Transcript PPT

Real Numbers and Algebraic
Expressions
• Recognize subsets of the real number.
•Use inequality symbols.
•Evaluate absolute value to express distance.
•Algebraic expressions and their evaluation
•Properties of algebraic operations
•Understand and use of integer exponents
•Properties of exponents
•Simplify exponential expressions
•Use of scientific notation.
H.Gevorgyan/1100/04
HamestGevorgyan
Departmen of Mathematics and CS
[email protected]
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Algebraic expression

A combination of variables and numbers using
the operations of addition, subtraction, division,
or multiplication, as well as powers or roots, is
called an algebraic expression.
Example: 7x, x+11, x-9, x4 -10,
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Definition of a Natural Number Exponent

If b is a real number and n is a natural number,
n
b  b  b b ... b

bn is read “the nth power of b” or “ b to the
nth power.” Thus, the nth power of b is
defined as the product of n factors of b.
Furthermore, b1 = b
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The Order of Operations Agreement
1.
Perform operations within the innermost parentheses and work
outward. If the algebraic expression involves division, treat the
numerator and the denominator as if they were each enclosed in
parentheses.
2.
Evaluate all exponential expressions.
3.
Perform multiplication or division as they occur, working from left
to right.
4. Perform addition or subtraction as they occur, working from left to
right.
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The Basics About Sets
The set {1, 3, 5, 7, 9} has five elements.

A set is a collection of objects whose contents can be clearly determined.
•
The objects in a set are called the elements of the set.
•
We use braces to indicate a set and commas to separate the elements of
that set.
For example,
The set of counting numbers can be represented by {1, 2, 3, … }.
The set of even counting numbers are {2, 4, 6, …}.
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The set of even counting numbers is a
subset of the set of counting numbers,
since each element of the subset is
also contained in the set.
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Set operations
The union of two sets is the set of all elements formed by combining all the
elements of set A and all the elements of set B into one set.
The symbolism used is The Venn Diagram representing the union of A and B is
the entire region shaded yellow.
A
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A B  {x x  A or x  B}
B
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Union of Sets




Combination of everything in both sets
A = {all tall children}
B = {all girls}
A union B = {all girls OR tall children} = {all girls and
all tall boys}
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Intersection of sets A and B

The intersection of sets A and B is the set of elements that is common to both
sets A and B. It is symbolized as


A B
{ x l x ∈A and x ∈B }


Represented by Venn Diagrams:
A
B
Intersection
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Intersection of Sets
What they have in common
 A = {all tall children}
 B = {all girls}
 A intersect B = {all tall girls}
 All children that are girls AND are tall

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A group of biology majors are taking Biology I & Chem. I. A group of chemistry
majors are taking Calculus, Chem. I and Physics I. The Physics majors enrolled in
Calculus, Physics I, and Chem I. What is the intersection of the 3 groups?
1.
2.
3.
4.
Students in biology, chemistry, & physics.
Students in chemistry.
Students in calculus.
Students in physics.
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Important Subsets of the Real Numbers
Name
Description
Examples
Natural Numbers
N
{1, 2, 3, …}
These are the counting numbers
4, 7, 15
Whole Numbers
W
{0, 1, 2, 3, … }
Add 0 to the natural numbers
0, 4, 7, 15
Integers
Z
{…, -2, -1, 0, 1, 2, 3, …}
Add the negative natural
numbers to the whole numbers
-15, -7, -4, 0, 4, 7
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Important Subsets of the Real Numbers
Name
Description
Examples
Rational
Numbers
Q
These numbers can be expressed as an
integer divided by a nonzero integer:
Rational numbers can be expressed as
terminating or repeating decimals.
17 
Irrational
Numbers
I
This is the set of numbers whose decimal
representations are neither terminating nor
repeating. Irrational numbers cannot be
expressed as a quotient of integers.
17
5
,5 
,3, 2
1
1
0,2,3,5,17
2
 0.4,
5
2
 0.666666...  0.6
3
2  1.414214
 3  1.73205
  3.142

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
2
 1.571
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The Real Numbers
Rational numbers
Irrational numbers
Integers
Whole numbers
Natural numbers
The set of real numbers is formed by combining the rational numbers and
the irrational numbers.
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The Real Number Line
The real number line is a graph used to represent the set of real numbers. An
arbitrary point, called the origin, is labeled 0;
Negative numbers
-4
-3
-2
Units to the left of the
origin are negative.
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-1
Positive numbers
0
the
Origin
1
2
3
4
Units to the right of
the origin are
positive.
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Graphing on the Number Line
Real numbers are graphed on the number line by placing a
dot at the location for each number. –3, 0, and 4 are
graphed below.
-4
-3
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-2
-1
0
1
2
3
4
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Ordering the Real Numbers
On the real number line, the real numbers increase from left
to right. The lesser of two real numbers is the one farther to
the left on a number line. The greater of two real numbers is
the one farther to the right on a number line.
-2
-1
0
1
2
3
4
5
6
Since 2 is to the left of 5 on the number line, 2 is less than 5. 2 < 5
Since 5 is to the right of 2 on the number line, 5 is greater than 2. 5 > 2
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Inequality Symbols
Symbols Meaning
Example
Explanation
a<b
3<7
Because 3 < 7
7<7
Because 7 =7
b is greater than or equal to a. 7 > 3
Because 7 > 3
b>a
a is less than or equal to b.
-5 > -5
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Because -5 = -5
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Absolute Value
Absolute value describes the distance from 0 on a real number
line. If a represents a real number, the symbol |a| represents
its absolute value, read “the absolute value of a.”
For example, the real number line below shows that
|-3| = 3 and |5| = 5.
|–3| = 3
-3
-2
-1
The absolute value of –3 is 3
because –3 is 3 units from 0
on the number line.
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|5| = 5
0
1
2
3
4
5
The absolute value of 5 is 5
because 5 is 5 units from 0
on the number line.
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Definition of Absolute Value
The absolute value of x is given as follows:
{
x if x > 0
|x| =
-x if x < 0
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Properties of Absolute Value
For all real number a and b,
1. |a| > 0
2. |-a| = |a|
3. a < |a|
4. |ab| = |a||b|
5.
a |a|
= , b not equal to 0
b |b|
6. |a + b| < |a| + |b| (the triangle inequality)
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Example
Find the following: |-3| and |3|.
Solution:
| -3 | = 3 and | 3 | = 3
Distance Between Two Points on the Real Number Line
If a and b are any two points on a real number line, then the
distance between a and b is given by
|a – b| or |b – a|
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Text Example
Find the distance between –5 and 3 on the real number line.
Solution Because the distance between a and b is given by |a – b|, the
distance between –5 and 3 is |-5 – 3| = |-8| = 8.
8
-5
-4
-3
-2
-1
0
1
2
3
We obtain the same distance if we reverse the order of subtraction:
|3 – (-5)| = |8| = 8.
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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.
Here are some examples of algebraic expressions:
x + 6, x – 6, 6x, x/6, 3x + 5.
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The Order of Operations Agreement
1.
Perform operations within the innermost parentheses and work
outward. If the algebraic expression involves division, treat the
numerator and the denominator as if they were each enclosed in
parentheses.
2.
Evaluate all exponential expressions.
3.
Perform multiplication or division as they occur, working from left
to right.
4. Perform addition or subtraction as they occur, working from left to
right.
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Text Example
The algebraic expression 2.35x + 179.5 describes the
population of the United States, in millions, x years after
1980. Evaluate the expression when x = 20. Describe what
the answer means in practical terms.
Solution We begin by substituting 20 for x. Because x = 20, we
will be finding the U.S. population 20 years after 1980, in the year
2000.
2.35x + 179.5
Replace x with 20.
= 2.35(20) + 179.5
= 47 + 179.5
= 226.5
Perform the multiplication.
Perform the addition.
Thus, in 2000 the population of the United States was 226.5 million.
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Properties of the Real Numbers
Name
Meaning
Examples
Commutative
Property of
Addition
Two real numbers can be added in
any order.
a+b=b+a
• 13 + 7 = 7 + 13
• 13x + 7 = 7 + 13x
Commutative
Property of
Multiplication
Two real numbers can be multiplied
in any order.
ab = ba
• x · 6 = 6x
Associative
Property of
Addition
If 3 real numbers are added, it
makes no difference which 2 are
added first.
(a + b) + c = a + (b + c)
• 3 + ( 8 + x)
= (3 + 8) + x
= 11 + x
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Properties of the Real Numbers
Name
Meaning
Examples
Associative
If 3 real numbers are multiplied,
Property of
it makes no difference which 2
Multiplication are multiplied first.
(a · b) · c = a · (b · c)
• -2(3x) = (-2·3)x = -6x
Distributive
Multiplication distributes over
Property of
addition.
Multiplication a · (b + c) = a · b + a · c
over Addition
• 5 · (3x + 7)
= 5 · 3x + 5 · 7
= 15x + 35
Identity
Property of
Addition
• 0 + 6x = 6x
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Zero can be deleted from a sum.
a+0=a
0+a=a
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Properties of the Real Numbers
Name
Meaning
Examples
Identity
Property of
Multiplication
One can be deleted from a product.
a · 1 = a and 1 · a = a
• 1 · 2x = 2x
Inverse
Property of
Addition
The sum of a real number and its
• (-6x) + 6x = 0
additive inverse gives 0, the additive
identity.
a + (-a) = 0 and (-a) + a = 0
Inverse
Property of
Multiplication
The product of a nonzero real
number and its multiplicative
inverse gives 1, the multiplicative
identity.
a · 1/a = 1 and 1/a · a = 1
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• 2 · 1/2 = 1
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Definitions of Subtraction and Division
Let a and b represent real numbers.
Subtraction: a – b = a + (-b)
We call –b the additive inverse or opposite of b.
Division: a ÷ b = a · 1/b, where b = 0
We call 1/b the multiplicative inverse or reciprocal of b. The quotient of
a and b, a ÷ b, can be written in the form a/b, where a is the numerator
and b the denominator of the fraction.
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Text Example
Simplify: 6(2x – 4y) + 10(4x + 3y).
Solution
6(2x – 4y) + 10(4x + 3y)
= 6 · 2x – 6 · 4y + 10 · 4x + 10 · 3y
= 12x – 24y + 40x + 30y
= (12x + 40x) + (30y – 24y)
= 52x + 6y
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Use the distributive property.
Multiply.
Group like terms.
Combine like terms.
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Properties of Negatives
Let a and b represent real numbers, variables, or
algebraic expressions.
1.
(-1)a = -a
2.
-(-a) = a
3.
(-a)(b) = -ab
4.
a(-b) = -ab
5.
-(a + b) = -a - b
6.
-(a - b) = -a + b = b - a
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Definition of a Natural Number Exponent

If b is a real number and n is a natural number,
n
b  b  b b ... b

bn is read “the nth power of b” or “ b to the
nth power.” Thus, the nth power of b is
defined as the product of n factors of b.
Furthermore, b1 = b
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The Negative Exponent Rule

If b is any real number other than 0 and n is a
natural number, then
b
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n
1
 n
b
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The Zero Exponent Rule

If b is any real number other than 0,
b0 = 1.
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The Product Rule
b m · b n = b m+n
When multiplying exponential expressions with
the same base, add the exponents. Use this sum
as the exponent of the common base.
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The Power Rule (Powers to Powers)
(bm)n = bm•n
When an exponential expression is raised to a
power, multiply the exponents. Place the product
of the exponents on the base and remove the
parentheses.
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The Quotient Rule
m
b
m n
n b
b

When dividing exponential expressions with
the same nonzero base, subtract the exponent
in the denominator from the exponent in the
numerator. Use this difference as the
exponent of the common base.
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Example

Find the quotient of 43/42
Solution:
3
4
3 2
1

4

4

4
2
4
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Products to Powers
(ab)n = anbn
When a product is raised to a power, raise each
factor to the power.
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Text Example
Simplify: (-2y)4.
Solution
(-2y)4 = (-2)4y4 = 16y4
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Quotients to Powers
n
a
a
   n
b
b

n
When a quotient is raised to a power, raise
the numerator to that power and divide by
the denominator to that power.
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Example

Simplify by raising the quotient (2/3)4 to the
given power.
Solution:
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4
2
16
2
   4 
3
81
3
4
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Properties of Exponents
1
0
m n
m n
m n
mn
1. b  n 2. b  1 3. b  b  b
4. (b )  b
b
n
m
n
a  a
b
mn
n
n n
5. n  b
6. (ab)  a b
7.    n
b  b
b
n
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QUIZ#1

If the sum of q-6, q-3, and q is 0, what is the
value of q.
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Scientific Notation
The number 5.5 x 1012 is written in a form called scientific notation.
A number in scientific notation is expressed as a number greater than
or equal to 1 and less than 10 multiplied by some power of 10. It is
customary to use the multiplication symbol, x, rather than a dot in
scientific notation.
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Scientific Notation

A number is written in scientific notation when
it is expressed in the form
a x 10n
where the absolute value of a is greater than or
equal to 1 and less than 10, and n is an integer.
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Text Example

a.
b.
Write each number in decimal notation:
2.6 X 107
b. 1.016 X 10-8
Solution:
a. 2.6 x 107 can be expressed in decimal notation by moving the decimal point in
2.6 seven places to the right. We need to add six zeros.
2.6 x 107 = 26,000,000.
b. 1.016 x 10-8 can be expressed in decimal notation by moving the decimal point
in 1.016 eight places to the left. We need to add seven zeros to the right of the
decimal point.
1.016 x 10-8 = 0.00000001016.
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Scientific Notation
To convert from decimal notation to scientific notation, we reverse the
procedure.
• Move the decimal point in the given number to obtain a number
greater than or equal to 1 and less than 10.
• The number of places the decimal point moves gives the exponent on
10; the exponent is positive if the given number is greater than 10 and
negative if the given number is between 0 and 1.
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Text Example
Write each number in scientific notation. a. 4,600,000 b. 0.00023
Solution
a. 4,600,000 = 4.6 x
b. 0.00023 = 2.3 x
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10?
10?
Decimal point moves 6 places
Decimal point moves 4 places
4.6 x 106
2.3 x 10-4
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