Post on 19-Dec-2015
transcript
Some Basics of Algebra• Algebraic Expressions and Their
Use
• Translating to Algebraic Expressions
• Evaluating Algebraic Expressions
• Sets of Numbers
1.1
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:
, 2 2 , .4
yt l w m x b
When an equal sign is placed between two expressions, an equation is formed.
Translating to Algebraic Expressions
per of less than more than
ratio twicedecreased byincreased by
quotient of times minus plus
divided byproduct ofdifference of sum of
divide multiply subtract add
DivisionMultiplicationSubtractionAddition
Key Words
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
8 for 2, 7, and 3.xz y x y z
Solution
8xz – y = 8·2·3 – 7
= 41
= 48 – 7
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.
1 1
2 2b h
Solution
10·3.1
= 15.5 square feet
h
b
Exponential Notation The expression an, in which n is a counting number means
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.
a a a a 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
Solution
2(x + 3)2 – 12 x2
Substituting
Simplifying 52 and 22
Multiplying and Dividing
Subtracting
2 22 3 12 for 2.x x x
= 2(2 + 3)2 – 12 22
2 22 5 12 2 2 25 12 4
Working within parentheses
50 3 47
Example
Evaluate the expression 24 2 for 3, 2, and 8.x xy z x y z
Solution
4x2 + 2xy – z = 4·32 + 2·3·2 – 8
= 36 + 12 – 8
= 40
= 4·9 + 2·3·2 – 8
Substituting
Simplifying 32
Multiplying
Adding and Subtracting
Part 2 of 1.1
Sets of Numbers
Sets of NumbersNatural 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,…}
Rational Numbers
Numbers that can be expressed as an integer divided by a nonzero integer are called rational numbers:
Sets of Numbers
is an integer, is an integer, and 0 .p
p q qq
Converting Fractions to DecimalsDivide the numerator by the denominator
6
5 .8 3 3 3
6 5.0 0 0
4 8
2 0
1 8
2 0
38.0
...8333.0
36.011
4
6.03
2
375.08
3
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.
1
1414
1
33
Look at the following conclusion.
Rationals include the following.
• All integers (-2, 5, 17, 0)• All fractions (proper,
improper, or mixed)
• All terminating decimals
• All repeating decimals
2
11,
8
7,
11
91
-2.34, 0.0456
784.3 ,2.1
Sets of Numbers
Real Numbers
Numbers that are either rational or irrational are called real numbers:
is rational or irrational .x x
Numbers like are said to be irrational. Decimal notation for irrational numbers neither terminates nor repeats.
5 and
Identify as natural,whole, integers, rational, or
irrational.
5 -3 105
4
0.457 16 0 6.2
3
25
2
342.0
9
400
Answers
• Natural: 10
• Whole: 10 0
• Integers: 10 0 -3
• Rational:
• Irrational:
16
16
16
9
400 ,342.0 ,
3
25 ,2.6 ,0 ,16 ,457.0 ,
5
4 ,10, 3
2 ,5
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 Notation12} and 5between number even an is |{ xx
}10,8,6{
7}least at number natural a is |{ xx
}7,6,5,4,3,2,1{
Write with Set-Builder Notation
1} and 6-between integer an is |{ xx
}17,15,13,11,9{ )2
18} and 8between number oddan is |{ xx
1) The set of all integers between -6 and 1
5)
33} and 21between 4 of multiple a is |{ xx
}32,28,24{
Elements and SubsetsIf 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
NW
ZH
RQ
Q
H
W
6
5
4
3.2
Answers
• False
• True
• False
• True
• True
• False