0.999...

0.999..., also written as 0.9 or 0..9, and read as "0 point 9 repeating", is one of the ways the number 1 can be written. Even though it does not have any ones in it, it is still equal in value to 1.

About

0.9 is a repeating decimal, which means the nines after the decimal point keep going forever. No matter how far you look, there will still be more nines.

It is hard for many people to understand why 0.9 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.

Examples

One simple way of showing that 0.9 and 1 are the same thing is to divide them both by the number 3. When 0.9 is divided by 3, the answer is 0.3. When 1 is divided by 3, the answer is also 0.3.

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

${\frac {1}{3}}=0.333\ldots$

If 0.9 and 1 divided by 3 are the same, then they must be the same even without the division.

Another way of thinking about it is if $1/3=0.333\ldots$ and $2/3=0.666\ldots$, then $3/3=0.999\ldots$ therefore, as $3/3=1$, 0.999... must also equal 1. There are many other ways of showing this.

Another simple way of proving that 0.9 = 1 is by accepting the simple fact that if two numbers are different, there must be at least one number between them. For example, a number between 1 and 0.9 is 0.95, and a number between 0.99 and 1 is 0.995. Since 0.9 has an infinite number of 9s, there cannot be another number after the "last" 9, because there is no last 9. Therefore, there is no number between 0.9 and 1. Therefore, they are equal.

A more mathematical proof can be found with the following equation:

$x=0.999\ldots$

$10x=9.999\ldots$

$10x-x=9.999\ldots -x$

$9x=9.999\ldots -0.999\ldots =9$

$x=9/9$

$x=1$

$0.999\ldots =1$

In popular culture

As the Internet developed, arguments about 0.9 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 0.9 is a "popular sport". It is also one of the questions in its FAQ.

Related pages

• Limit (mathematics)

Other websites

• Why does 0.9999... = 1?
• Ask A Scientist: Repeating Decimals Archived 2015-02-26 at the Wayback Machine
• A Friendly Chat About Whether 0.999... = 1