Download Section 9.2 – The Real Numbers

Chapter 9. Section 2
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Section 9.2 – The Real Numbers
Homework (pages 386-387) problems 1-9, 15-21
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So are there any numbers that are not rational?
What is so special about all the fractions we have looked at so far?
Every fraction we have seen, when converted to a decimal either terminates or repeats
But there are values that neither terminate or repeat
The book gives the definition of real numbers as: the set of all numbers that have an infinite
decimal representation. This is a bit misleading, because you don't consider 2.1 to have an infinite
decimal representation, but 'officially' it does, that is 2.10000000…
A better definition of real numbers is the set of all terminating, repeating, and nonterminating
decimals. This set includes:
Counting or natural numbers
Whole numbers
Fractions
Rational numbers
Recall our Venn diagram from Section 9.1, how would this new group fit into that Venn diagram?
Pulling it All Together:
• So during the semester we have looked at properties of addition
Closure
Commutative
Associative
Identity
Inverse
• And we have looked at properties of multiplication
Closure
Commutative
Associative
Identity
Inverse
Distributive
• We have shown along the way that each of our groups of numbers has satisfied these properties, that
means the real number system satisfies all of these properties!
Roots of Real Numbers:
• We have looked at integer exponents so far, that is
32 = 3 ⋅ 3
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3−2 = 2 =
3 3 ⋅3
• But what if we want to 'undo' the root?
Say we want to know what number, when squared, gives us 9?
We write this as 2 9 = 3 (most of the time we leave off the 2)
Chapter 9. Section 2
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This is actually known as a fractional exponent. Let's see how this makes sense by looking at the
above equation
(91/2 ) 2 = (3) 2
9=9
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Formally, if a , b ≥ 0, then n a = b if and only if bn = a
You cannot have one without the other because they are inverses
34 = 3 ⋅ 3 ⋅ 3 ⋅3 = 81
so it follows that 4 81 = 3
All of Our Old Rules Still Apply:
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a m ⋅ a n = a m+ n
( ab) m = a mb m
( a m )n = a mn
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a−m = m
a
Some Exercise Problems:
• Example, page 388 number 1h. Classify the number 5.626626662…
Real, it appears to be non repeating nonterminating
• Example, page 388 number 3, arrange the numbers (see book)
0.876000000000000
0.8766876667...
0.8766876687668766
0.876767676767676
0.876787677876...
0.87678776677887666...
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0.876876876876876
Example, page 388 number 4. Find an irrational number between 0.5777 and 0.5778
First of all, since it is irrational there is really no knowing what number I mean exactly, but…
0.577712345678954987…
For it to be irrational it must be nonterminating and non repeating
We could also find the fraction that the above two numbers represent
n = 0.5777
n = 0.5778
10000n = 5777.5777 and 10000n = 5778.5778
5777
5778
n=
n=
9999
9999
But then what? We can't represent our irrational value as some fraction between them, why?
Chapter 9. Section 2
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Example, page 386 number 7a. Find
4 × 9 = 2(3) = 6
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4 × 9 = 36 = 6
Why is this true?
4 × 9, 4 × 9