The Art of Proof pp 113-119 | Cite as

# Decimal Expansions

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## 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