Things, Names, and Numbers

Happy January 48th, everyone! (More about that strange date later.)

Mathematician Henri Poincaré once wrote “Mathematics is the art of giving the same name to different things,” and he wasn’t wrong, exactly. He was thinking about the way mathematics advances by generating new concepts that unify old ones. For instance, mathematicians noticed that adding 0 to a number, like multiplying a number by 1, doesn’t change the number you apply it to. Eventually they celebrated this resemblance between 0 and 1 by coming up with new vocabulary: nowadays we say that 0 and 1 are “identity elements” (the former for addition, the latter for multiplication).¹ Two different things, same name.

But giving different things the same name is only half the story. Mathematics also invites us – and frequently requires us – to give different names to the same thing.² Seventeen isn’t just 17. It’s also 10001_two. It’s the fraction 34/2 (or the mixed number 16 2/2, if we’re feeling goofy). It’s the real number 17.000… and the real number 16.999…. It’s the complex number 17 + 0i.

Actually, is 16 2/2 all that goofy? If I’m doubling 8 1/2, isn’t 16 2/2 a reasonable intermediate stage in the calculation? Carrying this idea further, we can conceive of calculation as the art of transforming names like 8 + 9 into names like 17. “17 × 23” may be a starting point for a school exercise, but it isn’t a question; it’s already an answer, or close to an answer. We just need to convert this number-name into a different sort of number-name that’s more useful for most purposes (though not for all purposes, since if our ultimate goal is to compute 17 × 23 + 17 × 77, it’s better to rewrite that expression as 17 × (23 + 77) = 17 × 100). The fundamental act of reckoning on a decimal abacus – trading ten beads in the ones column for one bead in the tens column – can be viewed as the act of trading one name for another. Adding fractions with different denominators requires us to rewrite one or both fractions, replacing a fraction by an equivalent fraction that names the same rational number. Allowing things to have more than one name is precisely what makes reckoning possible.

The example of adding fractions reminds us that having more Names than Things is useful when we’re building a new number system from an old one. We could just say that a fraction is specified by two whole numbers that have no factor in common (the numerator and the denominator), but it’s more convenient for both practical and theoretical purposes to allow rational numbers to have extra names in which the numerator and denominator have a common factor.

The modest price we pay for having more Names than Things is that we have to specify when two Names name the same Thing. For instance, in the case of fractions, we have a rule that says that the fractions a/b and c/d are different names for the same number when ad = bc (and that they represent different numbers otherwise). We can use this rule and the rules

to create a purely formal approach to fraction arithmetic whose rules teach us nothing about what fractions mean but tell us everything we could want to know about how to add, subtract, multiply, or divide fractions. William Rowan Hamilton, who went on to invent quaternions, was one of the first to approach fractions in this formal way.³ His point was not that this is a good way to think about fractions for practical applications; rather, he was demonstrating that you can create a new number system (the rational numbers) from a smaller number system (the integers) even before you have an interpretation for the new number system or know what it might be good for, just by specifying good rules.

What’s a good rule? Well, let’s look at a not-so-good one. If we define a/b ⊕ c/d = (a + c)/(b + d) (this is the mediant operation that I briefly mentioned last month), then 1/2 ⊕ 1/1 = 2/3 and 1/2 ⊕ 2/2 = 3/4. This is a problem: since 1/1 and 2/2 are two different names for the same number, we should get the same value for 1/2 ⊕ 1/1 and 1/2 ⊕ 2/2; yet we got 2/3 and 3/4, which are not two different names for the same number (check: 2 × 4 ≠ 3 × 3). A good rule is one that won’t lead to this kind of contradiction. Then again, maybe we shouldn’t one-sidedly put all the blame on ⊕; it might be better to say that there’s a mismatch between how ⊕ is defined and our rule for recognizing when two fractions name the same rational number.

Just as Hamilton built up the rational numbers from the integers, today I’ll show you how to build up the the real numbers from the rational numbers.

THE PROBLEM

Why aren’t mathematicians satisfied with the rational numbers? After all, rational numbers can be used to approximate any irrational number as closely as one might wish. And the rational numbers form a tidy system in which one can perform addition, subtraction, multiplication, and division on any two elements and get something sensible (as long as we don’t try to divide something by zero). So why use irrational numbers at all?

Here’s a paradox to explain part of what’s wrong with the rational numbers as far as coordinate geometry is concerned. Consider the circle of radius 1 centered at (0, 0). If you draw a line from (−1, −1) to (0, 0) and on to (1, 1), you start outside the circle, cross into the circle, and then cross back out of the circle. But where exactly do the two crossings take place? We can see them with our finite-resolution eyes when we look at our blobby sketches, but if we insist on infinite precision and we’re only allowed to use rational numbers, the crossing points are nowhere to be found. That’s because any intersection point (x, y) between the circle and the line would have to satisfy both x² + y² = 1 (the equation defining the circle) and y = x (the equation defining the line). That is, we’d need to have 2x² = x² + x² = x² + y² = 1, so that x² = 1/2. But there is no rational number x satisfying this equation; if there were, its reciprocal would be a rational number whose square was equal to 2, and we’ve already seen (back in my November 2022 essay “The Infinite Stairway”) that there is no such rational number. So in “rational geometry”, we find that a line can pass through the interior of a circle and out the other side without ever cutting the circle at a point! The Greeks would have been confounded by such a situation, since the basis of their geometry was drawing lines and circles and marking points of intersection. What’s going on is that if we use rational numbers as coordinates then the seemingly solid line and circle are riddled with gaps, as is the horizontal number line itself.

When we’re trapped in a number system that has gaps in it, we can’t use a famous result called the Intermediate Value Theorem. This theorem is a bedrock of the calculus, and is used to show that when we talk about the region bounded by a closed curve (for instance), the area of that region is a definite number whether or not we know how to compute it. In a more recreational vein, the Intermediate Value Theorem plays a key role in certain sorts of puzzles, such as the following: Show that, if you hike up a mountain on Monday and hike back down on Tuesday, there must be an instant on Tuesday when you’re at the exact same altitude that you were at exactly 24 hours earlier. The solution involves superimposing the graph that shows your altitude as a function of time on Monday with the graph that shows your altitude as a function of time on Tuesday; the puzzle is solved by observing that the graphs of the two functions (one increasing from some low number a to some high number b and the other decreasing from b to a during that same time interval) must cross. But if a line can pass into and out of a circle without piercing its skin, why must the two graphs cross? If our number system has gaps, the two graphed curves might pass through each other without crossing anywhere. This won’t do.

In 1872 mathematician Georg Cantor found a way to fill the gaps in the rational number line and to construct, not just some special irrational numbers like sqrt(2) and π, but all of them at once. He showed that the rational numbers already yearn to give birth to the irrational numbers by a process we call completion (metric completion, to be more specific). The roots of his construction go back thousands of years, when various cultures found systematic ways to generate better and better rational approximations to the square root of two. You saw some of these approximations if you read “The Infinite Stairway” though you might not have recognized the approximations as such. There I presented an ancient method of generating more and more solutions to the equation 2a² − b² = ±1. Since the numbers a and b quickly get large, in relative terms the pairs a, b satisfy 2a² − b² ≈ 0, or b² ≈ 2a², or b²/a² ≈ 2, or (b/a)² ≈ 2. Take the diagram from the section “Up the Down Staircase” of the November 2022 essay and flip it upside down, inserting some fraction lines:

You see the beginning of the infinite sequence of fractions 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, …: an interleaving of the increasing sequence 1/1, 7/5, 41/29, … and the decreasing sequence 3/2, 17/12, 99/70, ….

Here’s what we see when we plot 1/1, 7/5, 41/29, and 239/169 (the first four terms of the increasing sequence) on the number line:

At first glance it looks like just three dots, not four, but if you look carefully you’ll see that the blue blob on the right is two dots mostly superimposed. And here’s what we see when we plot the next four terms of that sequence on the number line:

Again, that rightmost blob is really two dots ever-so-slightly displaced from one another. It certainly seems as if this increasing sequence is “trying” to converge to something on the number line that’s outside of the rationals, and we could define sqrt(2) as the Thing the sequence is “trying” to converge to.

That definition seems circular, doesn’t it? But we’ve seen similar mathematical hocus-pocus before. Consider: “3 ÷ 2” is the name of a division problem that we can’t solve in the counting numbers, so we created 3/2, essentially naming it after the problem it lets us solve. Likewise, “0 − 2” is the name of a subtraction problem that we can’t solve in the counting numbers, so we created −2, again naming it after the problem it lets us solve. Solving a problem by giving the problem a name and then proposing the name as a solution to the problem sounds like answering-a-question-with-a-question at best and circular-reasoning sophistry at worst. But we’ve seen that if we do it properly, it works, at least in math. So if we can’t find a limiting value of the sequence 1/1,7/5,41/29,… within the set of rational numbers, let’s define a new mathematical beast called lim(1/1, 7/5, 41/29, …) (where the dots represents all the terms of the sequence beyond 41/29) and give good rules for working with it.

Hence, our brave first draft for a model of the set of real numbers is Names of the form lim(a₁, a₂, a₃, a₄, …) where a₁, a₂, a₃, a₄, … is some arbitrary infinite sequence of rational numbers. Call this Draft #1. It’s a nice try, but it won’t do. Let’s see what’s wrong with it.

CAUCHY TO THE RESCUE

We want to help 1/1,7/5,41/29,… converge to something, but that doesn’t mean we want to help just any old sequence converge to a limit in our envisaged real number system. I mean, the sequence 1,2,3,4,5,6,… isn’t even trying to converge. And 1,0,1,0,1,0,…. is just messing with us.

What makes one feel that the sequence 1/1, 7/5, 41/29, … deserves to have a limit to converge to, but that the sequences 1, 2, 3, 4, … and 1, 0, 1, 0, … don’t?

French mathematician Augustin-Louis Cauchy had an answer to this question fifty years before Cantor asked it.⁴

From some point onward, all the terms of the sequence 1/1, 7/5, 41/29, … differ from one another by less than 0.1. Also: from some point onward (further on in the sequence), all the terms of the sequence 1/1, 7/5, 41/29, … differ from one another by less than 0.01. In fact, no matter what tiny (but positive) constant c I pick, you can always find a place in the sequence with the property that all the terms to the right of the place you found differ from each other by less than c. This “bunching up” property is called the Cauchy property. The sequence 0.3, 0.33, 0.333, …, consisting of the successive decimal approximations to 1/3, has the Cauchy property. So does our sequence 1/1, 7/5, 41/29, … of increasing approximations to the square root of 2. So does the sequence 1, 1.4, 1.41, 1.414, … of increasing decimal approximations to the square root of 2. On the other hand, the sequences 1, 2, 3, 4, … and 1, 2, 1, 2, … don’t satisfy the Cauchy property. So let’s cut our universe of infinite sequences of rational numbers down to size, or at least make it a bit more manageable, by culling the sequences that don’t satisfy the Cauchy property. Or if “culling” sounds too ruthless, let’s imagine stationing a bouncer at the door of “Club Cantor” and instructing our bouncer to reject all sequences that don’t observe Cauchy’s rule of decorum.

We thus arrive at Draft #2 of the set of real numbers: all names of the form lim(a₁, a₂, a₃, …) where a₁, a₂, a₃, … is a sequence of rational numbers that satisfies the Cauchy property. (Mathematicians call such sequences “Cauchy sequences” for short.)

This proposal is also a failure, but for a totally different reason.

GET OUT YOUR GLUE GUNS

We can see the problem if we look at lim(1/1, 7/5, 41/29, …) and lim(3/2, 17/12, 99/70, …). The former sequence is increasing toward the gap in the number line where we’re going to construct sqrt(2) – the construction site, you might call it – while the latter sequence is decreasing toward that very same gap. Surely we want them to be the same real number! For that matter, lim(1, 1, 1, 1, …) and lim(.9, .99, .999, .9999, …) and lim(1.1, 1.01, 1.001, 1.0001, …) are different Names, so in Draft #2 of the real number system they are different Things. We’re having doppelgänger-management issues again, but in a much bigger way than last month: now every rational number has infinitely many doppelgängers, and so do all the gaps in the number line (the future homes of the irrational numbers) that we’re trying to fill!

Just as Hamilton (and others before him, of course) gave us a way to recognize when two different fractions represent the same rational number, we’ll need a way to recognize when two different sequences represent the same rational or irrational number. Hamilton said that a/b and c/d are two different names for the same thing when ad = bc; what’s the corresponding trick for recognizing when lim(a₁, a₂, a₃, …) and lim(b₁, b₂, b₃, …) are two different names for the same thing?

We can get a clue by considering the concrete case of 1/1, 7/5, 41/29, … and 3/2,17/12,99/70, …. Here’s what we get when we plot the first three terms of the increasing sequence in blue and the first three terms of the decreasing sequence in red:

The two sequences are getting closer and closer to each other. If we subtract the first sequence from the second sequence term-by-term, the differences 3/2 − 1/1, 17/12 − 7/5, 99/70 − 41/29, … get closer and closer to 0.

So now we can fix what was wrong with Draft #2: for any two Cauchy sequences a₁, a₂, a₃, … and b₁, b₂, b₃, …, decree that lim(a₁,a₂,a₃,…) and lim(b₁,b₂,b₃,…) are two different Names for the same Thing if the differences a₁ − b₁, a₂ − b₂, a₃ − b₃, … approach zero. That’s Draft #3. Notice that when we do this, lim(.9,.99,.999,…) and lim(1,1,1,…) and lim(1.1, 1.01, 1.001, …) get glued together, which is a good sign.

If decreeing that things that look different are to be regarded as the same seems too much like double-think to you, you might prefer the bag theory way to think about what’s going on. Bag theory was mostly the creation of Georg Cantor, except that he called it set theory (“Mengentheorie” in German), and it’s a good thing that he did, since otherwise mathematicians like his nemesis Leopold Kronecker might have dismissed his work even more than they actually did. Nobody really calls it bag theory, but it’s really about bags – not physical bags, but the kind that exist in your head. Sets are bags that can contain any objects of thought that you care to put into them, including other bags. Cantor’s bags have been just as consequential in math as paperclips and manilla folders were in the evolution of the early 20th century office. Paperclips and folders give you a way to stick things together whether they want to be stuck together or not, and Cantor’s bags do that for math. You want to make a bag that contains just the number 7, the number 24, and the number 365? Poof! No sooner have you thought of it than it exists. That is, it exists in your mind, because you put it there. And if you write ”{7,24,365}” on a blackboard in front of three friends, then poof-poof-poof! It exists in their minds too. In Draft #3, we put lim(a₁,a₂,a₃,…) and lim(b₁,b₂,b₃,…) into the same bag when the differences a₁ − b₁, a₂ − b₂, a₃ − b₃, … approach zero.

Notice that we wind up with fewer Things in Draft #3 than we had in Draft #2, but not because we’re throwing any of our Cauchy sequences away; rather, we are lumping many of them together into our bags. Culling and lumping are different processes, but they work in the same direction: making collections smaller and more manageable. Moving from the rationals to Cantor’s model of the reals was sort of a one-step-forward, two-steps back process, featuring a huge initial step forward (when we introduced infinite sequences of rational numbers) followed by two smaller steps back (first culling the non-Cauchy sequences, then lumping together those Cauchy sequences into bags). The end result was progress.

In Club Cantor terms, the bouncer will no longer admit individual sequences but will only admit complete gangs consisting of all the Cauchy sequences that have the same vibe, where two sequences are said to vibe together if the difference between the nth term of one and the nth term of the other goes to zero as n gets large. These gangs (like the bags) correspond precisely to the real numbers.

The square-root-of-two bag contains infinitely many names for sqrt(2). One of them is lim(1/1,3/2,7/5,17/12, 41/29,99/70,…). Each successive numerator and denominator are obtained from the previous numerator and denominator by following a simple recipe. What could be nicer?

PREJUDICE

Well, some people aren’t happy. They learned about the real numbers via their decimal expansions, and if you can’t show them a pattern in the decimal expansion of a number they think there’s something fishy about the number. “Irrational numbers don’t exist!” they cry. They forget that decimal expansions are only one way of understanding real numbers, and a recent one at that; the ancient Greeks knew a lot about rational and irrational numbers (or as they called them, ratios) with nothing like the modern decimal system. Just as it’s silly to say that negative numbers don’t exist because in some contexts (say, counting sheep) negative numbers don’t apply, it’s silly to say that the square root of two doesn’t exist because in some representations it looks random. Listen: I know a way of writing numbers in which the ordinary number 1/2 gets represented by .0100000100100101000…, an unending sequence of 0s and 1s in which no patterns have been found; does that mean that there’s something suspicious about the ontological status of the number 1/2? Surely the lack of pattern is best seen as a case of a mismatch between the number we are trying to express and the number-system in which we are trying to express it.⁵ Focusing on what numbers look like in base ten is a decimal-centric prejudice that a properly-tutored young mathematician should be steered away from as early as possible, certainly by the age of ten. When it comes to representing numbers, God gave us the unary system; all the rest is human contrivance for human convenience.

Perhaps some of the modern animus against irrational numbers stems from the fact that they outnumber the friendly, familiar rational numbers. You may think “outnumber” is an odd word to use here; aren’t there infinitely many of both, and doesn’t infinity equal infinity? The surprising answer is that sometimes, infinity doesn’t equal infinity, or rather (to put things in a less provocative way) there’s more than one size of infinity. Cantor was the one who taught us this, and I’ll talk about his infinities in a future essay. But one thing I’ll say now, which sort of hints at why “most” real numbers are irrational, is that if you generate a real number at random by using successive decimal digits chosen by throwing a 10-sided die, you’re virtually certain to generate an irrational number. That’s because the sequence of digits you generate is virtually certain to be an “infinite monkey sequence”, that is, a sequence that contains every possible finite sequence of digits. On the other hand, the digits of a rational number must eventually settle down into a pattern which they repeat forever after. It’s not hard to show that an eventually-repeating infinite sequence of digits can’t be an infinite monkey sequence.⁶

ARITHMETIC WITH REALS

Okay, so we’ve constructed the real numbers, or something that we hope will behave like the real numbers; how do we know that we’ve succeeded? We claim to have constructed the square root of two as a bag marked “square root of two” on the outside, with infinitely many names of the form lim(a₁, a₂, a₃, ….) on the inside. But in what sense can we “square” this bag and obtain the bag marked “two” as the result?

Reach your hand into the bag and pull out some name lim(a₁, a₂, a₃, ….). The sequence is guaranteed to satisfy the Cauchy property. Moreover the sequence is going to bunch up around the same gap in the rational numbers where the sequence 1/1, 3/2, 7/5, 17/12, … does. “Square” the sequence a₁, a₂, a₃, … by squaring all the terms: a₁², a₂², a₃², …. It can be proved (with a bit of work or with a bit of cleverness) that this new sequence will satisfy the Cauchy property too, so the name lim(a₁², a₂², a₃², ….) must be in one of our bags somewhere. And which bag will that be? The bag marked “two”! For instance, if the name that we pulled out of the square-root-of-two bag was lim(1/1, 3/2, 7/5, 17/12, …) itself, then squaring all the terms gives 1/1, 9/4, 49/25, 289/144, …, and since this sequence converges to 2, the name lim(1/1, 9/4, 49/25, 289/144, …) must be in the bag marked “two”.

Here’s a general prescription for multiplying bags: To multiply two real numbers (call them r and s), reach into the “r bag” and pull out some name lim(a₁, a₂, a₃, …); likewise reach into the “s bag” and pull out some name lim(b₁, b₂, b₃, …); then form the infinite sequence c₁, c₂, c₃, … , with c₁ = a₁b₁, c₂ = a₂b₂, , c₃ = a₃b₃, , etc.; and then find the bag that contains the name lim(c₁, c₂, c₃, …). The same prescription applies to adding two real numbers, except now we put c₁ = a₁+b₁, c₂ = a₂+b₂, c₃ = a₃+b₃, etc.

I won’t take you through the details, but with this definition it’s easy to show that all the algebraic properties of the rational numbers carry over to the new number system we’ve built. For instance, we can use the fact that a + b = b + a for all rational numbers to prove that r + s = s + r for all real numbers. All the algebraic properties of our old number system are satisfied by the new number system. The principle of the permanence of form has been vindicated!

What’s more, we get a new non-algebraic property, the one we were hoping for: gap-free-ness, more properly called completeness. The real number system, unlike the rational number system, satisfies the Intermediate Value Theorem. So a line that passes into and out of a circle must cross the circle somewhere, and if you hike up a mountain one day and down the next, there must be a time of day when your altitude was the same on both days.

There’s only one more step, but in many ways it’s the most subtle one.

ARROWS

We’ve built a new number system, and it has lots of wonderful properties, but why do we call it an extension of the rational numbers? Isn’t it just some new number system, external to our old one?

It’s time for the final step in the process. We want to glue every rational number a to the corresponding bag containing lim(a, a, a, …), and we want to do this for every rational number. The rational numbers must step into the picture they’ve painted, becoming characters in the fictional world they helped create.

But if we do this with our mathematical glue gun, confusion ensues. We’ve defined real numbers as bags of infinite sequences of rational numbers, but if those rational numbers are bags too, then we have bags of bags of bags of …

A better approach is to use what in category theory are known formally as “morphisms” and informally as “arrows”. Arrows give us a way to say that two things are the same (in some ways) and yet not the same (in other ways). In this case, the arrow points from the set of rational numbers to the set of real-number bags, and associates with each rational number a to its avatar in the real number system: the bag that contains the name lim(a, a, a, …). So, even though we didn’t construct our real numbers as a superset of the rational numbers, this arrow lets us think of the rational numbers as a subset of the real numbers.

I want to point out how truly sneaky all this is. What rescued our solve-the-problem-by-naming-it tactic from utter sophistry was that the new number system we constructed – the one that’s essentially made up of the names of problems that we can’t solve in our old number system – was external to the old number system. But then we magically inserted the old number system into the new one.

Part of what category gives us is an appropriately relaxed attitude about what things “really are”. This can be especially useful if we want to consider Cantor’s approach to constructing the real numbers alongside a different construction that was proposed by Richard Dedekind at almost the exact same time. Dedekind’s idea was that if you want to specify a particular irrational number, which is to say, a particular gap in the rational number line, it’s enough to specify which rational numbers are to the left of the gap and which rational numbers are to its right, so why not just define the irrational number as that particular way of breaking the rational number line into a left piece and a right piece? These are the famous “Dedekind cuts” (though the core idea derives from Eudoxus two millennia earlier). This is a different definition of the real numbers, and you might worry that from different definitions, different consequences will follow. But there’s nothing to worry about. Each “Cantor real number” corresponds to one and only one “Dedekind real number”, so the two constructions are only different in their internal workings, not different in terms of how they interface with the rationals. Cantor and Dedekind didn’t construct two different number systems; they constructed the same number system in two different ways.

Category theory gives us a way to say that the two systems are the same without speaking nonsense. There are arrows between the Cantor reals and the Dedekind reals, giving us a weakened form of equality which is all we really need. Category theory provides a sophisticated framework for voicing your apathy toward questions like “Are real numbers really infinite decimals, or Dedekind cuts, or gangs of Cauchy sequences with a common vibe?” You get to say “Who cares? They’re all isomorphic anyway.”

(Decades ago, President Bill Clinton tried to evade accusations of dishonesty by saying that the interpretation of an assertion he’d made depended “on what the meaning of the word ‘is’ is.” This may have been lawyerly weaseling, but it sounds a little bit like mathematics. Was Clinton a category-theorist?)

Neither Cantor’s construction nor Dedekind’s plays much of a role in the day-to-day work of mathematicians who study real numbers. Like the plumbing in a house, the details of how real numbers work are normally hidden from view so we can focus on other things. If we need to dig into the infrastructure and wrestle with specific real numbers, we’re likely to use infinite decimals as our go-to model of what real numbers “are”.

BUILDING NUMBER-UNIVERSES

So why bother constructing the real numbers at all, if you’re never going to use the details? One answer has to do with the 19th century crisis of faith in the foundations of mathematics. Sure, you could just posit all the properties you think real numbers should satisfy as axioms, but how do you know your axioms don’t harbor some subtle self-contradiction? The real number system definitely seems weirder than the rational number system; how do we know it hangs together logically? The payoff of building up the reals from the rationals is that it provides a proof of relative consistency: as long as your axioms for the rational numbers are consistent, your axioms for the real numbers must be consistent too.

But the main reason I’ve spent so much time on building the reals as a completion of the rationals is that it’s a trick that can be used in other contexts. In particular, if we adopt a different notion of what we mean by words like “distance”, “limit” and “converge” in the context of the rational numbers, we can construct infinitely many new number systems (the p-adic number systems, where p is any prime you like). We’ll do that later, but we don’t have to wait till then before seeing Cantor’s ideas construct a novel number system. Let’s go back to Club Cantor and give the bouncer a new, more permissive admission criterion. It’ll still be Cauchy-ish, but instead of measuring the distance between two rational numbers in the ordinary way our kinder-gentler bouncer will measure the vertical distance between them in the following curiously warped number line:

This particular graph is the graph of the function f (x) = (sqrt(x² + 1) − 1)/x (with f(0) = 0 by special stipulation), but the specific equation isn’t important; what matters is that it’s an increasing function with horizontal asymptotes y = 1 and y = −1. In this warping, the numbers 100 and 1000 are really close together (because the latter is only slightly higher than the former) and the numbers 1,000,000 and 1,000,000,000 are even closer together. Now sequences like 1, 2, 3, 4, … that go off to infinity are allowed into Club Cantor. Likewise 1, 10, 100, 1000, … and −1, −2, −3, −4, … and lots of other sequences that didn’t pass the Cauchy test before but do pass it now that we’re measuring distance in a different way.

The sequences that head off to the right like 1, 2, 3, 4, …  get a new bag, which we can call +∞, and the sequences that head off to the left like −1, −2, −3, −4, … get a different new bag, which we can call −∞. Nothing else changes; all the other bags are as before. But now we get a number system called the extended real numbers. Want to know what +∞ times −∞ is in this new number system? Reach into the +∞ bag and the −∞ bag and pull out a sequence from each; multiplying term-by-term, you’ll find you get a sequence from the −∞ bag, so +∞ times −∞ is −∞. What about +∞ plus −∞? Now you’ll find that the answer is indeterminate; the term-by-term sum of the two sequences you pulled from the two bags might belong to either of the two bags, or to one of the bags labeled by a real number, or to none of the bags at all (since it might not satisfy the Cauchy property).

This last caveat hints at a detail I’ve swept under the rug (one of many such details): how do we know that in Cantor’s constructon of the ordinary real numbers, the sum or product of two sequences satisfying the Cauchy property must also satisfy the Cauchy property? In presenting Cantor’s construction as glibly as I’ve chosen to do, I run the risk of making it seem simpler than it actually is.

I referred to the extended real number system as “new”, meaning not yet described by me in this essay, but it wasn’t new even back in Cantor’s day; Euler and others had made extensive informal use of +∞ and −∞ as a kind of shorthand for describing the behavior of functions. But it’s nice to know that we can extend the real numbers in such a way that what was formerly informal becomes literally true.

(As we’ll see, Cantor came up with a new theory of infinity, but it was much more original and outrageous than the extended real number system!)

MORE BIRTHDAYS

Now that you know the theme of this essay, maybe you can guess why I started by wishing you a happy January 48th. January 48th is an extra name we might give to February 17th. Normally we don’t think of January as going past the 31st, but if we posit that January the (n+1)st should always refer to the day after January the nth, then January the 32nd should be another name for February 1, January the 33rd should be another name for February 2, and so on. To convert January dates beyond the 31st into proper dates, just subtract 31 and replace “January” by “February”. And if the number you got by subtracting is bigger than the number of days in February, keep going with more subtraction. On the other hand, January 0 is another name for December 31, and you can go deeper into negative January from there.

This kind of half-nonsense is actually useful when you’re doing mental calendar calculations, as intermediate steps in figuring out when something in the past happened or when something in the future is going to happen. Suppose you go into a shop today and buy something today that has a 30 day return policy. What’s the last day you can return it? Today is February 17th, so 30 days from now is February 47th; subtracting 28 (the number of days in February this year) we find that February 47th coincides with March 19th. (Note: I advise you to return your item no later than March 18th, since the shopkeeper may insist that February 17th counts as day 1, hence March 19th counts as day 31, which is one day too late. To read about a time this actually happened to me, see my essay “Impaled on a Fencepost”.)

A different application of allowing months to have supernumerary days is that it gives you a second chance (and a third, and maybe more) at celebrating golden birthdays. In case you don’t know, a golden birthday has traditionally been defined as the unique birthday in which the age you have just reached equals the day of the month on which you were born. So for example if you were born on the first day of the month, you had a golden birthday when you turned one. And that golden birthday – which I’m sure you don’t remember since you were only one at the time – is the only golden birthday you’ll ever have.

Or, it would have been like that, until now! (I think my proposal to redefine golden birthdays is original with me.) With supernumerary days, you’ll get a golden birthday every thirty year or so. For instance, if you were born on January 1st, you were also born on December 32nd, and November 62nd, and October 93rd, and you’ll get extra golden birthdays when you turn 32, 62, and 93.

So now you see a practical application of assigning different names to the same thing, and you don’t have to ask me “When will I ever be able to use this stuff?” My answer is: in thirty years or less. And when you prepare the cake and candles, don’t worry about how you’ll fit dozens of candles on your cake or stress about how much wax they’ll drip: even if you live to 127, seven candles will suffice. That’s because you can represent your age in binary instead of unary, using lit candles to represent 1s and unlit candles to represent 0s. Actually, Cantor (born on the 184th day of September) has a golden birthday coming up soon so when he turns 184, put an eighth candle on that cake. 184_ten can also be written as 10111000_two. A different name for the same number. Cantor would approve.

ENDNOTES

#1. The original term was German; the identity property of 1 was called “Einheit”, or one-ish-ness. Accordingly, the generic symbol for an identity element is the letter e.

#2. After defining Mathematics as the art of giving the same name to different things, Poincaré defined Poetry (by way of contrast) as the art of giving different names to the same thing. I think his dichotomy is unfair to both poetry and mathematics.

#3. Well, Hamilton was the first to treat fractions in a purely formal manner on purpose. One could credibly and depressingly argue that meaning-free manipulation of fractions has been taught and learned in classrooms for centuries.

#4. Cauchy was actually answering a different question when he formulated his convergence criterion in 1821; he just wanted a way of assessing whether or not a sequence converges if one doesn’t know in advance what the supposed limit is. Meanwhile, working on his own, Bernard Bolzano arrived at the same criterion in 1817 and had a clearer understanding than Cauchy of what the criterion was good for, so by rights it should be called the Bolzano property; but Bolzano did not publish his work and his advances did not come to light until long after his death.

#5. This system is credited to Alfréd Rényi, and is called the β-expansion of a number, where β can be any real number greater than 1. When β is ten, the β-expansion of a number is just its ordinary decimal expansion, and when β is a positive integer, the β-expansion of a number is its base-β expansion. Regardless of what β is, to find the β-expansion of a number r, subtract off the biggest power of β from r that you can, and then do the same to the remainder, and so on. For instance, to derive the β-expansion of 1/2 with β = 3/2, we subtract (3/2)⁻² = 4/9 from 1/2, leaving us with 1/18; then we subtract (3/2)⁻⁸ from 1/18, leaving us with 217/13122; and so on. Recording which powers of 3/2 got subtracted gives us the β-expansion of 1/2: .0100000100100101000…. (See entry A360649 in the Online Encyclopedia of Integer Sequences for more terms.)

#6. Suppose the eventually-repeating sequence “clears its throat” for M digits and then launches into a never-ending repeating pattern whose length is N digits. Then a block of M + N consecutive 0s cannot occur in the infinite sequence unless the repeating pattern of length N consists entirely of 0s, in which case a block of M + N consecutive 1s cannot occur in the infinite sequence. Either way, we see that an infinite sequence that eventually repeats forever after cannot be an infinite monkey sequence.