Why 0.999... = 1

Is anyone else inordinately annoyed that ⅓ doesn’t have a nice decimal expansion? ½ we can just write as 0.5, 1/10 as 0.1, even something like 1/512000 has 0.000001953125, which is difficult but at least it stops at some point. ⅓ has infinite 3’s, 0.33333… and they never stop. Why is it like that? The short answer is that it’s like that because 3 isn’t a factor of 10, and decimal notation is built around powers of 10, and for now that’ll have to suffice. Because I don’t want to talk about that today, I want to ask the question of why it makes sense to have infinite nonzero digits in the first place– what does that actually mean?

Then of course, whatever we find out, we’re going to generalize, because what’s the point of learning something new if you don’t stretch it out well beyond the original context in which it was found.

First, a refresher on decimal notation. When we write a number like, say 19683.71828, each digit represents a multiple of a certain power of ten. In this case,

This is exactly what’s meant by there being a 9 in the thousands’ place, or a 7 in the tenths’ place; this number includes 9 “groups of 1000” or of 10³, and 7 “groups of one tenth,” or 10⁻¹. All of this feels pretty reasonable and familiar, but then what if we look at ⅓?

But then you’re adding together infinitely many terms. Maybe that feels fine to you, but it should at least ruffle a few feathers. Consider for example, this other infinite sum

What’s the value of that? Infinity? Sure, we can roll with that. But then what about

Now we’re adding up the squares of all the numbers. Is this also infinity? It seems like it should be substantially bigger than the first one, at the very least. But whatever, infinity is big, maybe it has space for multiple different infinite sums. But then what about something like

What’s that equal to? If you take finite sums, it just keeps bouncing up and down, getting super negative, then super positive, and back and forth forever. What is it at the end?

Maybe you’re rolling your eyes at this point. I clearly just pulled that last one out of a hat to be annoying, but it still illustrates my point. There are some infinite sums that make sense to add up and get a value, and some that don’t. How can you look at one and know for sure that it makes sense? And to show that this is generally difficult, take a look at the so-called “harmonic series.”

Would you believe me if I told you this sum actually went off to infinity, just like 1+1+1+1...? You can even limit it to prime numbers

And this still goes off to infinity. Moreover, we have the surprising (or possibly cliché) result

So pray tell: what’s so special about infinite sums like that one, or like the one for ⅓ that make them summable to a finite value, but things like 1+1+1… or 1+½+⅓+… that aren’t? The answer lies in the idea of convergence.

Convergence is, roughly speaking, the way mathematicians formalize what a process or object does “at infinity.” Sometimes, as with our alternating squares sum above, it’s not really possible to say what that is. But frequently it is. Today we’re just going to look at a few simple examples in the real numbers, but suffice it to say there’s plenty I’m not sharing.

A sequence of objects is an infinite list of those objects. A sequence of integers is a list of integers, a sequence of real numbers is a list of real numbers, a sequence of frog genera is an infinite list of frog genera. The elements of a sequence don’t need to be distinct; something like

is a perfectly valid sequence of integers, even though some are repeated. We often denote a sequence like (aₙ), where n denotes the index of the sequence. That is, a₀ is the 0ᵗʰ element of the sequence (sometimes we’ll start our index at 0, sometimes at 1), a₁₆ is the 16ᵗʰ element, and so on. We can use this notation to specify a sequence: for example, our alternating sum of squares can be described by aₙ = (-1)ⁿn².

A series is an infinite sum of objects. Basically the same as a sequence, but added together. For example, we’ve been somewhat looking at the sequence

or aₙ = 3×10⁻ⁿ. But what we’re actually more interested in is the series that you get by adding all of those terms together:

As a point of notation, we sometimes use the Greek letter Σ, read “sigma,” to denote a sum. For the sum above, we have

Where Σ means “add all these things together”, the n=1 on the bottom means “start the sum with n equal to 1”, the ∞ means “and make it an infinite sum; go on forever,” and 3×10⁻ⁿ are the terms being added. Plug everything from 1 to infinity into that expression, and add them all up.

Frequently, we will identify a series with its sequence of partial sums. That is, we make a sequence out of the series by only adding the first however many terms. With Σ notation, that looks like this.

Here we’re using a capital S to evoke the capital sigma, and to stand for “sum.” For the ⅓ example, it’s something like this

And now we’re ready to dive into convergence

We say a sequence (aₙ) of real numbers converges to a limit L if, for any ε>0, there exists some N such that if n>N, then |aₙ – L|<ε.
That definition is chock full of technical jargon so let’s unpack it a little bit.

We know what a sequence is now, so that’s fine. L is just the name we give to the “limit” of the sequence, which we can basically think of as what aₒₒ “should be.”

Then we have ε, which is another Greek letter: “epsilon.” In general, whenever you see ε in math, you should think of it as being very small. When we say “for any ε>0,” what we mean is “ε can get as small as it wants, as long as it’s positive.” ε is the Greek equivalent of the letter E, so here you can imagine it as standing for “error.” This is especially prudent as we move on to the next part of the definition: N.

If you think of ε as being an error term that you make very very small, N exists as a kind of threshold. If (aₙ) converges to L, then no matter how small you make ε, if you go out long enough you’ll find some large value N such that from then on, your sequence is always within ε of L, in the sense that the absolute value of the difference of the terms of your sequence and L will be less than ε. For any value of n larger than N, the terms of your sequence will be “sufficiently” close to L, and you can make this happen no matter how stringently you define “sufficiently.”

Now with this definition in our pocket, let’s prove something a little weird. I claim that

There are lots of proofs of this fact, many of which are much simpler than this. For example, a popular one goes like this

Then by the transitive property, 0.999…=1. This is a delightful little proof, but it hinges on the fact that our sequence converges at all; that 0.999… has a limit. As we saw earlier, some sequences that seem like they should have limits actually don’t. To resolve this, we need to fiddle with convergence:

To this end, first we’ll take our sum and extract our partial sums

To compare these to 1, we’ll write them as fractions

In general, we have Sₙ equal to a string of n 9’s in the numerator, and 10ⁿ in the denominator. It happens that n 9’s is just 10ⁿ-1, so we have more generally that

Now, if we want our sum to converge to 1, we need |Sₙ– 1| to get small. Indeed, we need it to be eventually less than ε for any value of ε. So let’s compute it really quick. Note that 1>Sₙ so we’ll actually compute 1-Sₙ, since that has the same absolute value.

Now, for arbitrary ε, we need to find for our value of N. Thankfully, that’s just a bit of algebraic manipulation.

Whatever ε is, if it’s positive, it has a base-10 log, so we can find our N. And since 10⁻ⁿ is decreasing in n, which is to say it gets smaller as n gets bigger, we can conclude that if n>N then 10⁻ⁿ<ε, so long as N>-log₁₀(ε). Altogether, this proves that the sum converges to 1, and since the sum is precisely 0.999…, that means 0.999...=1.
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I think it’s also instructive to also show a non-example, investigating a sequence that doesn’t converge. To make it easy on ourselves though, we’re going to pick a rather simple example

That is, the infinite sum 1+1+1+1… . It sequence of partial sums is, as is easily found, 1,2,3,4,5,… . We’re going to show that it doesn’t converge to any limit by contradiction.

Say that 1,2,3,4… does converge to a limit L. L is finite of course, so there’s some integer above it. If it’s an integer, just add 1. If it’s not, round it up to the next integer, and call that integer M. We’re now going to consider a specific value of ε: specifically, let ε=M-L.

Our sequence of partial sums just walks up the positive integers, so at some point it will be equal to M. Right then, we have |Sₙ– L|=ε, by definition. But then |Sₙ₊₁– L|=ε+1, and |Sₙ₊₂ – L|=ε+2, and so on. The distance from Sₙ to L will always increase, so it will never again get below ε.

So then the sum doesn’t converge to L, but remember that L could have been anything. So the sum doesn’t converge to anything – it just doesn’t converge, just as we sought out to prove.
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And to close it out, let’s try out the example that started us thinking about this whole mess: ⅓. We have this sum ere

Which begets its sequence of partial sums (in decimal and fractional form)

In order to work with this more conveniently, we’ll do a bit of algebraic manipulation. The numerator of Sₙ is a string of n 3’s, which is a third of a string of n 9’s, which is 10ⁿ-1. As such, the general term should be

Just like with the .999… example, we need |Sₙ – ⅓| to get small, so we compute |⅓ – Sₙ|

So in absolute value, |Sₙ–⅓| = ⅓×10⁻ⁿ. As before, this value is decreasing in n, which means as n gets bigger, this gets smaller, so all we need to to is find a value of n such that ⅓×10⁻ⁿ<ε. Pushing some algebra around, we get

No matter what ε is, we can pick an N at least that big, and we’re done. We’ve shown, just as we set out to, that

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There’s a lot more that can be said about sequences and their convergence. You could just study the behavior of sequences of real numbers for an entire semester. Indeed, that’s what a real analysis course is. But this isn’t meant to replace a real analysis course, it’s meant to be a fun little foray into convergence.