0.999…
0.999...

context: zh_tw: x = b0.b1b2b3… 在這種形式中，1 = 1.000… 而且 1 = 0.999… 的事實，反映了 1 既位於 [0, 1]，又位於 [1, 2]，所以我們在尋找它的數字時，可以選擇任意一個子區間。為了保證這種記法沒有濫用 “=” 號，我們需要一種辦法來為每一個小數重新構造一個唯一的實數。這可以用極限來實現，但是還有其它的方法。一個簡單的選擇，是區間套定理，它保證只要給出了一個長度趨近於零的閉區間套序列，那麼這些區間套的交集就正好是一個實數。這樣，b0.b1b2b3… 便定義為包含在所有的區間 [b0, b0 + 1]、[b0.b1, b0.b1 + 0.1]，依此類推的唯一的實數。而 0.999… 就是位於所有的區間 [0, 1]、[0.9, 1]、[0.99, 1]、[0.99…9, 1]（對於任意有限個 9）的唯一的實數。由於 1 是所有這些區間的公共元素，因此 0.999… = 1。區間套定理通常是建立在一個更加基本的實數特徵之上的：最小上界的存在。為了直接利用這些事物，我們可以把 b0.b1b2b3… 定義為集合 {b0，b0.b1，b0.b1b2，…} 的最小上界。然後我們就可以證明，這種定義（或區間套的定義）與劃分的過程是一致的，再一次證明了 0.999… = 1。湯姆・阿波斯托爾得出結論：
en: In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.One straightforward choice is the nested intervals theorem, which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their intersection. So b0.b1b2b3... is defined to be the unique number contained within all the intervals [b0, b0 + 1], [b0.b1, b0.b1 + 0.1], and so on. 0.999... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or suprema. To directly exploit these objects, one may define b0.b1b2b3... to be the least upper bound of the set of approximants {b0, b0.b1, b0.b1b2, ...}. One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,
qas: question: zh_tw: 這個數字的特性是什麼？
en: What is the characteristic of the number mentioned?
answers: zh_tw: text: 1]（對於任意有限個 9）的唯一的實數。
en: text: the unique number

context: zh_tw: 有些方法用公理集合論明確把實數定義為一定的建立在有理數上的結構。自然數──0、1、2、3，依此類推──從零開始並繼續增加，這樣每一個自然數都有一個後繼者。我們可以把自然數的概念延伸到負數，得出所有的整數，並可以進一步延伸到比例，得出所有的有理數。這些數系伴隨著加法、減法、乘法和除法的算術。更加微妙地，它們還包括排序，這樣一個數就可以與另一個進行比較，並發現是大於、小於，還是等於。
en: Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include ordering, so that one number can be compared to another and found to be less than, greater than, or equal to another number.
qas: question: zh_tw: 他們用哪種理論定義實數？
en: What theory do they use to define real numbers?
answers: zh_tw: text: 公理集合論
en: text: axiomatic set theory