Download Real Numbers

Real Numbers
Subsets, Properties, Order of Operations, Radical
Simplification
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Subsets of Real Numbers
 Natural Number Set
 Rational Number Set
 N: {1, 2, 3, 4, ….}
 Q: {
 Whole Number Set
 These are Fractions!
 W: {0, 1, 2, 3, 4, ….}
 Decimal #’s that repeat or
 Integer Number Set
p
: where q  0 and p, q  Z
q
}
terminate.
 Z: {…, -4, -3, -2, -1, 0, 1, 2,
3, 4, ….}
 Irrational Number Set
 Q c :{ Real numbers that are
not rational; nonterminating, non-repeating
decimals} For example:
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2  1.4142135623 731
The Real Number System
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Properties of Real Numbers
 Let a, b, and c be real numbers.
 Property
 Closure
 Commutative
 Associative
 Identity
 Inverse
Addition
Multiplication
a  b is a real #
a  b is a real #
abba
ab  ba
(a  b)  c  a  (b  c)
(ab)c  a(bc)
a  0  a, 0  a  a
a  1  a, 1  a  a
a  ( a )  0
a
1
 1, a  0
a
 The following property involves both addition and multiplication
 Distributive
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a(b  c)  ab  ac, (b  c)a  ba  ca
Identify the real number property that
is being demonstrated…
3  (2  5)  (3  2)  5
1
2 1
2
606
2  3  3 2
2  (3  4)  (2  3)  4
2(4  7)  2  4  2  7
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1
1
 1  
3
3
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Order of Operations
 Recall that the four basic math operations are addition,
subtraction, multiplication, and division.
 There is an order in which operations need to be carried out.
 Order of Operations:
 (1) Grouping Symbols: [], ()
 (2) Exponents
 (3) Multiplication/Division from left to right
 (4) Addition/Subtraction from left to right
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Original Problem
Take care of any exponents
Start with parenthesis, inner most
parenthesis first
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Continue with parenthesis, follow
multiply/divide and add/subtract rules.
Multiply/Divide from left to right
Add/Subtract from left to right
Simplify, Show all of your steps and label the steps
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 
 Original problem
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 3  3 2  4  3
3
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 3  32  7
3
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 3  9  7 
3
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 3  2 
3
18
1
3
6 1
5
 
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
Inner most parenthesis
Exponent
Inner most parenthesis
Parenthesis
Multiply/divide left to right
Add/subtract left to right
Simplify
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Anatomy of a Root
Radical Sign
n
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Index of the
Radical: “The
root you are
taking”
Radicand
What it means to find an nth root
 When finding an nth root, you are actually finding the base of an
exponential expression.
When evaluating exponential expressions the base grows really big really fast
SO
When finding nth roots the radicand gets really small really fast
81  9 because 9(9)  81
3
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27  3 because 3(3)(3)  27
Nth Roots Key properties
n
x n  x, when n is odd
n
x n  x , when n is even
Absolute value is not needed
if variables are assumed to be
positive
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x
x2
x3
x4
x5
x6
x7
x8
x9
x10
1
1
1
1
1
1
1
1
1
1
2
4
8
16
32
64
3
9
27
81
243 729 2187 6561
4
16
64
256 1024 4096
5
25
125 625 3125
15625 78125
6
36
216 1296 7776
46656
7
49
343 2401
15
16807
128 256 512 1024
16384 65535
19683
Simplify
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Radical Simplification
 In order to simplify a radical you need to pull any perfect nth
root factors that are in the radicand using the following rule:
a b  a  b
Simplifying works out much better if this is the
biggest perfect nth root factor! (some radicands
will have more than one perfect nth root factor)
 Take the nth root of a and leave b under the radical sign.
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Real Nth Roots
 Let n be an integer greater than 1 and let a be a real number.
 If n is odd, then a has one real nth root: n
a
 If n is even and a > 0, then a has two real nth roots:
 If n is even and a = 0, then a has one nth root:
n
0 0
 If n is even and a < 0, then a has no real nth roots.
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n a
Odd nth roots and negative radicands
 A negative radicand with an odd nth root can be simplified.
3
8
  2  2  2
 2
5
1
  1  1  1  1  1
 1
 CAUTION! You can not simplify a negative radicand with an
even nth root!
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Writing a radical in simplest radical form
Example
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 25  2
 55  2
5 2
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Simplify
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Simplifying radicals with variables
x
3
 x
x
2
x
x x
 x
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2
x
 xx x
x
Simplify
12 x 3
5
2
x
2
x
9x 6 y 3
 12 x 3
 9 x6
 4 3 x2 x
 9 x2 x2 x2
 2 3x x
 3xxxy y
 2 x 3x
 3x 3 y y
y3
y2
y
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Combining like objects

How many oranges do you have?
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How many apples do you have?

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Combining like terms is just like combining like objects. You
just need to tell how many of a particular variable you have.

How many x’s are there?
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Combining like terms
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6x  9x
12 y  7 y
6x  9x
 12 y  7 y


How many apples and oranges do you have?
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What to do when the variables aren’t the same
6x  9x  3y
6 x  12 y  7 y
 6x  9x  3y
 6 x  12 y  7 y
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Adding/Subtracting Like Radicals
Combining like radicals is just like combining like terms. Add
the coefficient and bring the radical along for the ride.
3 57 5
9 2 7 2
 10 5
2 2
In order to add/subtract radical you
have to have the same radicand.
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Unlike Radicals…
 If you don’t have like radicands then you have to simplify
where possible and then add/subtract like radicals
2 8
 2 4 2
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4 3  27
4 3 9 3
 22 2
 4 3 3 3
3 2
 3
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Simplify Even Roots of Negative Numbers
 Recall in the last unit we talked about finding nth roots of negative radicands and you
were told you couldn’t take the nth root of a negative radicand if the nth root was even.
 NOW we will explore how to simplify even nth roots of negative radicands.

The imaginary unit i is a complex number whose square is 1.
i  1
 And use this idea to simplify even nth roots of negative radicands

So we can say that
 For Example:
 81
 81  1
 9i
8
 1 4 2
 1 2  2 2
 i2 2
 2i 2
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