Download irrational number

5.4
The Irrational Numbers and the
Real Number System
Copyright © 2009 Pearson Education, Inc.
Slide 5 - 1
Pythagorean Theorem


Pythagoras, a Greek mathematician, is credited
with proving that in any right triangle, the square
of the length of one side (a2) added to the
square of the length of the other side (b2)
equals the square of the length of the
hypotenuse (c2) .
a2 + b2 = c2
Copyright © 2009 Pearson Education, Inc.
Slide 5 - 2
Irrational Numbers


An irrational number is a real number whose
decimal representation is a nonterminating,
nonrepeating decimal number.
Examples of irrational numbers:
5.12639573...
6.1011011101111...
0.525225222...
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Slide 5 - 3
Radicals

2, 17, 53 are all irrational numbers.
The symbol
is called the radical sign. The
number or expression inside the radical sign
is called the radicand.
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Slide 5 - 4
Principal Square Root


The principal (or positive) square root of a
number n, written n is the positive number
that when multiplied by itself, gives n.
For example,
16 = 4 since 4  4 = 16
49 = 7 since 7  7 = 49
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Slide 5 - 5
Perfect Square


Any number that is the square of a natural
number is said to be a perfect square.
The numbers 1, 4, 9, 16, 25, 36, and 49 are the
first few perfect squares.
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Slide 5 - 6
Product Rule for Radicals
a  b  a  b,

a  0, b  0
Simplify:
a)
40
40  4 10  4  10  2  10  2 10
b)
125
125  25  5  25  5  5  5  5 5
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Slide 5 - 7
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Slide 5 - 8
Addition and Subtraction of Irrational
Numbers


To add or subtract two or more square roots
with the same radicand, add or subtract their
coefficients.
The answer is the sum or difference of the
coefficients multiplied by the common radical.
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Slide 5 - 9
Example: Adding or Subtracting
Irrational Numbers

Simplify: 4 7  3 7

Simplify: 8 5  125
4 7 3 7
8 5  125
 (4  3) 7
 8 5  25  5
7 7
8 5 5 5
 (8  5) 5
3 5
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Slide 5 - 10
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Slide 5 - 11
Multiplication of Irrational Numbers

Simplify:
6  54
6  54  6  54  324  18
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Slide 5 - 12
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Slide 5 - 13
Quotient Rule for Radicals
a
b
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
a
,
b
a  0, b  0
Slide 5 - 14
Example: Division

Divide:
16

Divide:
4

Solution:
16
16

 42
4
4
144
2

Solution:
144
144

 72
2
2
 36  2  36  2
6 2
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Slide 5 - 15
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Slide 5 - 16
Rationalizing the Denominator


A denominator is rationalized when it contains
no radical expressions.
To rationalize the denominator, multiply BOTH
the numerator and the denominator by a
number that will result in the radicand in the
denominator becoming a perfect square. Then
simplify the result.
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Slide 5 - 17
Example: Rationalize


8
Rationalize the denominator of
.
12
Solution:
8
8


12
12

2
3

2
3
2
3

3
3
6

3
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Slide 5 - 18
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Slide 5 - 19
Homework

P. 249 # 9 – 66 (x3)
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Slide 5 - 20