What is Infinity?

There’s an interesting blog discussion going on about the age-old question of whether .99999…, where the nines go on forever, is actually equal to one. The answer is: Yes, it does, and if you think it does not then you are mistaken. Polymathematics got the ball rolling with several arguments establishing the equality of the infinite decimal on the one hand and the number one on the other. Mark Chu-Carroll offered some follow-up thoughts here.

One way to prove that .9999… repeating equals one is to realize that the notation “.9999…” is really just a short-hand way of writing the infinite geometric series:


(9/10)+(9/100)+(9/1000)+…

There is a simple formula for evaluating the sum of such a series, and when that formula is applied in this case we get 1 as the result.

A commenter to Mark’s post (Paul S.) provided a nice summary of this argument, but then asked:

Have I got that right so far? If so, then I guess the fundamental philosophical question is, as n approaches infinity, can it ever actually reach infinity? If the answer is yes, then the sum of the series equals 1. If no, then the sum only converges on 1.

Not so?

To resolve the questions raised by the commenter, think for a minute about basic arithmetic. Addition is something that is done to two numbers. You take two real numbers and add them together. But suppose you had three numbers, x, y, and z? What does it mean to add three numbers together?

Very simple. You would begin by adding x to y. Then, you would take the result and add that to z. The resulting number can be said to be the sum of x, y and z. The point is that at each step of the process we only added two numbers together. The fact that addition (and all of the other standard arithmetic operations like subtraction, division and multiplication) can only be carried out with two numbers is what is intended by describing addition as a “binary operation.”

The procedure just outlined can be extended to any finite collection of real numbers. Thus, if I hand you ten numbers you can add them all together by starting with any two of them and proceeding from there. But what does it mean to add up infinitely many numbers? You certainly can’t go two at a time any more. You will never come to the end of the process if you try it that way.

The answer is that adding up infinitely many numbers is a fundamentally different process from adding up finite collections of numbers. The only way to give meaning to the idea of finding the sum of an infinite series is to use the idea of a limit. The basic idea is this:

Suppose you have a series like:


(1/2) + (1/4) + (1/8) + (1/16) + (1/32) +…

The bottom of each fraction is twice the number that appears on the bottom of the previous fraction.

Notice that the first fraction is (1/2). The sum of the first two fractions is (1/2)+(1/4)=(3/4). The sum of the first three fractions is (7/8), the sum of the first four is (15/16), and the sum of the first five is (31/32).

We then line up these sums and stare at them for a moment:


1/2, 3/4, 7/8, 15/16, 31/32, …

Since each term in this sequence represents the sum of part of the series, we refer to it as the sequence of partial sums.

You might notice that each term in the sequence is closer to one than the term preceding it. In fact, it seems reasonable to say that if I continued evaluating partial sums, I would produce fractions that become arbitrarily close to one. And from there it seems reasonable to take the plunge and say that since the limit of the sequence is one, we may as well say that our original series adds up to one.

I have left out a fair amount of technical detail here. The term “limit” has a precise definition usually expressed with a multitude of Greek letters (epsilons and deltas to be specific). But the important point is that summing an infinite series is a fundamentally different process from adding up finite collections of numbers. The former case is adequately modeled by thinking about how many objects you have when collections of various sizes are put together. The latter involves computing a certain sequence of numbers and trying to determine the limit (if, indeed, the limit exists at all).

That may seem like a subtle distinction, but it is crucial. Many of our intuitions about finite arithmetic break down when you try to apply them to infinite sums. I wrote a lengthy post a while back exploring one bizarre consequence of the way the sum of an infinite series is defined. Specifically, consider the series:


1 – (1/2) + (1/3) – (1/4) + (1/5) – (1/6) + …

Take my word for it that this series adds up to log 2, where the log refers to the natural (base e) logarithm.

(Which reminds me of a little math joke. What’s a logarithm? It’s a birth control method for lumberjacks! ha ha ha ha ha.)

Now for the bizarre part. I can rearrange the terms of that series so that when you add it up, instead of getting log 2, you get the number five instead. Don’t like five? Well, I can rearrange the series so that you get seven, or eleven, or minus eighteen or the square root of two or any other real number you care to mention. Really! The details are given in the post linked to above.

Sums of infinite collections of numbers are fundamentally different from sums of finite collections of numbers.

Now we can clear up the confusion in the comment left to Mark’s post. There is no distinction between saying the sum of the series equals one on the one hand, but merely converges to one on the other. When talking about infinite sums, convergence is the only game in town. Saying the sum equals one and saying the series converges to one are two different ways of saying the same thing (with the second formulation a bit more precise).

The commenter asks if n actually reaches infinity or if it merely approaches infinity. The answer is that infinity is not a final destination for wandering variables. The entire phrase “as n approaches infinity” has a precise definition. You should not think of this phrase as indicating that n is the sort of thing that goes places, infinity is a place for it to go, and the word “approaches” means the same thing here as it means in every day speech.

Going back to the problem that started it all, I would express things as follows: The expression .9999… repeating is a short-hand way of writing the number obtained when the infinite series


(9/10) + (9/100) + (9/1000) + …

is evaluated. It is a consequence of the way the sum of an infinite series is defined that the series above converges to one. Therefore, it is meaningful to say that the expression .99999… repeating is another way of writing the number one.

The final point is that there is no philosophical question here. That .9999… repeating is a logical consequence of the way various terms are defined, and that is all.

Which brings me, finally, to the question asked in the title of this post. What is Infinity? Well, the answer is that infinity, by itself, is not a word that mathematicians use very much. It is definitely not a number. You can’t add infinity to other numbers or multiply a number by infinity. And if you ever come across a math book where it appears the author is doing precisely that, I can assure you that specific rules were established for doing so in the context presented in the book.

It is often said that infinity is a concept. It expresses the idea of boundlessness, or of something that never ends. That’s a reasonable way of thinking about it. I would repeat, however, that infinity as an abstract concept just isn’t something mathematicians talk about very much.

In calculus people often say “as x approaches infinity” or in geometry you migh talk about the “point at infinity” but again, in these contexts it is the whole phrase that gets defined, not infinity by itself.

What mathematicians do talk about quite a bit is the infinite, especially infinite sets. It seems meaningful to say that there are infinitely many positive integers but only finitely many people on Earth. A great number of fascinating issues arise when you think seriously about infinite sets, but we’ll save that for a different post.

Comments

  1. #1 Karl
    June 20, 2006

    Hey, while you’re having fun talking about infinity why don’t you tell them about the three different infinities. Or just refer them to George Gamow’s (old) book “1,2,3, infinity”?

  2. #2 Jason Rosenhouse
    June 20, 2006

    Karl-

    What three infinities do you have in mind?

  3. #3 Karl
    June 21, 2006

    Well, it’s been a long time. Let’s see:
    I think they are called Aleph1, Aleph2, and Alepn3.
    Aleph1 is the infinity of the integers which is equal to the infinity of the rational numbers because the set of rational numbers can be ordered and therefore can be put into one-to-one correspondence with the natural numbers. Aleph2 is the infinity of the Real numbers which is greater than the infinity of rational numbers because between any two rational numbers there must be an irrational number. So the infinity of irrationals must be greater than the infinity of rationals.
    I don’t seem to remember what set has Aleph3 infinity.
    Maybe it’s the set of Complex Numbers. Actually, I’m not even real sure about the first two. As I said, it’s been a loooong time – we’re talking 40 or 50 years here – an infinity of time as it were.
    The significance of the title is that certain primitive peoples only have names for one, two and three. anything else is “more” or “many”. And now that we have learned about infinities, we can only count three of them. That’s about as good as I can do. I thought that you would be up on that.

  4. #4 Karl
    June 21, 2006

    Some googling has revealed that I, at the very least, have the names wrong. The rational numbers are aleph0, and the reals are aleph1. I can find no mention of what aleph2 is.

  5. #5 Corkscrew
    June 21, 2006

    Aleph2 is the infinity of the Real numbers which is greater than the infinity of rational numbers because between any two rational numbers there must be an irrational number. So the infinity of irrationals must be greater than the infinity of rationals.

    Your logic is a bit dodgy – after all, between any two irrational numbers there lies a rational number.

    I don’t seem to remember what set has Aleph3 infinity.
    Maybe it’s the set of Complex Numbers.

    The set of complex numbers has the same cardinality as the set of reals – you can construct a mapping from one to the other. An example: take a complex number x+iy and construct a real number by taking the first digit from the expansion of x, the first digit from the expansion of y, the second digit from the expansion of x, the second digit from the expansion of y, and so on. So, for example, 0.1234+0.5678i would map to 0.15263748.

    There are actually an infinite number of infinities, not just three. They can be recursively defined by taking a set of cardinality aleph_i, taking the set of all possible subsets of that set, and calling the cardinality of that set aleph_{i+1}.

    This appears to be a good page.

  6. #6 Corkscrew
    June 21, 2006

    Regards the 0.999999… thing: my understanding was that, if you define 1 to be the supremum of the set of numbers less than 1, it’s obvious that 0.9999… is greater than any such number and hence must be >= 1. Since it’s obviously not greater than 1, it must equal 1.

    The problem, of course, lies with the “it’s obvious that” – for example, nonstandard analysis defines a “number” that is greater than any real number less than 1 but itself less than 1. That completely mucks up the argument.

  7. #7 DV8 2XL
    June 21, 2006

    Hello, has anyone here taken Cal I? I recall this being dealt with in the first month.

  8. #8 polymath
    June 21, 2006

    hi…thanks for taking up my cause. this is an excellent discussion. in case any one is interested, i’ve tried to deal with some of the arguments on a new post on my blog.

  9. #9 Norman Levitt
    June 21, 2006

    I recommend that participants in this discussion look at a standard introductory work on axiomatic set theory to orient themselves properly. A key fact about cardinality:

    Given a set X , let P(X) denote the set of all subsets of X (including the empty set as well as X itself). Then the cardinality of P(X) is strictly greater than the cardinality of X itself. This is demonstrated by the Cantor diagonal argument.

    If N denotes the natural numbers, N = {0,1,2, …}, then Z= the set of integers and Q = the set of rationals have the same cardinality as N . However, the set of real numbers R (in the conventional construction) has the cardinality of P(N).

    The central mystery of set theory is the so-called “continuum hypothesis”: there is no cardinality strictly between that of N and that of R. In a more general version, this becomes the conjecture that for infinite sets X , there is no cardinality strictly between that of X and that of P(X). It is a mystery because it is possible to show rigorously that this conjecture can neither be proved nor disproved from the standard axioms of set theory.

    This is a two-part result. The first part shows that the continuum hypothesis is “relatively consistent” with the standard set theory axioms, that is, if the axioms, together with the CH, create inconsistency, then the same is true of the axioms all by themselves. This part was proved by Godel.

    The second part asserts that the negation of the CH is also relatively consistent with the standard axioms; as before, if the axioms together with the negation of CH produce inconsistency, so will the axioms alone. This is a result of Paul Cohen.

    This seemingly paradoxical result stems from an ingenious and rather simple basic idea. One creates within standard set theory a “sub-theory” which, under a suitable “interpretation”, satisfies the axioms of set theory plus the CH. Then one does the same for the negation of the CH. Thus the logical structure of axioms-with-CH is embedded in the logical structure of the axioms alone, and similarly for the negation of CH.

    This approach closely parallels the development of non-Euclidean geometry in the 19th century. There, the joker in the deck was the famous “parallel postulate”. What Riemann showed was that one could construct, within standard Euclidean geometry, a theory that, under the appropriate interpretation, satisfied all the Euclidean axioms, with the parallel postulate replaced by the proposition that any two “straight lines” must meet. In brief, one does this by considering the standard 2-dim sphere. Define an R-point to mean a pair of antipodal points in this sphere, e.g., north and south pole. Define a R-straight-line” to mean the set consisting of all antipodal pairs that lie on a given great circle. Then the axioms of Euclid, as modified by the alternative to the parallel postulate, hold if we now understand “point” to mena R-point and “straight line” to mine R-Straight-line. This system is called “elliptic geometry”.

    Poincare did something analagous, constructing what is now called the “Poincare plane”. This is a little more technical than Riemann’s construction, so I omit details. The bottom line however, is that this system, with the new interpretation of “point” and “straight line” satisfies the Euclidean axioms with the parallel postualte replaced by the proposition that there are infinitely many straight lines through an outside point that fail to intersect a given straight line. This system is called hyperbolic geometry.

    Note that both elliptic and hyperbolid geometry are, strictly speaking, subsystems of standard Euclidean geometry. This gives you the general idea of what goes on in the logic of the Godel-Cohen result, usually called, the Independence of the Continuum Hypothesis.

    Naturally, this all leaves the actual “truth status” of the continuum hypothesis as a deep conundrum, about which point philosophers of mathematics argue endlessly.

    Norman Levitt

  10. #10 Karl
    June 21, 2006

    Norman:
    As I said in my original post: It’s been a long time – so please forgive my ignorance.
    I thought that the definition of a line in elliptic geometry was a great circle – any circle on the surface whose center is the center of the sphere. Is that not so? Was that so previously, and has it changed?
    AND, I thought that plane geometry would be a subset of elliptic geometry – on a small section of a sphere (a subset of a sphere) plane geometry works (is true). Is that not so?

  11. #11 Dom
    June 21, 2006

    Suppose a candy bar costs 10 cents, and the wrapper can be used as a coupon to purchase your next candy bar. 10 wrappers will get you a free candy bar.

    How many candy bars can you buy with 90 cents?

    Answer 1: 90 cents will buy you 9 bars plus 9 coupons; these 9 coupons coupons will get you .9 bars plus .9 coupons; these .9 coupons will get you … and so on to infinity. In short, 90 cents will get you 9.999… candy bars.

    Answer 2: 90 cents will get you 10 candy bars even. Take the 10 candy bars, and pay the clerk 90 cents plus the 10 coupons you just got on the candy bars. There are no coupons left over.

    Thus, 9.999… equals 10 exactly.

    Dom

  12. #12 Jason Rosenhouse
    June 21, 2006

    Norman-

    I’m not sure if anyone was ever made less confused by reading a book on axiomatic set theory :) Seriously though, thanks for the interesting comment. As it happens, you’ve stolen my thunder a bit since I was planning to do a separate post dealing with different orders of infinity, which is what I had in mind with my closing paragraph.

    For anyone else reading this, I prepared a set of notes for a discrete math class I taught a few years ago in which I spelled out the details for many of the things Norman says. My intent was to make them accessible to people without much math experience, but with a willingness to work a bit. If you’re interested, you can find them in PDF format here. Please forgive the typos.

    Dom-

    Cool argument. I like it.

    Polymathematics-

    Glad I could help. Thank you for your excellent initial post on this subject.

    Karl-

    By now you’ve probably gotten more than you bargained for. As Corkscrew points out, there is actually an endless supply of different kinds of infinities. Given any infinite set S, the set of all subsets of S has strictly larger cardinality than S itself. That’s proved in the set of notes I linked to above.

    Concerning elliptic geometry, in Norman’s presentation the straight lines are, effectively, the great circles. His presentation is merely a bit more formal than what you find in elementary descriptions of spherical geometry.

    With regard to your final paragraph, keep in mind that in Euclidean geometry, words like point and line are undefined terms. The axioms allow you to make certain assumptions about the properties they have, but they tell you nothing about what they actually are in any given situation.

    So in elliptic geometry we have one notion about what points and lines are, and in plane geometry we have a different notion. Once you have provided a definition of these terms, you can then try to verify that the standard Euclidean axioms hold, given your definitions. That’s why Norman began his description of elliptic geometry by stating precisely what is meant by a point and a line in his environment.

    So I don’t think it really makes sense to say that on a small section of a sphere, plane geometry works. I think what you have in mind is that in practical terms, if you live on a very large sphere but are only concenred with some small region on that sphere, it is acceptable to apply standard plane geometry. So, for example, you can figure out the area of a rectangular football field by applying standard Euclidean geometry, without worrying about the ffects caused by the curvature of the Earth.

    DV8 KXL-

    Actually, sums of infinite series is generally second semester calculus.

  13. #13 argystokes
    June 21, 2006

    The .999… question always seemed pretty obvious to me, though I am no mathhead.

    1/9 = .1111…
    .1111… x 9 = .9999…
    1/9 x 9 = 1
    .9999… = 1

    At least that’s how my elementary-school mind sees it.

  14. #14 Karl
    June 21, 2006

    Jason:
    Not more than I bargained for, but more than I had hoped for. Many, many years ago I got an MS in math and taught math for 15 years – including Calc 1. Since then, for 25 years, I was in Application Programming and Data Base Administration. So I haven’t had a math discussion in a long time. I love it.
    But I have yet another question – maybe a quibble.
    You said:
    “So I don’t think it really makes sense to say that on a small section of a sphere, plane geometry works. I think what you have in mind is that in practical terms, if you live on a very large sphere but are only concenred with some small region on that sphere, it is acceptable to apply standard plane geometry. So, for example, you can figure out the area of a rectangular football field by applying standard Euclidean geometry, without worrying about the ffects caused by the curvature of the Earth.”
    I don’t see the difference. Isn’t “you can figure out the area…” the same thing as saying “on a small section…plane geometry works”?

  15. #15 Doormat
    June 22, 2006

    Corkscrew brought this point up: if you want to be annoying, then in non-standard analysis one can both define the number 0.99999… and show that it doesn’t equal 1. Of course, I somehow doubt that this is what most people mean when they ask “Is 0.9999… repeated equal to 1?”

    For maths people out there, I strongly recommend having a play with non-standard analysis. It’s kinda fun, in a slightly not-useful way…

  16. #16 Andy
    June 22, 2006

    Doormat’s post is incorrect. In non-standard analysis, there are numbers that are greater than all of .9, .99, .999, .9999, and so forth, but are less than 1. For example, any number of the form 1 – epsilon, where epsilon is an infinitesimal, satisfies these inequalities. But the statement

    “The limit of the sequence .9, .99, .999, .9999, is 1″ is still just as true in non-standard as in standard analysis, as is guaranteed by the transfer principle.

  17. #17 Paul S
    June 24, 2006

    Jason, since I’m the commenter to whom you’re replying, I thought you might like to see my latest addition to the saga.

    I Am Not A Mathematician, so I don’t know how close that is to correct. But it seems to jibe with what you have here, so I’m pleased to note that I worked it out before I read your blog.

  18. #18 David Harmon
    June 25, 2006

    “The answer is that infinity is not a final destination for wandering variables.”

    LOL! Great phrase there!

    Karl, the “third infinity” you were probably thinking of is F, the cardinality of functions. (Gamow describes it as the cardinality of “curves”, but I’m pretty sure that’s equivalent).

    Gamow also pointed out that since N had been proven to be the “smallest” infinity, Aleph-0 had was defined as equal to N. Aleph-1 was originally meant to represent “the next infinity after Aleph-0″, but “we” (he at the time) didn’t actually know whether that was, in fact, equal to C. According to Norman Levitt’s explication above, the question has since been “proved to be unprovable”, so I don’t know what’s actually happened to the Aleph notation.

  19. #19 Polymath
    June 29, 2006

    for anyone still interested, i’ve posted a final essay about it here.

  20. #20 Polymath
    June 29, 2006

    uhh…i mean on my blog, of course.

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