# 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 $latex 0.999 \cdots = 1$. The $latex \cdots$ symbol indicates that there are infinitely many $latex 9$’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.

Theorem: $latex 0.999 \cdots = 1$

Proof 1

We know that

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

Multiplying both sides by $latex 3$, we have

$latex 0.999 \cdots = 1. \blacksquare$

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

Proof 2

Let $latex x = 0.999 \cdots$.

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

Subtracting, we have

$latex 10x – x = 9.999 \cdots – 0.999 \cdots$.

$latex 9x = 9$

$latex x = 1$.

Therefore, $latex 0.999 \cdots = 1. \blacksquare$.

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Photo Credit: Marco Arment via Flickr Creative Commons