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st be flawed. b) Some numbers have two different decimal representations. Specifically, any decimal expansion that terminates can also be written with repeating 9's. For instance, 1/2 can be written as 5 or as 4999.... Doesn't this cause some problems?

st be flawed. b) Some numbers have two different decimal representations. Specifically, any decimal expansion that terminates can also be written with repeating 9's. For instance, 1/2 can be written as 5 or as 4999.... Doesn't this cause some problems?

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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number has a decimal expansion, so we could apply this same argument to show that the set of rational numbers between 0 and 1 is uncountable. However, because we know that any subset of Q must be countable, the proof of Theorem 1.6.1 must be flawed. (b) Some numbers have two different decimal representations. Specifically, any decimal expansion that terminates can also be written with repeating 9's. For instance, 1/2 can be written as 5 or as 4999.... Doesn't this cause some problems? 34 Chapter 1. The Real Numbers Exercise 1.6.4. Let S he the set consisting of all sequences of 0's and I's. Observe that is not a particular sexquence, but rather a large set whose ele- namely
Transcribed Image Text:number has a decimal expansion, so we could apply this same argument to show that the set of rational numbers between 0 and 1 is uncountable. However, because we know that any subset of Q must be countable, the proof of Theorem 1.6.1 must be flawed. (b) Some numbers have two different decimal representations. Specifically, any decimal expansion that terminates can also be written with repeating 9's. For instance, 1/2 can be written as 5 or as 4999.... Doesn't this cause some problems? 34 Chapter 1. The Real Numbers Exercise 1.6.4. Let S he the set consisting of all sequences of 0's and I's. Observe that is not a particular sexquence, but rather a large set whose ele- namely
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