Is 0.999... = 1?

Fred Richman


Few mathematical structures have undergone as many revisions or have been presented in as many guises as the real numbers. Every generation reexamines the reals in the light of its values and mathematical objectives.

F. Faltin, N. Metropolis, B. Ross and G.-C. Rota, The real numbers as a wreath product


A serious question

Arguing whether 0.999... is equal to 1 is a popular sport on the newsgroup sci.math---a thread that will not die. 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 stake is the sanctity of the current party line regarding the nature of real numbers.

Many believers in the equality think that we may no longer discuss how best to capture the intuitive notion of a real number by formal properties. They dismiss any idea at variance with the currently fashionable views. They claim that skeptics who question whether the real numbers form a complete ordered field are simply ignorant of what the real numbers are, or are talking about a different number system.

One argument for the equality goes like this. Set

x = 0.999...

multiply both sides by 10

10x = 9.999...

subtract the first equation from the second

9x = 9

so

x = 1.

Essentially you are observing that 9x + x = 9 + x, which is true, and then concluding that 9x = 9. That's a valid inference, if x is cancellable. A skeptic would say that 9x = 8.999..., which is different from 9, even though when we add x to each of them we get the same thing. The skeptic's intuition that 0.999... is not equal to 1 conflicts with the intuition that we should be able to cancel x. Which of those intuitions should take precedence?

An even simpler argument is

1/3 = 0.333...

multiplying by 3 gives

1 = 0.999... .

The second step is pretty hard to fault, so a skeptic must challenge the first equation. This argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking.

Yet a third kind of argument proffered by believers is that

0.999... = 9/10 + 9/100 + 9/1000 + ...

and the sum of the geometric series on the right is

(9/10)/(1-1/10) = 1.

A skeptic who accepts the series interpretation could say that 0.999... converges to 1, or that it is equal to 1 in the limit, but is not equal to 1. There is an ambiguity in standard usage as to whether the expression on the right stands for the series or for its limit. The fact that we use that notation whether the series converges or not argues in favor of the series interpretation. Also, we talk about the rate of convergence of such expressions. So some distinction between convergence and equality in the present case might well be appropriate.

Perhaps the situation is that some real numbers can only be approximated, like the square root of 2, whereas others, like 1, can be written exactly, but can also be approximated. So 0.999... is a series that approximates the exact number 1. Of course this dichotomy depends on what we allow for approximations. For some purposes we might allow any rational number, but for our present discussion the terminating decimals---the decimal fractions---are the natural candidates. These can only approximate 1/3, for example, so we don't have an exact expression for 1/3.

Decimal numbers

By a (nonnegative) decimal number, I mean an infinite string of digits with a decimal point, like

1247.4215347528... .

As usual, we don't allow the string to start with a 0 unless the decimal point comes immediately after. There are a couple of notational conventions. An infinite string of 6's (without a decimal point) is denoted by 6*. So the number at issue, 0.999..., is denoted by 0.9*. The number 120.450* is said to terminate, and is denoted simply by 120.45, while 120.0* is denoted by 120 (with no decimal point).

The decimal numbers are ordered in the standard way. Line up the decimal points and compare corresponding digits. At the first place where the digits differ, the number with the bigger digit is the bigger number. So 999.999... is less than 1247.421... because the initial 1 of the latter number is the first place where the digits differ, while 1247.430... is bigger than 1247.421... because the 3 in the former number is bigger than the corresponding digit 2 in the latter, and that is the first place they differ. In particular, 0 = 0.0* is the smallest decimal number, and 0.9* is less than 1.

How do you add two decimal numbers? There is a problem because carries can propagate over arbitrarily long stretches, and we can't start adding at the far right! But the carry can never be bigger than 1, so if the sum of the two digits in a given place is less than 9, or if it is greater than 9, then we can compute the digits in the sum up to, but not necessarily including, that place. If the sum of the digits is exactly 9 from some place on, then there will be no carry at, or past, that place.

What is the nature of this algebraic structure---decimal numbers under addition---that we have defined? You can check that the addition is commutative and associative, and there is an identity 0. The cancellable elements are precisely the terminating decimals because

0.9* + x = 1 + x for all nonterminating x.

Is 1/3 = 0.3*? Clearly if a sum is cancellable, then each addend is cancellable, so there is no decimal number x such that x + x + x = 1. That is, 1/3 is not a decimal number. More generally, no nonterminating decimal number x can satisfy an equation of the form mx = n with m and n positive integers.

What about multiplication of decimal numbers? It is convenient to define multiplication in terms of cuts, and we will want to look at cuts in any case.

Dedekind Cuts

Dedekind cuts are usually defined in the ring of rational numbers, but if we are interested in decimal numbers, we will want to work with a different ring. Let D be any dense subring of the rational numbers. That is, D is any subring of the rational numbers other than the ring of integers. We have in mind the ring of decimal fractions, those rational numbers that can be expressed with denominator a power of 10. A Dedekind cut in D may be defined as a nonempty proper subset S of D such that if x < y, and y is in S, then x is in S.

This is essentially Dedekind's definition in [1]. Dedekind then identified the cut {x in D : x < r} with the cut {x in D : x < r or x = r}, for each r in D, saying they were "only unessentially different." A similar move, made for example in [4, Definition 1.4], is to restrict to Dedekind cuts that do not have a greatest element, so {x in D : x < r or x = r} is not considered to be a cut. Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. Indeed, 0.9* corresponds to the cut {x in D : x < 1} while 1 corresponds to the cut {x in D : x < 1 or x = 1}. In general, we may identify an element d in D with the cut {x in D : x < d or x = d} (we call these principal cuts). So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning. That is why anyone who challenges that equation is, in fact, challenging the traditional formal view of the real numbers.

If D is the ring of decimal fractions, then each decimal number u gives rise to a Dedekind cut

{x in D : x < u or x = u}

in D. Note that this cut contains 0. Conversely, any Dedekind cut S in the ring of decimal fractions, that contains 0, is associated with a unique decimal number u as follows. For fixed m, the largest element of S that can be written as a fraction with denominator the m-th power of 10 gives the digits in u up to the m-th place to the right of the decimal point. The nonterminating decimal number 3.1415926535... corresponds to the cut consisting of all those decimal fractions r such that r < 3, or r < 3.1, or r < 3.14, or r < 3.141, or ... .

Let cut D denote the set of all Dedekind cuts in D. Define the sum of two cuts in the usual way

u + v = {x + y : x is in u and y is in v}.

It is easily shown that the commutative and associate laws hold. The additive identity is the principal cut 0 = {x in D : x < 0 or x = 0}. The elements of D, in the guise of principal cuts, form a subgroup of cut D. In fact D consists precisely of the (additively) cancellable elements of cut D. This is because

u + d = {x + d : x is in u}

while if 1¯ = {x in D : x < 1}, then u + 1¯ = u + 1 for any cut u that is not principal. However, the nonprincipal cuts are cancellable among themselves, and are closed under addition, so they also form a subgroup of the monoid, cut D. This group may be identified with the traditional real numbers, as Rudin does with cuts in the rational numbers. Recall that any traditional positive real number has a unique nonterminating decimal expansion. Note that 0¯ = {x in D : x < 0} is the identity element of the group of nonprincipal cuts.

The order on cut D is given by inclusion of cuts. The weakly positive cuts are those that contain the rational number 0. These correspond exactly to the decimal numbers if D is the ring of decimal fractions. The product of two weakly positive cuts u and v is defined to be {st : s is in u and t is in v}. This multiplication on weakly positive cuts shows how to multiply any two decimal numbers. It's straightforward to show that the associative, commutative and distributive laws hold. So the decimal numbers form a positive, totally ordered, commutative semiring in the sense of [3].

The picture here is the traditional real numbers, in the form of nonprincipal cuts, living uneasily together with the ring D, in the form of principal cuts. For each element d of D, there is a traditional real number d¯ just below it, and u + d¯ = u + d for each traditional real number u. That, for traditionalists, is a complete description of the additive structure of cut D. Note that d¯ = d + 0¯.

Clearly 0.9* = 1 + 0¯, so 0¯ is a sort of negative infinitesimal. On the other hand, you can't solve the equation 0.9* + X = 1 because, in cut D, the sum of a traditional real with any real is a traditional real.

Another point of view

We looked at Dedekind cuts in order to describe multiplication of decimal numbers, and to see another way of describing the decimal numbers themselves. Because we looked at cuts in the ring of decimal fractions, both positive and negative, we got some numbers in addition to the decimal numbers. While the number 1¯ corresponds to the decimal number 0.9*, there is no decimal number corresponding to (-1)¯, which is the cut {x in D : x < -1}. Nor is there a decimal number corresponding to -1 itself.

However, cut D can be characterized in the following way (for D the nonnegative decimal fractions). It contains the decimal numbers and all the decimal fractions, both positive and negative. Each element of cut D can be written as a difference u - d of a decimal number u and a (nonnegative) decimal fraction d. We may think of cut D as being obtained from the decimal numbers by adjoining the negative decimal fractions, and taking sums. This construction is legitimate because the decimal fractions are cancellable in D.

Instead of extending the decimal numbers so as to include additive inverses of those decimal numbers that are cancellable under addition, we could extend them so as to include multiplicative inverses of those decimal numbers that are cancellable under multiplication. These are exactly the positive decimal fractions, because (0.9*)x = x whenever x is a nonterminating decimal number. This construction is an instance of forming a semiring of fractions, see [3]. It is not hard to verify that the result is (isomorphic to) the weakly positive elements of cut Q, where Q is the ring of rational numbers.

Open problems

Although we can introduce negative decimal fractions, negative numbers in general present a serious problem because we don't have cancellation in cut D. We can't simply write them as additive inverses of positive numbers. Moreover, we have no interpretation for the number -3.14159265... because this represents a process of approximation from above, -3.14159 being greater than -3.14159265..., whereas in cut D all real numbers are approximated from below. Of course we could just introduce symbols like -3.14159265..., but it's not clear how to get a satisfactory coherent system that incorporates them.

Because of this, multiplication of arbitrary real numbers is also a serious problem, if for no other reason than that we don't know how to multiply -1 by 3.14159265.... Even in the traditional approach, multiplication is awkward. The elegant treatment of addition is replaced by an ugly division into cases: one defines how to multiply positive numbers, and extends to negative numbers according to the usual rules [4, pages 7--8].

References

[1] Dedekind, Richard, Continuity and irrational numbers, (1872), in Essays on the theory of numbers, Dover, 1963.

[2] Faltin, F., N. Metropolis, B. Ross and G.-C. Rota, The real numbers as a wreath product, Advances in Math., 16(1975), 278--304.

[3] Golan, Jonathan S., The theory of semirings with applications in mathematics and theoretical computer science, Longman Scientific and Technical, 1992.

[4] Rudin, Walter, Principles of mathematical analysis, McGraw-Hill, 1964.



Last modified June 8, 1999