Wikipedia:Reference desk/Archives/Mathematics/2023 November 19

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November 19

0.999 and limits

I know that 0.999... equals 1, even though it feels wrong and I don't like it. But I'm confused by a reference to it in the limit (mathematics) article: it says that 1 is the limit of the 0.9 + 0.09 + 0.009 etc. function, and the introduction reminds me that a limit is a value that a function approaches. If it approaches it, doesn't that mean that it doesn't quite reach it? (It not, I'd use reaches rather than approaches when writing such a statement.) These three statements sound self-contradictory — a limit is never completely reached by a function; 1 is the limit for 0.9 + 0.09 + 0.009 etc.; 0.999... equals 1 — so where am I misunderstanding something? Nyttend (talk) 05:11, 19 November 2023 (UTC)

The issue is that 0.999... is a (mis)representation in decimals and that limits are not "approaching" per se. The limit in this case would use an Epsilon-delta proof of something like $\lim _{x\to +\infty }\sum _{i\mathop {=} 1}^{x}{9 \over {10^{i}}}$. Let $\sum _{i\mathop {=} 1}^{x}{9 \over {10^{i}}}=j_{x}$, then for any x, there's always value k > x where $j_{x}<j_{k}<1$. We visualize this as "approaching" because we can see the first steps of the summation but it's not really. The end product of x being infinitely large is 1.

The other way to think of it is just as a limitation of the notation itself. 0.999... doesn't have a terminal digit, but we try to approximate it this way and we again think of it as a series of evermore 9s. Again, like a decimal representation of pi, it's only approximate. This can be seen when covering fractions to decimals: $1={1 \over 3}+{1 \over 3}+{1 \over 3}=0.333...+0.333...+0.333...=0.999...$ EvergreenFir (talk) 06:17, 19 November 2023 (UTC)

We have an article 0.999.... The concept of limit is applied in mathematics to two kinds of entities: we have the limit of a function and the limit of a sequence. Here we are dealing with sequences. When we state that the decimal representation of π is 3.14159..., the statement is equivalent to the statement that π is the limit of the sequence 3, 31/10, 314/100, 3141/1000, 31415/10000, 314159/100000, etc. Given any infinite decimal representation n.d₁d₂d₃...(an integer n before the decimal point and a stream of digits d₁, d₂, d₃ ... following the decimal point) we may ask which number it represents, and the answer is given by: the limit of the sequence n, nd₁/10, nd₁d₂/100, nd₁d₂d₃/1000, ... . While none of the approximants constituting this sequence may reach the limit, the meaning of the infinite decimal representation is by definition the limit of the sequence.  --Lambiam 08:14, 19 November 2023 (UTC)

The definition of real numbers implies they have the Archimedean property. One can have systems like the real numbers but with iinfinitely large or small numbers and don't have that property. NadVolum (talk) 08:09, 21 November 2023 (UTC)

An example are the surreal numbers. There exists a surreal number, let's call it λ, such that 0.999···9 < λ < 1, regardless of how many 9s are filled in for the three dots. There are in fact unfinitely many such surreal numbers, because if λ has this property, so does 1 − λ + λ². However, the surreal numbers, although linearly ordered, do not constitute a metric space, so the concept of limit does not apply to sequences of surreal numbers, which means there is no basis for defining the meaning of infinite decimal representations.  --Lambiam 11:27, 21 November 2023 (UTC)