Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. 5.4 The Irrational Numbers and the Real Number...
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Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. 5.4 The Irrational Numbers and the Real Number System
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Transcript of Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. 5.4 The Irrational Numbers and the Real Number...
Slide 5 - 1Copyright © 2009 Pearson Education, Inc.
5.4
The Irrational Numbers and the Real Number System
Slide 5 - 2Copyright © 2009 Pearson Education, Inc.
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
Slide 5 - 3Copyright © 2009 Pearson Education, Inc.
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...
Slide 5 - 4Copyright © 2009 Pearson Education, Inc.
are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.
Radicals
2, 17, 53
Slide 5 - 5Copyright © 2009 Pearson Education, Inc.
Principal Square Root
The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n.
For example,
16 = 4 since 44 =16
49 = 7 since 77 = 49
n
Slide 5 - 6Copyright © 2009 Pearson Education, Inc.
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.
Slide 5 - 7Copyright © 2009 Pearson Education, Inc.
Product Rule for Radicals
Simplify:a)
b)
ab a b, a 0, b 0
40 410 4 10 2 10 2 10
125 255 25 5 5 5 5 5
40
125
Slide 5 - 8Copyright © 2009 Pearson Education, Inc.
Slide 5 - 9Copyright © 2009 Pearson Education, Inc.
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.
Slide 5 - 10Copyright © 2009 Pearson Education, Inc.
Example: Adding or Subtracting Irrational Numbers
Simplify: Simplify: 4 7 3 7
4 7 3 7
(4 3) 7
7 7
8 5 125
8 5 125
8 5 25 5
8 5 5 5
(8 5) 5
3 5
Slide 5 - 11Copyright © 2009 Pearson Education, Inc.
Slide 5 - 12Copyright © 2009 Pearson Education, Inc.
Multiplication of Irrational Numbers
Simplify:
6 54
6 54 654 324 18
Slide 5 - 13Copyright © 2009 Pearson Education, Inc.
Slide 5 - 14Copyright © 2009 Pearson Education, Inc.
Quotient Rule for Radicals
a
b
a
b, a 0, b 0
Slide 5 - 15Copyright © 2009 Pearson Education, Inc.
Example: Division
Divide:
Solution:
Divide:
Solution:
16
4
144
2
16
4
16
4 4 2
144
2
144
2 72
362 36 2
6 2
Slide 5 - 16Copyright © 2009 Pearson Education, Inc.
Slide 5 - 17Copyright © 2009 Pearson Education, Inc.
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.
Slide 5 - 18Copyright © 2009 Pearson Education, Inc.
Example: Rationalize
Rationalize the denominator of
Solution:
8
12.
8
12
8
12
2
3
2
3
2
3
3
3
6
3
Slide 5 - 19Copyright © 2009 Pearson Education, Inc.
Slide 5 - 20Copyright © 2009 Pearson Education, Inc.
Homework
P. 249 # 9 – 66 (x3)
