Hopefully Interesting

November 15, 2007

On Limits II

Filed under: math.LO — Pietro @ 12:18 pm
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(This is a continuation of On Limits I.)

Think of Pelé. When somebody asks “who is Pelé”, you might reply, “the best football player ever”. Notice that you are implicitly defining Pelé in terms of his relationship to other footballers; an observation which is often obscured by the fact that there is also an actual person you can point to, and say “that’s Pelé”.

Now suppose a toy company came out with a football trading card game, where each card represents a famous player, and contains various numerical ratings like precision, speed, stamina etc. You might now explain that Pelé (meaning the Pelé card) is “the card with highest speed and precision”. Even though there is still an actual card you call “Pelé” (and children might think of it this way), it is now clear that the thrust of the definition is quite another: what is important about the Pelé card is its relationship to the other cards, not its particular shape, or whether it’s made of paper or plastic.

Similarly, if you ask a child what is a chess pawn, they will most likely point to the actual, physical chess piece: “that’s a pawn”. Which is fine if we had asked “what is a chess pawn, in the physical world?” However, the deeper content of our question was “what is a chess pawn, in the game of chess?” Since grandmasters are able to play entire games in their minds, a pawn can’t be just a wooden piece. In fact, even novice players experience no confusion if pawns are replaced with (say) beans, as long as they agree to it beforehand. So we see that the main defining feature of chess pawns is not their wooden incarnation, nor their color, but their relation to other pieces; that is, the rules they obey in the game of chess. The physical pawn is just a memory aid.

Many of these points are obscured in everyday life because we usually have material counterparts to various concepts which would be better described in abstract terms: a card for Pelé, a wooden piece for the pawn, coins and bills for money. In mathematics, however, such counterparts are fewer in number; decimal representations are an example, but you won’t find many more. Furthermore, sticking to naïve concrete views (e.g. that ‘1’ actually is the number one) quickly leads to conceptual conundrums which would not arise if one took a proper view of things.

I will now give a simple mathematical example of the method, exemplified above, of defining something by its relationship to other things. Consider the sequence

S = (0.9, 0.99, 0.999, 0.9999, \ldots)

It is, implicitly, an infinite set of numbers (that’s what ‘…’ means), but one whose structure is easily grasped. Given such a sequence S, there are two obvious numbers that one can try to define, in terms of their relationship to the members of S:

Minimum. The smallest number in all of S.

Maximum. The biggest number in all of S.

These are very like the Pelé definition. One of them is easy: the smallest number is 0.9. The other, however, is no good. Each member of the sequence is strictly larger than the previous one; therefore, there is no largest element. Thus the second definition does not pick out any number. And that’s fine! Remember how we thought about the expression ‘1/0’: it tries to specify a number by a sequence of steps, but fails. ‘The biggest of them all’ tries to specify a number by its relationship to some others, and it also fails. No big deal.

(The maximum is not 1 because 1 is not a number in the sequence. Look at it again. It’s quite explicitly composed only of terminating decimals less than one. It is of the utmost importance, all through our reasoning, to hold off the impulse of saying things about “infinity”. We argue only about particular elements of the sequence; only they are “really there” for us to see. If we want to come up with new ways of speaking about numbers, we must base them firmly on what is “really there”. Otherwise all speech will be devoid of meaning.)

Alright. Here’s a not-so-obvious number we can define by its relationship to our sequence S above:

Popular number. We say a number a is “popular with our sequence” if, for every distance d > 0, however small, all elements of our sequence are less than d units away from a, except maybe for a finite number of them. (Intuitively, “almost all” elements of S are close to a.)

Let’s see what this means. Pick a random number, say -1. How far from -1 are the elements of our sequence S? Well, the first element, 0.9, is at a distance of 1.9. The next, 0.99, is 1.99 units away; the third, 1.999; and so on:

distances from -1 to elements of S: (1.9, 1.99, 1.999, \ldots)

Is -1 popular with our sequence? It seems not: the terms get farther away from -1 as we progress into S, which is not what we would expect if S “liked” the number -1. Let’s check this intuition against the definition above. It says that, in order for -1 to be popular with S, all but finitely many elements of S must be within a distance d of -1, no matter what tiny d > 0 we choose. To really test the popularity of -1, let’s pick an extremely tiny d, say 0.01. Is it the case that all but finitely many elements of S are within 0.01 of -1? Well, no. In fact, no elements of S are that close to -1. Therefore, -1 is not a popular number with S. If you ask whether -2, 0 or 0.5 are popular with S, the same kind of reasoning shows that they’re not.

For practice, and to understand how “all but finitely” really works, let’s try 0.998. Is it popular? Here are the distances from it to S:

(0.098, 0.008, 0.001, 0.0019, 0.00199, \ldots)

Notice how S starts out far from 0.998, gets pretty close, but then moves away. This suggests that S likes 0.998 a bit more than -1, but not enough to make it popular. Let’s prove that it’s not, like we did before. Choose d=0.005. Are all but finitely many elements of S less than 0.005 units away from 0.998? Whoops. Yes they are. The first two elements of S are farther than that: their distances are 0.098 and 0.008, respectively. But all the others are no further than 0.002 units away, which means 0.998 does pass the check for d=0.005: all but finitely many (in this case, all but two) elements of S are less than 0.005 units away from 0.998.

Even so, we shouldn’t give up. Recall that, for a number to be popular, it must pass the d-check for any positive d. Just because 0.998 passed one d-check, doesn’t mean it’s popular. (It’s just like the trading card game: just because a card has higher ratings than one other card, doesn’t mean it’s Pelé; it must have higher ratings than every other card.) In fact, as you can check, 0.998 fails the d-check already for d=0.001.

At this point we might worry that a popular number doesn’t exist. Given our experience with meaningless expressions like ‘1/0’, this wouldn’t be too surprising. It’s entirely possible to write something down, like an expression or a definition, which doesn’t actually stand for any number at all; we already saw this happen when we defined the maximum of S.

There are, in fact, some sequences other than our S for which there is no corresponding popular number. For example, the sequence (1, 2, 3, 4, …) eventually moves past any given number, and never comes back. In fact, any candidate popular number fails all possible d-checks, even for d as large as a billion. This sequence really hates every single number there is. (Again, it is important to resist talking about “infinity”. The sequence (1, 2, 3, 4, …) does not “go to infinity” in this context, simply because we have not defined what “going to infinity” means. We might as well say that it goes to the beach.)

Fortunately, our initial S isn’t as misanthropic: the number 1 is quite popular with it. You may like to prove this for yourself. Here are

distances from 1 to S: (0.1, 0.01, 0.001, 0.0001, \ldots)

the same, in a helpful way: (10^{-1}, 10^{-2}, 10^{-3}, 10^{-4}, \ldots)

The proof consists in showing that 1 passes all d-checks. This happens because, for each d > 0, there is some negative power of ten, say 10^{-n}, which is less than d. This means that all elements of S, except the first n (finitely many) are no farther than d units from 1.

As you probably noticed, a popular number is nothing but the limit of a sequence. Compare its definition with those for the Pelé card and the chess pawn.

Based on all this, I’ll introduce a new way of referring to numbers. Given a sequence

A = (a_1, a_2, a_3, \ldots)

I define the symbol

\lim \ a_n

to mean “the number which is popular with the sequence A”. In the particular case where A is a sequence of growing decimal expansions (like our original S), ie

A' = (0.b_1 , 0.b_1 b_2, 0.b_1 b_2 b_3, \ldots)

(each b_i a digit between 0 and 9) we use the alternative symbol ‘0.b_1b_2b_3\ldots‘. Though the symbol is different, we define it to stand for the exact same thing: the number which is popular with A’, or, in more standard terms, the limit of A’.

If you’re with me so far, you probably already have a much better understanding of limits than most beginning calculus students worldwide. (And we hardly used any fancy jargon!) For example, now that I’ve defined ‘0.999…’ to be the limit of the sequence ‘(0.9, 0.99, 0.999, …)’, you would never make B’s mistake:

Person A: 0.999… = 1.

Person B: It’s not equal. The left-hand side gets infinitely close to 1, but never equals 1.

Of course, this makes as much sense as saying that Pelé isn’t the best player because there are other players which are not the best. Or that the pawn can’t be traded in for a queen, because the other pieces can’t. This kind of confusion is the result of conflating two very different things:

  1. A collection of objects (the sequence ‘0.9, 0.99, 0.999, …’, football players, chess pieces);
  2. An object defined by its relationship to that collection (the number ‘1’, Pelé, the pawn).

Alright, we’re pretty much done. If you understand everything above well enough to explain it to somebody else, then all that’s left to do is go back into the mainstream. Understanding definitions in a deep way is a vital part of mathematics, but it is also important to see why they are interesting, where they lead. Indeed, the definition of limit is an unusually fruitful one. There are

  1. Foundational theorems, like existence and uniqueness: a certain kind of sequence (monotonic and bounded) always has a limit, and any sequence has at most one limit. You can prove it from the definition, and it justifies our constantly saying the limit instead of a limit.
  2. “Niceness” theorems: if you take two sequences, S and S’, which have limits, and you define a new sequence S+S’ from the addition of corresponding terms of S and S’, then the limit of S+S’ is the sum of the limits of S and S’ separately. The same goes for multiplication (S\cdotS’).
  3. Interesting uses of the definition: starting with derivatives and integrals, up to nets and filters in topological spaces, the basic idea of a “limit” underlies a lot of modern analysis.

Enjoy!

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