The Art of Proof pp 113-119

# Decimal Expansions

Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM, volume 0)

## Abstract

Our goal in this chapter is to prove that every real number can be represented by a decimal and that every decimal represents a real number. Before You Get Started. In Chapter 7 we discussed base-ten representation of integers. In fact, all real numbers can be given base-ten representations: this is what you called “decimals” in grade school. Do you remember repeating and nonrepeating decimals? For example, the decimal expansion 0.1111... is supposed to represent the number $$\frac {1}{9}.$$ Why is this a valid expression for $$\frac {1}{9}$$ ? What is a decimal expansion of $$\frac{1}{8}\ {\rm or}\ \sqrt 2 ?$$

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© Matthias Beck and Ross Geoghegan 2010

## Authors and Affiliations

1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
2. 2.Department of Mathematical SciencesBinghamton University State University of New YorkBinghamtonUSA