Is .999... = 1? A Non-standard View

I already had an opportunity to explain my point of view on a question of whether 0.999... equals 1 or not. One aspect of it is that an attempt to answer such a question should start with an inquiry as to what the symbol 0.999... stands for. As was observed in an 1999 article from the Mathematics Magazine,

Arguing whether 0.999... is equal to 1 is a popular sport on the newsgroup sci.math. It seems to me that people are often too quick to dismiss the idea that these two numbers might be different. The issues here are closely related to Zeno's paradox, and to the notion of potential infinity versus actual infinity. Also at the stake is the orthodox view of the nature of real numbers.

Actual infinity is a multifaceted mathematical (and philosophical) concept whose existence has been proved or taken for granted. The intention here is to associate 0.999... with a sum of an infinite series defined as the limit of its partial sums. (This is a well defined number which happens to be 1, showing the reasons for the dismissive attitude of a great number of mathematicians.) Potential infinity is rather an unending process, or in this case as the author put it, Simply the unending possibility of writing down more nines. Whether a process is a number is quite questionable and is a great source of bewilderment. In mathematics education a novel term - procept - has been introduced by Gray and Tall to denote a combination of a process, its outcome, and a symbol used to represent either of the two. ("2 + 4" is a symbol that points to the operation of addition, i.e., the process of adding two numbers, and to the outcome of this operation.) In connection with the decimal expansions the notion of procept was mentioned by M. Katz. Katz also observes the inherent confusion in the vernacular interpretation of the ellipses "..." that follow a finite number of decimal 9s as an unbounded number of repeated digits 9, noting that the expression "infinitely many 9s" is only a figure of speech, as "infinity" is not a number in standard analysis. His foray into the domain of the non-standard analysis confirms that in no way the symbol .999... may be interpreted as the "infinitely many 9s". This leaves the limit of a series of finite sums as the only consistent interpretation of that expression.

Following [Lightstone] one may introduce decimal expansions of the hyperreal numbers. For numbers between 0 and 1, these look like

0.d₁d₂d₃...;...d_H-1d_Hd_H+1...,

where H is an arbitrary hyperinteger. The semicolon separates the digits with finite indices (which are simply the natural,natural,real,rational,integer numbers) from the infinite indices which come from N^* - N. By the construction, hyperreals possess all the algebraic properties of the real numbers. In particular, the expansion 0.000...;...010..., with the lone 1 in the H's place is a rightful representation of the inverse of 10^H. However, there are significant differences between the decimal expansions of the real and hyperreal numbers.

First of all, whereas any sequence of digits 0.d₁d₂d₃... represents a real number, this is not true of every hyperinteger expansion 0.d₁d₂d₃...;...d_H-1d_Hd_H+1... Indeed, the symbol κ = 0.000...;...999..., with all the digits after the semicolon being 9 denotes no hyperreal number. To see that note first that if it were, it would be less than any positive real, for it has only 0s to the left of the semicolon. So it should be an infinitesimal. On the other hand, adding an infinitesimal to κ would spill a "carry" to the left of the semicolon, thus producing a number with a non-zero standard part. This would contradict the fact that the sum of two infinitesimals is infinitesimal itself. It follows that κ∉R^*.

The second point is that the decimal notation 0.d₁d₂d₃...;...d_H-1d_Hd_H+1... does not meet an expectation of showing a unique real number (before the semicolon) and an infinitesimal (after the semicolon) parts of a hyperreal number. For x = 0.d₁d₂d₃...;...d_H-1d_Hd_H+1..., a = 0.d₁d₂d₃...;...0... and ε = 0.000...;...d_H-1d_Hd_H+1..., one is not necesasry real, the other is not necessary infinitesimal. In fact neither even has to be a (hyperreal) number. To see that, consider the expansion of 1/3. Since its standard expansion consists of all 3s, the same holds for its non-standard expansion:

1/3 = 0.333...;...333...

But what follows after the semicolon,three 3s,symbol =,semicolon is not a hyperreal, let alone its being an infinitesimal. If it were, taken three times it would produce the above κ which, as we have seen, is not a hyperreal number. In the non-standard analysis, 1/3 is strictly more than 0.333...;...000... but we are not sure by how much. The last sentence was written with a tongue-in-cheek, for 0.333...;...000... is actually not a hyperreal number. On the other hand, 0.333...;...3330... is a hyperreal which is strictly smaller than 1/3.

Similarly, we see that 1 = 3×1/3 is strictly greater than 0.999...;...000... [Katz] shows even more. For any finite integer n > 0, 1 = 0.999...9 + 10^-n, where there are n 9s in the expansion. Expanding 0.999...9 = 1 - 10^-n to the hyperintegers,

0.999...;...9 = 1 - 1/10^H < 1,

where there are exactly H 9s. That is, for every hyperinteger H, the decimal expansion that consists of H 9s (even when H is infinite) is strictly less than 1!.

To repeat, for any infinite number of digits the expansion 0.999...;...9 is strictly less than 1. Whether this vindicates the oft mentioned students' intuition that 0.999... < 1 is questionable because 0.999..., as we just saw, does not correspond to an expansion with an infinite number of 9s! This is just a symbol that stands for a limit of a sequence of finite sums - neither less nor more. On the other hand, for any infinite H, st(1 - 1/10^H) = 1 such that for any infinite number of digits H st(0.999...;...9) = 1.

Back to What Is Infinity?

References

E. Gray, D. Tall, Duality, Ambiguity, and Flexibility: A "Proceptual" View of Simple Arithmetic, Journal for Research in Mathematics Education 25(2) 1994, 116-140
A. H. Lightstone, Infinitesimals, The American Mathematical Monthly, Vol. 79, No. 3 (Mar., 1972), pp. 242-251
F. Richman, Is .999... = 1?, Mathematics Magazine, Vol. 72, No. 5 (Dec., 1999), pp. 396-400
K. Usadi Katz, M. Katz, A Strict Non-Standard Inequality 0.999... < 1, February 23, 2009

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