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Transcript
Chapter 9. Section 2 Page 1 Section 9.2 – The Real Numbers Homework (pages 386-387) problems 1-9, 15-21 • • • • • • 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 1 1 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 Page 2 • 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 • • 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: • a m ⋅ a n = a m+ n ( ab) m = a mb m ( a m )n = a mn 1 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... • 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 Page 3 • Example, page 386 number 7a. Find 4 × 9 = 2(3) = 6 • 4 × 9 = 36 = 6 Why is this true? 4 × 9, 4 × 9