0.999...

0.999... repeating infinitely.

This Wikibook is a reference guide to the theory behind the mathematical identity 0.999... = 1. It includes select topics in real analysis, calculus, and the arithmetic of infinite decimals, culminating in several proofs of the identity itself. The material cuts across the study of pure mathematics, mathematics for scientists and engineers, and high-school and middle-school mathematics.

Introduction

A traditional textbook aimed at a single grade level will often present a proof that 0.999... = 1, but the textbook will rely upon assumptions whose proofs are deferred to more advanced textbooks. The reader who wants a full proof starting from basic algebra is therefore required to consult multiple references. However, different books take different approaches to real analysis, often presenting the material in different orders. Some textbooks assume 0.999... = 1 in their development of the real numbers. So a full proof that is pasted together introduces a real danger of circular reasoning. This book aims to alleviate worries about circular reasoning by staying self-contained and clearly indicating which results are used in which proofs.

The mathematics literature contains a wide variety of proofs that 0.999... = 1. This book provides enough background to understand all commonly encountered proofs.

This book sits somewhere between high school and university in difficulty. It will be most relevant to you if you've already completed an algebra curriculum, but not a course in real analysis. A working knowledge of the rational numbers from basic algebra is assumed throughout. A background in set theory will help for the more advanced modules.

How to read this book

Depending on what you're aiming to get out of this book, there are many ways to read it.

• If you read from start to finish, you'll encounter topics in their logical order, and each proof will rely on concepts you've already seen. You'll also gain an understanding of all the most popular proofs of 0.999... = 1. However, you'll be hit with the most abstract concepts right away, and it won't be clear why each concept is introduced until later. This strategy is recommended for a more advanced reader.
• You can start by reading a proof of 0.999... = 1 from the last chapter of the book. When you encounter a step in the proof that you're unsure about, you can then go back to an earlier chapter and read the justification for that step. This way, you'll encounter topics in the same order as in a traditional mathematics curriculum, and you won't waste time on steps that are irrelevant to your chosen proof. This strategy is recommended for a less advanced reader.
• If you're already familiar with 0.999... = 1, but you've always wondered about a particular ingredient in a proof, you can jump directly to that ingredient in the below table of contents.

Contents

Real analysis

• Construction of the real numbers from Dedekind cuts
• Construction of the real numbers from Cauchy sequences
• Axioms of the real numbers
• Archimedean property of the real numbers
• Nested Intervals Theorem

Calculus

• The limit of a sequence
• Term-by-term operations on sequences
• The limit of a geometric sequence
• The sum of a series
• Term-by-term operations on series
• Shifting a series
• The geometric series formula

Definitions of infinite decimals

• Definition from Dedekind cuts
• Definition from Cauchy sequences
• Definition from series
• Definition from sequences
• Definition from nested intervals

Properties of infinite decimals

• Decimal addition and subtraction
• Decimal multiplication by a small number
• Decimal multiplication by 10
• Decimal long division

Proofs that 0.999... = 1

• Proof from 0.333...
• Proof from 0.111...
• Proof by multiplication by 10
• Proof by the geometric series formula
• Proof by limit manipulation
• Proof by nested intervals
• Proof by equality of Dedekind cuts
• Proof by equivalence of Cauchy sequences

Related resources

This book is narrowly focused on a handful of results that are directly useful for understanding 0.999.... It omits plenty of topics that are essential for a balanced understanding of calculus and real analysis. For Wikibooks that develop a broader background, see:

• Real Analysis and other books in Subject:Mathematical analysis
• Calculus and other books in Subject:Calculus

There are also independent learning activities and materials in these areas at Wikiversity:

• Topic:Real analysis
• Topic:Calculus

For an encyclopedic overview of 0.999..., including history, education, applications, and popular culture, see Wikipedia:

• 0.999...