Section 6.4 Real Numbers
Rational
Numbers
if converted to decimal will form terminate or repeat
Irrational
Numbers
As decimals - are NOT rational
- Do NOT repeat or terminate
Examples: 0.718629371
. 3.40300400030004
.
Also includes Radicals 
Also include 
**Note: book for kids - Sir Circumference and the Dragon of Pi
**Cartoon
Page 406
Principal Square
Root
is a nonnegative number
b, such that 
EXAMPLE: 
Compare:
Look at
drawing of right triangles (pg 401) to see the length of various
: 
Other
Roots 
Find 
** When 
exists, what must be
true of n?
Estimating Square
Roots
Squeeze
Method to
approximate square roots -
Approximate 
to two decimal places
by the Squeeze Method
Archimedes Method to approximate 
(Pretend
you are using a 4 function calculator it adds, subtracts, multiplies and divides
by does not have a square-root key)
1.
Divide n by Guess1
2.
Find Guess2 by averaging Guess1 and the Quotient answer to step
1.
3.
Divide n by Guess2
4.
Find Guess3 by averaging Guess2 and the Quotient answer to step
3.
. . . Until desired accuracy is
achieved.
Use
Archimedes Method to approximate 
to three decimal
places.
Simplify
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Real
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Rational
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Irrational
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Integers |
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Whole
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Natural
Numbers |
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Indicate
with an X which sets include the number:
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Natural |
Integer |
Rational |
Irrational |
Real |
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-3 |
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Properties
of Real Numbers
Closure Properties
For real numbers a
and b, 
are unique real
numbers
Commutative Properties
For real numbers a and b, 
Associative Properties
For real numbers a, b and c, 
Identity Properties
The number 0 is the unique additive
identity and the number 1
1 is the unique multiplicative identity, such that, for any real number 
Inverse Properties
1)
For every real number 
is its unique additive
inverse; 
2)
For every nonzero real number 
is its unique
multiplicative inverse; 
Distributive Property of
Multiplication over addition
For real numbers 
Denseness
Property
For real numbers 
, there exists a real number 
such that 
Find three
irrational numbers between 1 and 3.
The
following exponential function approximates the number of bacteria after t
hours: 
a) What is the
initial number of bacteria that is when t = 0?
b) After Ό hour, how
many bacteria are there?
c) After ½ hour, how
many bacteria are there?
Pi 
is an irrational
number. Could 
? Why or why not?
More Properties of Exponents
Let r and s be any rational numbers, x
and y be any real numbers, and n be any nonzero integer,
a)
b)
c)
d)
Simplify
each of the following if possible.
