Real Numbers and Algebraic Expressions

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Real Numbers and Algebraic Expressions
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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.

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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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• Real Numbers
• The real number line
• The set of natural (counting) numbers
• The set of Whole numbers

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Integers

• The set of integers

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Important Subsets of the Real Numbers
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• Definition
• Rational Numbers A number that can be written
  in the form a / b where a and b are integers.

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The Real Numbers

• Rational numbers

Irrational numbers

Integers
Whole numbers
Natural 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

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Graphing on the Number Line

• Which numbers are plotted?

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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 lt 5
Since 5 is to the right of 2 on the number line,
5 is greater than 2. 5 gt 2
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Inequality Symbols
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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.

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Definition of Absolute Value

• The absolute value of x is given as follows

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Properties of Absolute Value

• For all real number a and b,
• 1. a gt 0

2. -a a
3. a lt a
5. , b not equal to 0
4. ab ab
6. a b lt a b (the triangle inequality)
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Example

• Find the following -3 and 3.

Solution
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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.
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CE

• Review
• Evaluate

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CE

• Review
• Evaluate the expression for x 3 and y 2

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CE

• Find the distance between the following points.
  (use absolute value)
• -2 and 5

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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.

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CE

• List 3 examples of algebraic expressions
• x 6
• x 6
• x/6

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CE

• Please Excuse My Dear Aunt
• Sally!

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The Order of Operations Agreement

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

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CE

• 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.

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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 Perform the multiplication.
226.5 Perform the addition. Thus, in 2000 the
population of the United States was 226.5 million.
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CE Properties of Real Numbers

• For all real numbers a,b, and c
• Closure Properties
• Addition Multiplication
• a b is real a b is real

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Continued

• Commutative Property
• Addition Multiplication
• a b b a ab ba

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Continued

• Associative Property
• Addition Multiplication
• (ab)ca(bc) (ab)ca(bc)

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Continued

• Distributive Property
• a(bc) abac

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• Zero Product Property
• If ab 0 , then a 0 or b 0
• Or a b 0

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• Inverse Property
• a (-a) (-a) a 0

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SEE HANDOUT
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Properties of the Real Numbers
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Properties of the Real Numbers
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Properties of the Real Numbers
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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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CE

• Simplify 6(2x 4y) 10(4x 3y).

Solution 6(2x 4y) 10(4x 3y)
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• 6(2x 4y) 10(4x 3y)

6 2x 6 4y 10 4x 10 3y
Distribute
12x 24y 40x 30y Multiply.
(12x 40x) (30y 24y) like
terms.
52x 6y Combine like terms.
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Properties of Negatives

• Let a and b represent real numbers, variables, or
  algebraic expressions.
• (-1)a -a
• -(-a) a
• (-a)(b) -ab
• a(-b) -ab
• -(a b) -a - b
• -(a - b) -a b b - a

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Section P.2

• Exponents and Scientific Notation

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Definition of Positive Exponents

• If n is a positive integer and b is any real
  number, then
• Where b is the base and n is the exponent.

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Rules of Exponents
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CE

• Evaluate

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CE

• Evaluate

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CE

• Evaluate

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Definition

• If b is a real number not equal to zero, then

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CE

• Evaluate

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CE

• Evaluate

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CE

• Evaluate

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CE

• Evaluate

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Definition

• If n is an integer and b is a real number not
  equal to zero, then

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CE

• Evaluate

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CE

• Evaluate

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CE

• Evaluate

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CE

• Evaluate

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Definition

• Scientific Notation
• Place the decimal point behind the first nonzero
  digit.
• Count the number of places you moved the decimal.

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Continued

• If you moved to the left the exponent is
  positive.
• If you moved to the right the exponent is
  negative.

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CE

• Write 1,575,000,000,000 in scientific notation.

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CE

• Write in scientific notation
• 3,450,000

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CE

• If your calculator displays this
• 1.23456 E9
• What is the equivalent number?

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