september 3, 2014 notes #p.1 real...
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What you’ll learn about Representing Real Numbers Order and Interval Notation Basic Properties of Algebra Integer Exponents Scientific Notation … and why These topics are fundamental in the study of mathematics and science.
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September 3, 2014
Notes #P.1 – Real Numbers Section P.1
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Real Numbers
A real number is any number that can be written as a decimal.
Subsets of the real numbers include: •The natural (or counting) numbers: {1,2,3…}
•The whole numbers: {0,1,2,…} •The integers: {…,-3,-2,-1,0,1,2,3,…} •The rational numbers
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Copyright © 2011 Pearson, Inc.
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Rational Numbers
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, are integers, and 0a
a b bb
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Copyright © 2011 Pearson, Inc.
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Irrational Numbers
A number is irrational if it is not rational. The decimal form of an irrational number is infinitely nonrepeating.
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Copyright © 2011 Pearson, Inc.
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Example 1: Classifying Numbers
Rat
ion
al
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The Real Number Line
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Order of Real Numbers
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Trichotomy Property
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Example 2: Interpreting Inequalities
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Example 2 solution
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The double inequality represents all real numbers between 3 and 5, including 3 and excluding 5.
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Bounded Intervals of Real Numbers
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Interval Notation
Inequality Notation
Graph
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Unbounded Intervals of Real Numbers
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Copyright © 2011 Pearson, Inc.
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Example 3: Converting between inequality and interval notation
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Properties of Algebra
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Let , , and be real numbers, variables, or algebraic expressions.
Addition:
Multiplication
Addition: ( ) ( )
Multiplication: ( )
u v w
u v v u
uv vu
u v w u v w
uv w u
1. Communative Property
2. Associative Property
( )
Addition: 0
Multiplication: 1
vw
u u
u u
3. Identity Property
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Properties of Algebra
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Let , , and be real numbers, variables, or algebraic expressions.
Addition: (- ) 0
1Mulitiplication: 1, 0
Multiplication over addition:
( )
u v w
u u
u uu
u v w uv uw
4. Inverse Property
5. Distributive Property
( )
Multiplication over subtraction:
( )
( )
u v w uw vw
u v w uv uw
u v w uw vw
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Example 4: Using the Distributive Property
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Copyright © 2011 Pearson, Inc.
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Example 5: Using the properties
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Properties of the Additive Inverse
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Let , , and be real numbers, variables, or algebraic expressions.
1. ( ) ( 3) 3
2. ( ) ( ) ( 4)3 4( 3) 12
u v w
u u
u v u v uv
Property Example
3. ( )( ) ( 6)( 7) 42
4. ( 1) ( 1)5 5
5. ( ) ( ) ( ) (7 9) ( 7) ( 9) 16
u v uv
u u
u v u v
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Exponential Notation
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n factors... ,
Let be a real number, variable, or algebraic expression and
a positive integer. Then where is the
, is the , and is the ,
read as " to
n
n
a a a a
a n
a n
a a
a
exponent base th power of n a
the th power."n
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Properties of Exponents
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Let u and v be a real numbers, variables, or algebraic expressions
and m and n be integers. All bases are assumed to be nonzero.
Property Example
1. umu n umn 53 54 534 57
2. um
u n umn
x 9
x 4 x 94 x5
3. u0 1 80 1
4. u - n 1
u n y -3
1
y 3
5. (uv)m umvm (2z)5 25 z 5 32z 5
6. (um )n umn (x 2 )3 x 23 x 6
7. u
v
m
um
vm
a
b
7
a7
b7
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Copyright © 2011 Pearson, Inc.
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Example 6: Simplifying Expressions Involving Powers
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Simplify u2v3
u1v2.
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Copyright © 2011 Pearson, Inc.
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Copyright © 2011 Pearson, Inc.
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Scientific Notation
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Example 7: Converting to Scientific Notation
Convert 0.0000345 to scientific notation.
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-50.0000345 3.45 10
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Example 8: Converting from Scientific Notation
Convert 1.23 × 105 from scientific notation.
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123,000