0.999...

0.999... is one of the ways the number 1 can be written. Even though it is written like this, it is still the same as one.

About

0.999... is a repeating decimal, which means the digit "9" is repeated forever. It is different from 0.999, which only has three 9s.

0.999... can also be written as $0.9 with bar over the 9$, $0.9 with dot over the 9$ or $0.(9)$.

It is hard for many people to understand why 0.999... is the same as 1. There are many proofs that show why they are the same number, but many of these proofs are very complex.

Example

One simple way of showing that 0.999... and 1 are the same thing is to divide them both by the number 3. When 0.999... is divided by 3, the answer is 0.333..., which is the same as ¹⁄₃ (the fraction one third).

${0.999\ldots \over 3} = 0.333\ldots = \frac 13$

When 1 is divided by 3, the answer is ¹⁄₃. Since the answers are the same, that means that 0.999... and 1 are the same. Another way of thinking about it is if ¹⁄₃ = 0.333... and ²⁄₃ = 0.666..., then ³⁄₃ = 0.999... = 1. There are many other ways of showing this.

In popular culture

As the Internet developed, arguments about 0.999... are often on newsgroups and message boards. Even newsgroups and message boards that do not have much to do with math argue about this. In the newsgroup sci.math, arguing about 0.999... is a "popular sport". It is also one of the questions in its FAQ.

Other websites

• Why does 0.9999... = 1 ?
• Ask A Scientist: Repeating Decimals
• A Friendly Chat About Whether 0.999... = 1