There is an elementary proof of the equation 0.999... = 1, which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. The proof, given below, [2] is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line there is no room left for placing a number between them and 1. The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so 0.999... = 1. This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer. This is the Archimedean property, that is verified for rational numbers and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, with infinitely small numbers ( infinitesimals) and infinitely large numbers ( infinite numbers). When using such systems, notation 0.999... is generally not used, as there is no smallest number that is no less than all 0.(9) n. (This is implied by the fact that 0.(9) n ≤ x< 1 implies 0.(9) n–1 ≤ 2 x – 1 < x< 1).

This is the part that matches your specification. The ?: is needed only if you want to keep the matched groups "clean", in the sense that there will be no group(2) for the middle case (?![0-9.]) If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1. is a look-behind that prevents us from ripping out pieces of number-literals in multi-line input, e.g. 10000010.0 should not be matched. (0|(?:[1-9]\.[0-9])|(?:10\.0)) All of the following will match: 0, 1.1, 1.0, 1.9, 2.0, 2.1, 9.0, 9.1, 9.9, 10.0, but all of the following will not: 0.1, 0.2, 0.9, 1.11, 1.20, 1.01, 10.05, 110.05. Does not require one-number per line, can extract numbers embedded in text.For example, Minecraft 1.0.9 users can visit End City. This structure consists of several towers. There are the shulkers who live in this area. These aggressive creatures shoot shells with the effect of levitation. Discussion on completeness: I honestly didn't understand what it meant, but in the next paragraph it says the previous paragraph isn't proof.

Therefore, if 1 were not the smallest number greater than 0.9, 0.99, 0.999, etc., then there would be a point on the number line that lies between 1 and all these points. This point would be at a positive distance from 1 that is less than 1/10 n for every integer n. In the standard number systems (the rational numbers and the real numbers), there is no positive number that is less than 1/10 n for all n. This is (one version of) the Archimedean property, which can be proven to hold in the system of rational numbers. Therefore, 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc., and so 1 = 0.999.... Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc.: a smallest number that is greater than all of the terms of the sequence. One of the axioms of the real number system is the completeness axiom, which states that every bounded sequence has a least upper bound. This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations. The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers in fact has a least upper bound, and that this least upper bound is equal to one.

On the other hand, the terms of the associated sequence, 0.9, 0.99, 0.999, 0.9999, …, etc, do get arbitrarily close to 1, in the sense that, for each term in the progression, the difference between that term and 1 gets smaller and smaller as the number of 9s gets bigger. No matter how small you want that difference to be, I can find a term where the difference is even smaller. But", some say, "there will always be a difference between 0.9999… and 1." Well, sort of. Yes, at any given stop, at any given stage of the expansion, for any given finite number of 9s, there will be a difference between 0.999…9 and 1. That is, if you do the subtraction, 1−0.999…9 will not equal zero. On this territory, you can also see a rare structure – the End ship. The player should carefully inspect the building, because there may be elytra there. With this item, Steve can fly. Ender Dragon