0.999...
0.999... (also written as 0.9 or 0.Template:Overset, and read as "0 point 9 repeating") is one of the ways the number 1 (one) can be written. Even though it is written like this, no matter how many nines there are before the ellipsis, it is still equal in value to 1.
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 [math]\displaystyle{ 0.\bar{9} }[/math] or [math]\displaystyle{ 0.\dot{9} }[/math].
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.[1]
Examples
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 [math]\displaystyle{ \frac13 }[/math] (the fraction one third).
- [math]\displaystyle{ \frac{0.999\ldots}{3}=0.333\ldots=\frac13 }[/math]
When 1 is divided by 3, the answer is [math]\displaystyle{ 1/3 }[/math]. Since the answers are the same, that means that 0.999... and 1 are the same. Another way of thinking about it is if [math]\displaystyle{ 1/3=0.333\ldots }[/math] and [math]\displaystyle{ 2/3=0.666\ldots }[/math], then [math]\displaystyle{ 3/3=0.999\ldots }[/math] therefore, as [math]\displaystyle{ 3/3=1 }[/math], 0.999... must also equal 1. There are many other ways of showing this.[1]
Another way of proving that 0.999... = 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 2 is 1.5, and a number between 0.9 and 1 is 0.95. Since 0.999... has an infinite number of 9s, there cannot be another number after the "last" 9, meaning there is no number between 0.999... and 1. Therefore, they are equal.
One more common proof is such:
[math]\displaystyle{ x=0.999\ldots }[/math]
[math]\displaystyle{ 10x=9.999\ldots }[/math]
[math]\displaystyle{ 10x-x=9x }[/math]
[math]\displaystyle{ 9x=9.999\ldots-0.999\ldots=9 }[/math]
[math]\displaystyle{ x=9/9 }[/math]
[math]\displaystyle{ x=1 }[/math]
[math]\displaystyle{ 0.999\ldots=1 }[/math]
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".[2] It is also one of the questions in its FAQ.[2]
0.999... Media
The Archimedean property: any point x before the finish line lies between two of the points P_n (inclusive).
Limits: The unit interval, including the base-4 fraction sequence (.3, .33, .333, ...) converging to 1.
Nested intervals: in base 3, 1 = 1.000... = 0.222...
Positions of 1⁄4, 2⁄3, and 1 in the Cantor set
The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.
Related pages
References
- ↑ 1.0 1.1 Katz, K.; Katz, M. (2010a). "When is .999... less than 1?". The Montana Mathematics Enthusiast. 7 (1): 3–30. doi:10.54870/1551-3440.1381. S2CID 11544878. Archived from the original on 2011-07-20. Retrieved 2010-10-17.
- ↑ 2.0 2.1 As observed by Richman (p. 396). Hans de Vreught (1994). "sci.math FAQ: Why is 0.9999... = 1?". Retrieved 2006-06-29.
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