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?

Proving 1 + 1 = 2

Mathematics is the science where we always prove things. This has even entered into popular rhetorics: If we claim that some fact or other is “mathematically proven” it is the highest kind of authority. Usually this is sloppy phrasing or simply a lie, because sociological, psychological, political, even physical facts cannot be mathematically proven, because you can only mathematically prove mathematic statements, and they are not. Mostly they are, if at all, juristically proven (“true beyond a reasonable doubt”). But in our own sphere we mathematicians prove everything, and if we can’t we don’t assume that thing as true (be that painful as it may).

I have a few times explained this to students. Usually the consequence is that someone ask me how you can prove that 1 + 1 = 2, or something along that line. In these cases I have to admit I generally just answer that this is indeed possible to prove, but don’t show how. The reason is that it simply would take too long. But here in this medium I have all the time I need.

If you read this article (and others in the same vein, which are sure to come soon) you will know what lies at the heart of mathematics. This is the stuff they don’t teach at school and I promise, it won’t contain any of the stuff you learned to hate when you were 14. (Fractions, for example: A neat and immensely useful method, but how on earth could anybody like doing them? It’s the results and the proofs that give satisfaction, the nuts-and-bolts calculations is just work for performing monkeys.)

If you feel that despite this you cannot be interested just skip this article! Mathematics is just one of my topics in this blog.

We start by asking: what are 1 and 2 anyway, or to be more exact: what are numbers?

For quite some time mathematicians have been using a tool called the axiomatic method. This lies at the roots of all mathematics, and certainly deserves to be talked about in detail. But I want to keep this article as short as possible and so I am forced to defer this undertaking to some other time. But I promise I will make good for it soon!

The basic idea is this: We might think we all share a common notion of numbers (that is natural numbers, 1,2,3,4 etc.). But is it true? In prehistoric times people used to count “one–two–many…” (allegedly there is still a tribe in Brazil that does that), and I read somewhere that in Indo-European languages the words “nine” and “new” are etymologically related—because when nine was invented after centuries of people only knowing 1 to 8 it became the new number (about 5000 years ago).

So I think we certainly do not have an instinctive notion of numbers. We merely learn it at a young age because our world is filled with numbers.

Learning numbers that way resembles learning a game by watching other people play and trying for yourself until you are sure you are doing it right. The downside is that in this way you can never be sure you are doing it right. Maybe in the next match a situation might arise where you do not know what to do? Some special case you are not prepared for?

But mathematicians want to be sure. In this analogy, it would mean that we want to thoroughly read the rules of the game before we play.

So mathematicians (actually it was an Italian, Giuseppe Peano) wrote down a set of rules (or axioms) for natural numbers. They clearly state which (natural) numbers exist and some properties of them. Here are Peano’s axioms:

1. There is a natural number, call it 1.
2. For every natural number n there is a natural number n′ which is the successor of n.
3. The successor of any natural number is not 1.
4. If two natural numbers have the same successors, they are the same.
5. If a set of natural numbers contains 1 and, for every number n in it also contains its successor n′, then that set is the set of all natural numbers.

Once we have these rules we will never give a thought to what natural numbers actually are. We don’t know and we don’t need to know. All we need to know is that we can use these rules freely. We have no idea what numbers are, but we know what they do. (If I write any more on this, it is bound to get philosophical and I am going to reserve that for some other occasion. Suffice to say that however you may look on the matter, if you accept these rules as true you are still with me—and we will not use anything other than these rules.)

Adding up

Now can define a few numbers and give them names:

• 1 we already know.
• 2 = 1′
• 3 = 2′ = 1″

and so on …

Now we know what 1 and 2 are, what what do we mean by “+”?

The task at hand is to define addition using only the Peano rules of natural numbers. This is done by stating two simple laws of addition, which I am sure will concur with your “instinctive” notion of Addition. We state:

“+” is an operator between two natural numbers that satisfies the following laws:

1. n + 1 = n′ for all natural numbers n.
2. n + m′ = n′ + m for all natural numbers n and m.

Now what is 1 + 1? We simply use the first rule, setting n = 1 and get

1 + 1 = 1′ = 2.

That is a proof. If we consent that natural numbers satisfy the Peano rules and that addition satisfies the two rules stated above and 2 is the successor of 1 then 1 + 1 = 2 is mathematically proven.

Now that was disappointingly simple. Let’s try for something harder. What’s 2 + 2 (that is 1′ + 1′)?

This time we start with addition rule (2). We find 1′ + 1′ = 1″ + 1. After that we once again use rule (1) and obtain

2 + 2 = 1′ + 1′ = 1″ + 1 = 1′′′ = 4.

I think you will now have no problem calculating 5 + 9 or whatever takes your fancy.

Two little things

As you probably know an ancient Middle East people, the Sumerians famously built huge temple platforms called ziggurats—towers with stairs at the sides and a shrine on top. For example this is a reconstruction image of the Ziggurat of Ur (modern-day Iraq):

Concerning ziggurats I found a fascinating little idea recently. It was stated by the late Dr. David Neiman, rabbi, archeologist and scholar, although I think it did not originate with him.

By the way: If you can spare the time, watch some of his lectures at Youtube. Despite the fact that I feel that his love of antiquity made him cast an unrealistically positive light on several things (slavery in antiquity, to name one), so this has to be taken with a grain of salt, there is a host of interesting material there.

The theory he put forward was this: He claimed that the ancestors of the Sumerians had migrated to Sumer from a mountainous region. This could be inferred from the fact that the early cuneiform symbol for “land” showed a picture of three mountains. It seems these people were used to worship their gods on mountain tops, and built their shrines and temples there.

When they later came to Sumer, which is a very flat country, it felt wrong to them to build temples on the ground. So they simply created artificial mountains and placed the temples on top of them—ziggurats.

The other one concerns the number 0. It is well known that the ancient European peoples generally had no sign for 0 in their number systems. The Greeks and Romans basically had nothing like it—they did not think it was a number at all, so why find a symbol for it? The Babylonians used a positional system to write numbers (as we do), so they had to have some kind of zero to discern, for example, 12, 120 and 102. They did so, but it was not allowed to stand alone, as pure 0. Neither did the Egyptians have it.

The first people to really understand the concept of zero, to write it and use it in calculations were Indians, and so it got into our number system, which is Indian in origin. This is all common knowledge.

Once I read in a book on the history of mathematics that the Indians were the only ones (or the first ones, anyway) to come up with a theory of zero for a special reason: The concept of nothingness (śūnyatā in Sanskrit) has been a very important idea in Indian philosophy since the earliest times. On the other hand Greek philosophers like Parmenides argued that “nothing” could not exist. They could not get their mind around the idea. The Indians did not have this trouble. They were used to 0.

I find these ideas very neat. In fact I have the strange feeling they fit together somehow. Let me try to clarify my point …

These things do not make you want to wear a tin foil helmet. They do not make you hate your neighbor. They do not make you feel you have been lied to your whole life.

Still, they are interesting. They are a little bit outside the canon of common knowledge. Outside the mainstream. You won’t find this information on Wikipedia. It’s one of the things they did not tell us. They are a little bit dubious. They might be true or they might not be, but is nice to belive they are. They do not rock your world like a flying saucer on the White House lawn but they might make you smile (or me at least). They might make you feel, in a tiny way, that the earth is a fascinating place if you lift the curtain. I will always have a place in my heart for things like these.

[I have started to make things like these into a series of articles. Here is the second one. Or just click the search key “little things”!]

Dan Brown part 2: φ

Just as I recently announced I have written an article about one of the many things Dan Brown could and should have gotten right in his books but did not. Those things are legion; and people more patient than me have already had their shot at it. For example there is a really excellent (and really long) article in Danny’s (don’t know his surname, I’m afraid) blog No Loss for Words titled Dan Brown is a fraud: A list of errors in Angels and Demons, which deals with this one book alone, and to which I, some years ago, had the honor of contributing a little thing as well.

Despite this abundance of material I have decided to confine myself to a single passage in the book The Da Vinci Code, which deals with a mathematical fact, which naturally interests me.

For your enjoyment here is the passage in its whole atrocious glory:

He felt himself suddenly reeling back to Harvard, standing in front of his “Symbolism in Art” class, writing his favorite number on the chalkboard. 1.618 Langdon turned to face his sea of eager students. “Who can tell me what this number is?”

A long-legged math major in back raised his hand. “That’s the number PHI.” He pronounced it fee.

“Nice job, Stettner,” Langdon said. “Everyone, meet PHI.”

“Not to be confused with PI,” Stettner added, grinning. “As we mathematicians like to say: PHI is one H of a lot cooler than PI!”

Langdon laughed, but nobody else seemed to get the joke.

Stettner slumped.

“This number PHI,” Langdon continued, “one-point-six-one-eight, is a very important number in art. Who can tell me why?”

Stettner tried to redeem himself. “Because it’s so pretty?”

Everyone laughed.

And I laughed too, a kind of sad, hollow laugh. Who would have thought it possible to get so much wrong in so little a piece of text? We are indeed witnessing a master at work. (Please excuse my bitterness, this thing happens to touch the field of my profession—who wouldn’t be touchy there?)

Most of which can be said (complained) about the matter has been said by the world’s most intelligent human being, the uncomparable Cecil Adams, but I’d like to add a few points of my own.

First of all, we are talking about a Greek letter here, φ. Am I too demanding in assuming that it should be possible, in the 21st century, to print some Greek letters in a book instead of writing it as “phi”? I mean, obviously this blog can do it, why not Doubleday Group publishers? Alright, minor point. By the way, if anything it should be “phi”, not “PHI”. “PHI” is obviously uppercase, which in Greek is Φ. The Golden Section (which is the thing meant here), however, is always a lowercase φ.

Secondly, the main point of Cecil Adams, φ is not 1.618. Just like 1/3 is not 0.333. φ is an irrational number, which have a unending, nonrepeating sequence of decimals after the comma. You just cannot write them that way. How to do it, then? Well, simply write

φ

=

which is the definition of φ. If you like to have the decimal expansion, please at least write it with dots: φ=1.618… But this thing that Brown (or Langdon, depending on how you look at it) wrote down, simply is not φ.

But this has been said before. Now I come to that which has not.

It is this: There are much too few letter in the alphabet for mathematics. There are 26, each lowercase and uppercase. Letters from other latin script variants are not used (e.g. ä, ç, ñ). And then there are the greek letters, where those that look like latin letters are excluded (an uppercase α, for example, looks just like an A). There are even a couple of hebrew letters that are used. Still it is too little. Therefore all letters are reused all the time and each one is used to name at least half a dozen things from various areas of mathematics. φ too. The most important are as angle parameters in functions, and the Euler φ function.

The point I’m trying to make here is that the Golden Section is not an important mathematical concept, and even less a frequently encountered one. I must have written several thousand φs since I began studying mathematics, and at most 20 of them denoted the Golden Section. If you ask a mathematician what φ is, in 99 of 100 cases (or more) the answer will not be Brown’s. The concept of Brown is called the Golden Section or Golden Ratio. If we name it with a letter, it is traditionally φ, but most of the time φ denotes something else.

And finally:

“Not to be confused with PI,” Stettner added, grinning. “As we mathematicians like to say: PHI is one H of a lot cooler than PI!”

If that guy is a mathematician he must be very drunk. Or on drugs. Or both. Because what he say is utter, utter bosh.

How can anyone confuse φ with PI (π)? Once a tutor told us not to confuse “angles” with “angels”, but that was a joke. A singularly lame one, but still. Anyway, “As we mathematicians like to say: PHI is one H of a lot cooler than PI!”. No, we mathematicians don’t say things like that.

Dan Brown, as he has showed on other occasions, has a rather weird conception of the sense of humor of scientists. There are indeed mathematicians’ jokes and physicists’ jokes, which other people don’t find funny at all (neither do mathematicians, to be honest). Its an interesting phenomenon and I may write a short piece about it some time. [Here it is.] Anyway, these jokes are in no way like these things Brown puts into our mouths. Not a bit. It’s a completely different style.

And finally, no mathematician would say that φ is cooler than π because it isn’t. It simply isn’t. φ does have a couple of beautiful properties, mathematically, but they are rather trivial. There is nothing a high school student could not understand. For example, that there is a relationship between φ and the Fibonacci numbers is completely correct, but it is no difficult or deep thing. I have written a PDF document that shows how it is done and you will probably not need more than 10 minutes to read and understand all of it (it’s also in Wikipedia).

π, on the other hand, is a monster that has baffled mathematicians for millenia. It drove the ancient Greeks crazy. There’s still many unsolved question left that involve it. It pops up in unexpected places. It is the reason you cannot square the circle.

(Here is my first article about Dan Brown.)