Proving 0.999999… = 1

I always knew there were people who disputed or disbelieved this fact. It did in no way worry me, because for every fact there are people who disbelieve it. There is even a man out there who claims that (–1)·(–1)=–1.

But then I came across conversations on the discussion forums of the popular xkcd webcomic (about which I wrote in this article), and that did worry me. After all, the adherents of this comic are all (assumedly) scientifically educated or at least interested—or they could not appreciate the comic fully. So those educated people were discussing the matter as well, without reaching a concensus. So I concluded that it must be a point worth clearing up once and for all, and hence this article.

On some website of jokes in a section titled “You know you are mathematician if …” I read that you could recognize mathematicians by their interest in the question whether 0.999999… was 1 or not. This made me angry; in fact you can much rather recognize non-mathematicians by it. Mathematicians don’t ponder the question because they know the answer, and it is boringly simple; they certainly do not discuss it with their peers. If they ever talk about it, it is in order to explain it to non-mathematicians who have doubts. And this, by the way, is exactly why I am doing it here.

So I will now make a note of the observation that the argument usually brought forth against 0.999999… being 1 is this:

1 is a normal number, whereas 0.999999… is not a number in the usual sense, but a process.

and I will adress this in detail below.

There is a Wikipedia article on the topic, but I ran over the page and think it does not adress the arguments commonly brought up against 0.999999… being 1. It simply takes the things for granted that are disputed by people. I thought I could do better. If, for all that, you still feel that your objection was not adressed by my article, please leave a comment, and I will try to clarify that point!

Decimal literals

First let’s look at a finite decimal literal. We find what it means is a (finite) sum of fractions, e.g.

0.234 =

2

10

+

3

100

+

4

1000

Now a nonending literal like 0.999999… is therefore an infinite sum, which in mathematics is called a series. They can be written with the “Σ” symbol:

0.999999… =

9

10

+

9

100

+

9

1000

+

9

10000

+ … =

∞

Σ

i = 1

9

10ⁱ

And that is the point where the objection from above comes in. Summing up infinitely many numbers is a thing that simply cannot be done, because, indeed, it is a process that would never end. And all of mathematics only deals with operations that end at one point. So the argument is really to the point.

But notations as the ones above are used all the time in mathematics. How can we mathematicians use them if they are impossible? Simple: We mathematicians are a lazy lot, and we use infinite sums simply as a convenient shorthand notation for a quite different thing. In fact, all of mathematicians is a tower of shorthands of shorthands of shorthands of shorthands. If a mathematician sees that infinite sum he/she knows that it is just a shorthand for

∞

Σ

i = 1

9

10ⁱ

 = 

lim

N → ∞

N

Σ

i = 1

9

10ⁱ

which is the limit of the sequence of the finite sums, all of which can be easily calculated. This gives us

0.999999… = lim (0.9, 0.99, 0.999, 0.9999, …)

This is the actual, precise meaning of a literal like 0.999999…, which you probably already knew. But what is a limit actually? It is the issue all the question hinges on. You may not have learned this at school. I will show you now.

Limits of sequences

I think we have a fuzzy intuitive notion of a limit. If people are asked to describe it without mathematics they will say thinks like “it is the point where the sequence goes when the indices become infinite.” It is very much like the popular phrase that is fed to school kids about how parallel lines “meet at infinity”. It is a convenient way of visualizing the thing (in a fuzzy way), but if you think about it with any degree of stringency, you get into trouble, because it is simply wrong. You cannot track a sequence up to infinity, because it is a never-ending process. Also infinity is not a place nor a number. Parallel lines do not meet at infinity because there is no such place. They do not meet at all. Likewise the sequence 0.9, 0.99, 0.999, 0.9999, … never reaches 1.

No, limits are a mathematically precisely defined concept. For a sequence a₁, a₂, a₃, a₄, … we call a number a the limit of the sequence if and only if

∀ ε > 0: ∃ N_ε ∈ ℕ:  ∀ n > N_ε: |a – a_n| < ε

This is of course mathematical notation, which you might not understand, but I just could not resist putting it here because it is so beautiful. What it means in plain English is this:

A sequence is called converging to a number (which is then called the limit of that sequence), if any arbitrarily small vicinity of the limit is eventually entered by the sequence and never left any more.

With “vicinity” we mean an interval of real numbers containing the limit, but where the limit is not one of the borders. Usually we make these intervals symmetric, namely (if a is the limit) [a–ε, a+ε], which is the interval of all numbers whose distance from a is at most ε.

So there has to be a sequence index n, where the sequence element a_n has a distance from a less than ε and all subsequent elements do so too. For example the sequence

1, 0.1, 5, 0.01, 0.001, 5, 0.0001, 0.00001, 5, …

does not converge to zero because every third element is 5, and therefore for every distance smaller than 5 the sequence will break out of the vicinity around 0 at every third element.

So obviously not every sequence has a limit (i.e., is convergent), but if it has a limit, it has only one. There cannot be multiple different limits to one sequence, which is very easy to prove.

Note that this approach (which was invented by Cauchy in the early 19^th century) works in a way backwards from the intuitive notion. We can not take a sequence, do the limiting process, and arrive at a limit number. Instead we have to start with a number we think is the limit number (by educated guessing, intuition or one of many tricks that exist for this purpose), and then merely test whether it satisfies the condition for a limit as stated above. This may sound like a frustrating state of affairs, but it cannot be done better.

Let’s consider a stupid illustrative example. Imagine a turtle that crawls towards a goal line. After one minute it arrives at a position one meter before the line. But the run has made it tired, so now it is getting slower and in the next minute it only manages half of the distance remaining, ending up at half a meter from the line. It’s getting more tired still, so after the third minute it is still a quarter of a meter from the line. After the fourth minute one eighth of a meter remains and so on.

Now the intuitive approach would be to ride the turtle, so to speak, and see where it will arrive. As we know by now, it won’t work, because it will never arrive anywhere but will crawl on without end.

The Cauchy method means that we guess where the turtle is going (the goal line), so we just position ourselves on that line and consider: For any small distance from the line, will the turtle eventually arrive there and not go back? The answer is of course yes; we can easily give a formula when it will pass any point with distance ε>0 from the goal line (it involves a logarithm). Also we know it will never overshoot the goal, so the limit is not somewhere beyond the line. So we have proven that the goal line is precisely the limit of the turtle’s movement.

I think I have said enough now about limits, and you understand the concept. But there is another thing I want to mention because it is fun (for me, at least).

Famous Argentine author Jorge Luis Borges once wrote an essay titled The Eternal Footrace, which deals with the famous ancient greek logical paradox of the race between Achilles and a turtle. (Invented by Zeno of Elea, for us it is not a paradox anymore. We have solved it, precisely by the application of Cauchy’s definition of a limit.) To quote from that essay:

One only needs to numeralize the speed of Achilles as one meter per second to give the time he needs as follows:

10 + 1 +

1

10

+

1

100

+

1

1000

+

1

10000

+ …

The final value of the sum of this infinte geometric progression is twelve (more exactly, eleven and a fifth; more exactly eleven and three twenty-fifths), but it will never be reached.

The funny thing is that Borges correctly identified the series as a geometric one, but totally failed to give the correct limit. He gave several values, as you saw, but admitted they were only approximations. I have no idea why he could not do it. The correct formula to derive the limit is taught in schools, and has been known for thousands of years. In this case the result is eleven and a ninth. Also you might have noticed how turtles seem to be important mathematical animals.

Wrapping things up

Now we can apply this definition to our sequence 0.9, 0.99, 0.999, 0.9999, …. I claim the limit is exactly one; let’s test if the Cauchy criterion holds.

As with the turtle example, the sequence will never overshoot 1; therefore the limit is certainly not larger than 1. Take for example a distance of 0.0001 from 1: it will be reached by the sequence with the fouth element and the following ones will not drop back down again. A distance of 0.0000000001 will be reached with the 10^th element and so on. Generally a distance of ε>0 will be reached with the first element whose index is at least log(1/ε), where “log” is the common logarithm.

So that’s the proof and therefore

0.999999… = lim (0.9, 0.99, 0.999, 0.9999, …) = 1

What’s there to say more on the issue? I don’t know. You might be saying, “you claim a limit is that thing you described, but I say it is something else.” Well, that is exactly as if you said that “” is not a quarter note, but something else. I am sorry, but it is. It just is defined that way.

Mathematical expressions like “0.999999…” are just strokes on the paper (or screen) that were ascribed certain meanings to by mathematicians. And if you use these strictly defined meanings you arrive at 0.999999…=1 as shown. There’s no hidden notion there that transcends the definition. It’s just black strokes, and we define what they mean.

Now please note this: You may very well assign a different meaning to these penstrokes and symbols, which does not give you the value of 1. It is allowed , and mathematicians play with things all the time. There is no conspiracy or establishment or mainstream at work to keep you from it. The usual definition is in no way more true than any other (provided it is logically consistent!), just like baseball is not more true than softball. Just be aware that your results do not tell you anything about the results of people who are using a different definition. In addition, always clearly state your definitions, if they deviate from the usual, so others can see what you were doing. For example, the word “also” means “too” or “as well”, whereas in German “also” means “therefore”. Would you say the English word proves that German is wrong?

What mathematicians find funny

As I mentioned in my second Dan Brown article there are special mathematicians’ and physicists’ jokes but they are not good. Here is a particularly hideous example:

All the real functions are having a party, when suddenly someone shouts: “A differential operator is coming! Run for your lifes, he will kill or distort us all!”

The all scatter and run away except one who stays behind. When the differential operator arrives he confronts the remaining function: why it is not afraid like the others?

The functions answers: “I am exp(x), you cannot harm me.”

But the differential operator laughs: “But I am ∂/∂y, ha ha ha!”

This is a typical example. If you have forgotten your calculus you may not understand it, but if you do it’s still not funny. Do mathematicians laugh at that kind of joke? No. In fact, the only kind of enjoyment to be gotten from jokes like these is in the knowledge that you understand them. Some people have tried to collect mathematical jokes and all they could scrape together was a sad little pile which made no-one laugh.

There are some jokes with which mathematicians make fun of physicists and vice versa. They feed off an ancient rivalry and some of them are rather good.

The following one is a variant of the popular “how to catch a lion” jokes. I have read a lot of them, which go: “How does a [insert profession] catch a lion?” followed by a more or less funny answer. Other than the “How many […] does it take to change a light bulb?” jokes catching-lion jokes seem to focus on scientific professions and their various sub-categories—e.g. general physicists, newtonian physicists, relativity physicists, string theorists, theoretical physicists etc. Or different kinds of mathematicians. Or programmers of different programming languages. Many are stupid and shallow, but most require you to know the basic characteristics of the respective fields, languages etc. The one I’m going to show you may very well be the original one. It is a thing a physicist might tell to make fun of mathematicians.

How does a mathematician catch a lion?

He puts up a cage, steps inside, closes the door and declares “Let this be the outside!”

So much for mathematical jokes, but what about the actual sense of humour of mathematicians, or of scientists in general? What do we find funny?

Generally scientists (or at least those that I have met) are a sarcastic lot. Then again, most people seem to be these days, so it may be nothing of note.

The best way I can think of to describe the sense of humour of scientist is to talk about the cartoons that are pinned to walls at the university institutes. In the foreword to The Far Side Gallery, Pt.3, renowned biologist Stephen Jay Gould stated that, judging by the walls of his institute, the favorite cartoonist of his collegues was Gary Larson. I found this to be true at our university too.

Why Larson? Firstly, if you have never seen any of his work, do so today! It’s really great! The Far Side Gallery, parts 2 and 3 show him at his best, I think. Some years ago it was practically impossible to find any Larson on the internet: Gary Larson had put out a letter to the world (by way of his lawyer) asking people not to infringe his copyright that way. It was not the most friendly thing to do, but of course he was perfectly entitled to it. Amazingly people complied, at least for the time being. It seems Larson has given up on this endeavor since, as by now the Cartoons have returned to the web in droves. So, have fun with the search engine of your choice!

By the way: There seems to be only a single small black-and-white photograph availabe of Gary Larson the man. There’s a guy who values his privacy. Almost like Pynchon.

So why exactly do scientist like Larson so much? He drew a lot of cartoons featuring all kinds of scientist (always wearing white lab coats, don’t cha know) and these, small wonder, are the favorites. Now other cartoonists have drawn scientists in lab coats too, what makes Larson’s special? I think his basic shtick in these cartoons is to show that scientists are childish and silly just like everybody else. It ran counter to the public perception of us so we loved it.

But in the last few years clearly Larson has been superseded by a new artist. Despite the fact that Larson got a lot right about scientists from a psychological point of view, he badly blundered when it came to the nuts and bolts. As I mentioned his scientists wear lab coats at all times, and they write on chalkboards and what they write is utter gibberish. There were jokes about scientists but never jokes about science, because Larson did not have the kind of knowledge it took to do them.

Since then a new cartoonist has entered the scene who is an actual scientist (a rocket scientist, to be exact), and he gets everything right. His name is Randall Munroe and he draws the XKCD webcomic.

I suspect it may boring stuff for biologists but physicists, mathematicians and other computer creatures find it delightful. These days institute corridors are filled with prints of Munroes little stick people (male and female) and at the last mathematicians’ party I came across more than one XKCD T-shirt.

I don’t think anyone will get every one of Randall Munroe’s comic strips (I know I don’t), unless you know a thing about relativity, string theory, iPods, MMORP games, mathematics, chess, Unix, programming and 10.000 other things. It feels good if you know what some of the more obscure ones are about. Like a huge in-joke. I think this accounts for much of the strip’s appeal.

Also the people in it are geeks drawn by a geek behaving in an authentic geeky way (let me stress the word “authentic”) and so we real-life geeks can relate to it.

Randall Munroe must be credited for helping the female scientist (or the nerd girl, to put it bluntly) to get her rights. In XKCD’s world, science is an equal-gender business and there is a lot of love and romance between the little stick people. It is a long way from the all-male labcoat wearers. It’s not completely the way reality looks like today but it is what I (and many others) hope we will achieve one day, and I think we’re heading in the right direction.

Speaking of science and romance, these have always been held to be mutually exclusive by popular culture. Lab coat wearers are basically asexual. In mainstream movies the professor is never the one who gets the girl in the end (except if they never were much of a professor anyway, like Indiana Jones). Of course that is not true, scientists have the same feelings as other people. Still we are full of nerdy quirks. XKCD shows how it works out.

There a small downsides to Randall Munroes cartoons too, of course. He does indulge in a certain amount of soapbox preaching on the greatness of science, for example (“because it works, bitches!”, as he puts it), which I personally find a little menacing. Also he writes from his own point of view, which is physics, so for example biologists will feel severely underrepresented, not to mention linguists, sociologists, or musicologists. Still it is the best thing of the last years.