0.99999... = 1: A discussion of numbers

[Surlethe]

Quick note on this question, which pops up every once in a while. I'm soliciting comments before I bump this to the Library for it to die in peace.

Confusion over whether 0.9999999.... = 1 stems from a fundamental misunderstanding of what real numbers are. So, let's first review natural numbers, integers, and rational numbers (fractions).

Natural numbers
Natural numbers are, well, natural. We all know what the natural numbers are: {1, 2, 3, 4, ...}. Counting numbers are used to represent groups of discrete objects, so they are a generalization of concepts like "number of cattle in my herd" or "number of people in the city." The direct applications of the natural numbers made them one of the first mathematical ideas invented; virtually every primitive civilization created the natural number system. There is also an operation on the natural numbers, that is, a way of putting two numbers together and getting another out: addition. Everybody understands how this works (and a rigorous discussion of defining the natural numbers and addition in terms of the axioms of set theory is, of course, beyond the scope of this document).

Question: what if I have no cattle? What if no transaction occurs? What if nobody enters or leaves a room? Let's invent another number: 0. This innovation was developed in India and spread through the West.

Integers
Natural numbers are fine for simple counting, but for accounting we need a system a little bit more subtle. We need another operation: subtraction. (If I have fifty cattle and I owe you twenty, you add twenty cattle to your herd and I subtract twenty from mine.) What if, however, I have no cattle and I still owe you twenty cattle?

Mathematically, we call this state of affairs "incomplete under subtraction." To deal with this, we define another set of numbers and append them to the natural numbers and zero: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Now we can deal with the accounting of discrete objects. Note that this set is complete under addition and subtraction: if you add or subtract any two integers, you get another integer.

There is another operation on the integers: multiplication. This leads us to the next question: what if I have an inheritance, say, of thirty cattle and three sons to whom I wish to leave them? What if I have thirty acres of land and four sons? We can do the first operation by undoing multiplication and dividing: thirty cattle distributed among three sons leaves ten cattle per son. But the discrete number thirty does not divide evenly among four sons.

Rational numbers (fractions)
This is known as the "distribution problem". To solve it, we can invent more numbers! (Mathematically, we are appending an additional set of numbers to the integers in order to make it complete under division.) Let's say we have an integer (4 sons) and another integer we want to divide: (30 acres). We can define the number "30/4" as what each son gets. We know from experience that each son gets 7.5 acres. So we define our fractions (rational numbers) as all possible ratios of integers, save those with 0 in the denominator.

Straightforward so far. But there is a hidden subtlety: if there are 15 acres and 2 sons, each son still gets the same amount. What's the problem with this? 15/2 and 30/4 are, under our schema, two different numbers. (Pay attention here; this is an important detail.) But we know that they're the same in practice. So we make the rational numbers smaller: we define two fractions (p/q) and (r/s) to be the same if ps = qr. These, then, are the fractions we know well and love.

(For future reference: all fractions can be expressed as decimals, either terminating (e.g., 1.43 = 143/100) or repeating (e.g., 0.333333... = 1/3).)

We're making significant progress. But rational numbers are not adequate for describing things like distance. For instance, the diagonal of a square is not rational, i.e., cannot be expressed as the ratio of two integers. (Can you prove this, reader?)

Real numbers
The discovery that there existed quantities which could not be described by fractions created a bit of a rumpus (I've heard it said, variously, that the man who discovered it was killed or committed suicide; he was a devoted Pythagorean). But, to modern people with no ideological attachment to the purity of integers, we need only repeat the process: we define numbers which can describe distances and other continuous quantities, and then append those to the rational numbers.

We can't exactly describe things like distance, but we can approximate them as close as we like. For instance, it's well-known (although non-trivial to prove) that the number pi is a non-rational number. But we can approximate it as close as we might like: {3, 3.1, 3.14, 3.141, 3.1415, ...}. The further out in the sequence we get, the closer we get to where we think pi ought to be.

So why not define pi to be that sequence? In fact, let's just define a whole new set of numbers by the sequences that "converge" to them.

After a bit of mathematical formalism, we get the real numbers. Problem: for every real number, there are a whole boatload of sequences converging to it. For examples, {1/3, 1/3, 1/3, 1/3, 1/3, ...} and {0.3, 0.33, 0.333, 0.3333, ...} both go to 1/3 and {3, 3.1, 3.14, 3.141, ...} and {3, 3.2, 3.1, 3.15, 3.145, 3.1415,...} converge to pi. This is the same problem we had with fractions: lots of fractions are the same number. Same here: lots of sequences are the same number!

So we just define the sequences which converge to each other to be the same number! This is just like the fractions: when we have "numbers" which ought to be the same, we just define them to be the same and see if the new scheme is self-consistent. After some mathematical gymnastics to prove that it is self-consistent, we get: the real numbers we know and love.

Application: 1 and 0.999...
So let's clear up the confusion about 1 and 0.999... . Let's see about the decimal 0.9999 ... . It's a real number, so let's pick a sequence which converges to it: {0, 0.9, 0.99, 0.999, ...}. It converges to 1 if the sequence {1-0, 1-0.9, 1-0.99, 1-0.999, ...} converges to zero (that is, if 1 - 0.999... = 0). But the sequence {1, 0.1, 0.01, 0.001, 0.0001, ...} gets smaller and smaller and smaller, so it goes to zero. That means that 0.999... = 1.

The big confusion here is that 1 and 0.9999... are two different ways of writing the same number. This is just like fractions: 1/2 and 2/4 are the same because they are two ways of writing the same number. The only difference is that the real numbers require a little bit more mathematical sophistication to treat rigorously; however, as you (dear reader) prove, the conceptual construction of the real numbers is certainly not beyond the layman.

[adam_grif]

Might as well throw this in with your explanation:

1 / 3 = 0.333...
2 / 3 = 0.666...

Therefore,
3/3 = 0.999... = 1

and

x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9

Therefore,

x = 1 = 0.999...

[Starglider]

Good explanation. Infinitesimals are just inherently counter-intuitive, but at least you can get your head around them by thinking in terms of limits. Transfinite numbers though, I can see the basic utility of the concept, but it still seems like symbolic play utterly divorced from reality to me.

[Bottlestein]

@Surlethe: Minor nitpick: Real numbers are the set of points such that specific sequences (Cauchy) converge to a number in that set. Your points are definitely true, but you should add:
To define the whole set of real numbers, take any Cauchy sequence of rational numbers, or irrational limit points of Cauchy sequences of rational numbers (e.g. pi, or sqrt(2)), and then take the limit points of these sequences. The set that contains all such limit points is R. (From the definition of Cauchy, it's easy to prove limit points exist.) This requires some additional work, since we have to prove that the metric for rationals carries over to R, but this formulation helps with things such as proving rationals are dense in R, etc.

[Surlethe]

@Surlethe: Minor nitpick: Real numbers are the set of points such that specific sequences (Cauchy) converge to a number in that set. Your points are definitely true, but you should add:

To define the whole set of real numbers, take any Cauchy sequence of rational numbers, or irrational limit points of Cauchy sequences of rational numbers (e.g. pi, or sqrt(2)), and then take the limit points of these sequences. The set that contains all such limit points is R. (From the definition of Cauchy, it's easy to prove limit points exist.) This requires some additional work, since we have to prove that the metric for rationals carries over to R, but this formulation helps with things such as proving rationals are dense in R, etc.

Yes, I'm aware of the technicality. You're missing a subtlety, though: not all Cauchy sequences of rational numbers converge. (In fact, most of them do not.) Convergence in a set requires the limit point exist in the set, but what we're after is defining the limit points, so they don't exist until we construct R in such a way that they do and then prove that we can embed Q in it. The real numbers are the equivalence classes of Cauchy sequences of Q, with appropriately defined metric and operations.

[Bottlestein]

Might as well throw this in with your explanation:

1 / 3 = 0.333...
2 / 3 = 0.666...

Therefore,
3/3 = 0.999... = 1

This step is the problem with the proof. You have assumed most of the results needed to carry out this addition. Specifically, 0.333... is a (Cauchy) sequence, though the "..." shorthand makes it look like a number. When you write "=" for the first two fractions, you are saying the sum of the Cauchy sequence converges to the number 1/3. Using the sum of an infinite geometric series, this is easily shown. We can give 2/3 the same treatment.

Now we move to 3/3: on the left hand side, you are doing an algebraic operation - which is fine. On the right hand side, you are adding up two Cauchy sequences. This requires: Cauchy sequences and their limit points have the same metric as rational numbers and their limit point (otherwise we can't write '=" in the third line). The Cauchy sequence converges - which requires an infinite geometric series sum, although that's easy application of the formula.

Thus, this doesn't really bypass the points in my previous post - it assumes all of those results in the use of the "..." and "=" notation being used in an "intuitive" way. For the purposes of demonstrating "0.9 repeating equals 1", this is fine. However, when we have to answer things like "Why can we treat infinity as a point on the real line?" or 'How can we prove the set of irrationals is larger than the set of rationals?' or "If the set of rationals is smaller that the irrationals, how can it be dense in the irrationals?", this formalism is vital.

[Bottlestein]

@Surlethe: Minor nitpick: Real numbers are the set of points such that specific sequences (Cauchy) converge to a number in that set. Your points are definitely true, but you should add:

To define the whole set of real numbers, take any Cauchy sequence of rational numbers, or irrational limit points of Cauchy sequences of rational numbers (e.g. pi, or sqrt(2)), and then take the limit points of these sequences. The set that contains all such limit points is R. (From the definition of Cauchy, it's easy to prove limit points exist.) This requires some additional work, since we have to prove that the metric for rationals carries over to R, but this formulation helps with things such as proving rationals are dense in R, etc.

Yes, I'm aware of the technicality. You're missing a subtlety, though: not all Cauchy sequences of rational numbers converge. (In fact, most of them do not.) Convergence in a set requires the limit point exist in the set, but what we're after is defining the limit points, so they don't exist until we construct R in such a way that they do and then prove that we can embed Q in it. The real numbers are the equivalence classes of Cauchy sequences of Q, with appropriately defined metric and operations.

Sorry, my first line wasn't clear: I meant to say that real numbers are the set such that any Cauchy sequence constructed from elements of the real numbers, will converge to a point in the real numbers.

[Surlethe]

I see, but still disagree. The real numbers are not unique: it is not the case that the real numbers are the only complete metric space --- e.g., any set X in the discrete metric (d(x,y) = 1 for all x != y) is complete because the only Cauchy sequences are those with constant tail, which converge to that element.

[Bottlestein]

I see, but still disagree. The real numbers are not unique: it is not the case that the real numbers are the only complete metric space --- e.g., any set X in the discrete metric (d(x,y) = 1 for all x != y) is complete because the only Cauchy sequences are those with constant tail, which converge to that element.

True - my post above does seem to imply the uniqueness of R. I guess what I was trying to include in the definition was that any Cauchy completion of a metric space will induce a map that preserves the metric (an isometry), and the original space will be dense in the Cauchy completion, independent of the properties of rational numbers. However, I see that without including the rational numbers as the original metric space, my definition won't work.

[Surlethe]

Moving to the Library.

[Kuroneko]

Although a proof only to the extent that real numbers mirror geometric concepts*, representations of geometric series are potentially useful to intuition. For example, the usual picture of a square being continually bisected (first taking half of it, then half of the remainder, and so forth) to represent the sum 1/2+1/4+1/8+... = 1 is nothing but the binary version of the same problem: 0.111...₂ = 1.

It's easy, though fallacious, to claim that 0.999... < 1 with some "bit left over," because the quantities involved are some abstract entities that many don't have any intuitive feel for. That seems to be the most common reaction to comparing those numbers, anyway. But invoking visual intuition of the geometric series makes it easier to see that there is no such remainder.

*And they do, as the Cantor-Dedekind axiom is required for analytic geometry, though it is better to avoid outright dependence on geometry for this topic. Interestingly, Hilbert's axioms for Euclidean geometry include a line completeness axiom that corresponds exactly to the Dedekind cut construction of the real numbers.

We're making significant progress. But rational numbers are not adequate for describing things like distance. For instance, the diagonal of a square is not rational, i.e., cannot be expressed as the ratio of two integers.

Unit square, surely. Or that the ratio of diagonal to a side is irrational, rather than the diagonal.

To define the whole set of real numbers, take any Cauchy sequence of rational numbers, or irrational limit points of Cauchy sequences of rational numbers (e.g. pi, or sqrt(2)), and then take the limit points of these sequences.

That's completely unnecessary. Though Surlethe already responded to this, I think the same kind of misunderstanding is invited by the following wording:

So we just define the sequences which converge to each other to be the same number!

Sequences converging to sequences is not something discussed previously, and might not have an obvious sense to the layman which does not beg the question. Why not just say that sequences with a term-wise difference converges to zero are the same number? Especially since that is the reasoning invoked below for for comparing 1 and 0.9̅.

"Why can we treat infinity as a point on the real line?"

What? Are you referring to a compactification? I'm not sure of any relevance here.

[Bottlestein]

"Why can we treat infinity as a point on the real line?"

What? Are you referring to a compactification? I'm not sure of any relevance here.

Yes - I was referring to compactification: There's the "one-point" projective compactification (I think its called "Alexandroff's compactification"), where R ends up homeomorphic to S^2, which is pretty intuitive. But also, there's a compactifiation where +/- infinity are treated as two separate points, and to define these points requires Cauchy sequence formalism. I was trying to get across that this second compactification might be less intuitive (since someone may not think of infinity as a "point"), but it becomes clear with Cauchy sequence formalism.

[Kuroneko]

But also, there's a compactifiation where +/- infinity are treated as two separate points, and to define these points requires Cauchy sequence formalism.

It definitely does not; probably the most common approach is to simply adjoin two elements and explicitly define their order. That's perfectly fine--about the worst thing than can be said about it is that it's not particularly elegant. But since every Cauchy sequence of either rationals or reals is bounded, I can't see how they improve that situation any.

Probably the most natural way of defining the extended reals is the Dedekind cut approach.
Def. A Dedekind cut of a totally ordered set X is subset A such that (a) for every a∈A, x<a implies x∈A, (b) A contains no largest element, and (c) A is nonempty and not X.
If one adjoins the complement of that set, X\A, sans a least element if it has any, then one has the intuitive picture of a set X being 'cut' into two pieces: <---A|B--->. But this is not necessary in practice. The real numbers can be defined as Dedekind cuts of rationals. To form the extended reals ℝ∪{±∞}, all one has to do is remove condition (c) from the cuts.

[Bottlestein]

^ The Dedekind cut without (c) does seem to be a better way to define the extended reals than what I learnt. I guess the method that I was taught was the "adjoin" method. I thought Cauchy sequences were required in the latter case to prove the "addition rules" with infinity, but I might be wrong about this.