Proof that 0.999… = 1

Introduction

There are certain concepts in mathematics that are counterintuitive. In this post, we discuss one of these concepts — the elementary proof that 0.999 \cdots = 1. The \cdots symbol indicates that there are infinitely many 9‘s to the right hand side of the decimal point. The proof of this theorem is extremely easy;  however, to be able to appreciate it will require understanding of the concept of limits and infinity.

0.999=1

Theorem: 0.999 \cdots = 1

Proof 1 

We know that

0.333 \cdots = \frac{1}{3}.

Multiplying both sides by 3, we have

0.999 \cdots = 1. \blacksquare

If the proof above does not convince you, there is another proof below.

Proof 2

Let x = 0.999 \cdots.

Multiplying both sides by 10, we have 10x = 9.999 \cdots.

Subtracting, we have

10x - x = 9.999 \cdots - 0.999 \cdots.

9x = 9

x = 1.

Therefore, 0.999 \cdots = 1. \blacksquare.

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Photo Credit: Marco Arment via Flickr Creative Commons

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