# 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.

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