3 3
6 5.0 0 0
0.8333...
0.83
48
20
18
20
3
0.375
8
2
0.6
3
4
0.36
11
Any fraction can be
converted to a
repeating decimal or a
terminating decimal..
All integers can be written as
fractions. Insert a denominator of 1.
3
3
1
14
14
1
Look at the following conclusion.
Rationals include the following.
• All integers (-2, 5, 17, 0)
• All fractions (proper,
improper, or mixed)
• All terminating decimals
9 7 11
1 , ,
11 8 2
-2.34, 0.0456
• All repeating decimals
1.2, 3.478
Sets of Numbers
Numbers like 5 and are said to be
irrational. Decimal notation for irrational
numbers neither terminates nor repeats.
Real Numbers
Numbers that are either rational or
irrational are called real numbers:
x x is rational or irrational.
Identify as natural,
whole, integers, rational, or
irrational.
5
-3
2
5
3
10
16
0.457
2
4
5
0
0.234
6.2
400
9
Answers
16
• Natural: 10
• Whole: 10
• Integers: 10
0
16
0
-3
16
4
2
400
3 ,10, , 0.457, 16 , 0, 6.2, 5 , 0.234,
5
3
9
• Irrational: 5 ,
• Rational:
2
Set Notation
Roster notation:
{2, 4, 6, 8}
Set-builder notation:
{x | x is an even number between 1 and 9}
“The set of all x
such that x is an even number between 1 and 9”
Write with Roster Notation
{x | x is an even number between 5 and 12}
{6,8,10}
{x | x is a natural number at least 7}
{1,2,3,4,5,6,7}
Write with Set-Builder Notation
1)
The set of all integers between -6 and 1
{x | x is an integer between - 6 and 1}
2)
{9,11,13,15,17}
{x | x is an odd number between 8 and 18}
5)
{24,28,32}
{x | x is a multiple of 4 between 21 and 33}
Elements and Subsets
If B = { 1, 3, 5, 7}, we can write 3 B to
indicate that 3 is an element or member of
set B. We can also write 4 B to indicate
that 4 is not an element of set B.
When all the members of one set are members
of a second set, the first is a subset of the
second. If A = {1, 3} and B = { 1, 3, 5, 7}, we
write A B to indicate that A is a subset of B.
True or False?
Use the following sets: N= Naturals, W = Wholes,
Z = integers, Q = Rationals, H = Irrationals,
and R = Reals
2.3 W
4
H
5
6 Q
QR
H Z
W N
Answers
•
•
•
•
•
•
False
True
False
True
True
False