Talk:0.999.../FAQ
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- Q: Are you positive that 0.999... equals 1 exactly, not approximately?
- A: In the set of real numbers, yes. This is covered in the article. If you still have doubts, you can discuss it at Talk:0.999.../Arguments. However, please note that original research may not ever be added to a Wikipedia article, and original arguments and research in the talk pages will not change the content of the article—only reputable secondary and tertiary sources can do so.
- Q: Can't
be expressed as 
?
- A: No. is not a meaningful string of symbols because, although a decimal representation of a number has a potentially infinite number of decimal places, each of the decimal places is a finite distance from the decimal point; the meaning of digit d being k places past the decimal point is that the digit contributes
toward the value of the number represented. It may help to ask yourself how many places past the decimal point the "1" is. It cannot be an infinite number of places, because all places must be finite. Also ask yourself what would be the value of 
. If a real number divided by 10 is itself, then that number must be 0.
- Q: 0.9 < 1, 0.99 < 1, and so forth. Therefore it's obvious that 0.999...<1.
- A: No. Something that holds for various values need not hold for the limit of those values. For example, f(x) = x3 / x is positive (>0) for all values in its implied domain (
). However, the limit as x goes to 0 is 0, which is not positive. This is actually an important consideration in proving inequalities based on limits.
- Q: Is it possible to create a new number system other than the reals in which 0.999... < 1, the different being an infinitesimal amount?
- A: Yes, although such systems are neither as used nor as useful as the real numbers, lacking properties such as the ability to take limits (which defines the real numbers), to divide (which defines the rational numbers, and thus applies to real numbers), or to add and subtract (which defines the integers, and thus applies to real numbers). Furthermore, we must define what we mean by "an infinitesimal amount." These is no nonzero constant infinitesimal in the real numbers; quantities generally thought of informally as "infinitesimal" include ε, which is not a fixed constant; dx, which is not a number at all, dx, which is not a real number and has anticommutativity, that is, dx dy = - dy dx; 0+, which is not a number, but rather part of
, the right limit of x (which can also be expressed without the "+" as 
); and values in number systems such as dual numbers and hyperreals. In these systems, 0.999...=1 still holds due to real numbers being a subfield. As detailed in the main article, there are systems for which 0.999... and 1 are distinct, systems that have both alternative means of notation and alternative properties, and systems for which subtraction no longer holds. These, however, are rarely used and possess little to no practical applications.
- Q: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly?
- A: At the expense of abandoning many familiar features of mathematics, it is possible to construct a system of notation in which the string of symbols "0.999..." is different than the number 1. This object would represent a different number than the topic of this article, and this notation has no use in applied mathematics. Moreover, it does not change the fact that 0.999... = 1 in the real number system. The fact that 0.999... = 1 is not a "problem" with the real number system and is not something that other number systems "fix". Absent a perverse desire to cling to intuitive misconceptions about real numbers, there is little incentive to use a different system.
