Let alpha be a real number

PM (Pure Mathematician): Let alpha be a real number.

NJ (Me): What does that mean?

PM: Surely you are joking. What do you mean by such a question? Everyone uses this phrase all the time, probably you also.

NJ: I used to, but now I am not so sure anymore what it means. In fact I suspect it is nonsense. So I am asking you to clarify its meaning for me.

PM: No problem, then. It means that we are considering a real number, whose name is alpha. For example alpha = 438.0457897416622849… .

NJ: Is that a real number, or just a few decimal digits followed by three dots?

PM: It is a real number.

NJ: So a real number is a bunch of decimal digits followed by three dots.

PM: I think you know full well what a real number is, Norman. You are playing devil’s advocate. Officially a real number is an equivalence class of Cauchy sequences of rational numbers. The above decimal representation was just a shorthand.

NJ: So the real number alpha you informally described above is actually the following: {{32/141,13/55234,-444123/9857,…},{-62666626/43,49985424243/2,7874/3347,…},{4234/555,7/3,-424/55,…},…}?

PM: Well obviously that equivalence class of Cauchy sequences you started writing here is just a random collection of lists of rational numbers you have dreamed up. It has nothing to do with the real number alpha I am considering.

But now that I think about it for a minute, I suppose you are exploiting the fact that Cauchy sequences of rationals can be arbitrarily altered in a finite number of places without changing their limits, so you could argue that yes, my real number does look like that, although naturally alpha has a lot more information.

NJ: An infinite amount of more information?

PM: If you like.

NJ: What if I don’t like?

PM: Look, there is no use you quibbling about definitions. Modern pure mathematicians need real numbers for all sorts of things, not just for analysis, but also modern geometry, algebra, topology, you name it. Real numbers are not going away, no matter what kind of spurious objections you come up with. So why don’t you spend your time more fruitfully, and write some papers?

NJ: Have you heard of Wittgenstein’s objections to the infinite shenanigans of modern pure mathematics?

PM: No, but I think I am about to.

NJ: Wittgenstein claimed that modern pure mathematicians were trying to have their cake and eat it too, when it came to specifying infinite processes, by bouncing around between believing that infinite sequences could be described by algorithms, or they could be defined by choice. Algorithms are the stuff of computers and programming, while choice is the stuff of oracles and slimy intergalactic super-octopi. Which camp are you in? Is your real number alpha given by some finite code or by the infinite musings of a god-like creature?

PM: I think you are trying to ensnare me. You want me to say that I am thinking about decimal digits given by a program, but then you are going to say that that repudiates the Axiom of Choice. I know your strategy, you know! Don’t think you are the first to try to weaken our resolve or the faith in the Axioms. Mathematics has to start somewhere, after all.

NJ: And your answer is?

PM: Sorry, my laundry is done now, and then I have to finish my latest paper on Dohomological Q-theory over twisted holographic pseudo-morphoids. Cheers!

NJ: Cheers. Don’t forget to take alpha with you.


18 thoughts on “Let alpha be a real number

  1. Tom Holroyd

    How do you feel about computable reals using e.g. continued fraction representation?
    It seems to me that there is a decision procedure for telling if two continued fractions are the same, which (*) works for numbers like e, which have simple, computable patterns. I wouldn’t mind not using uncomputable reals, since I can’t compute them anyway.

    I know you are OK with some of that, because the rationals have the same problem, i.e., given two rationals p/q and r/s, you often can’t tell if they are the same just by looking, you need to run a little subroutine that computes the cross product. Since rationals can’t even be represented uniquely, you might think that continued fractions, which are unique and have simple ways to manipulate them, would be OK.

    (*) “only”. So it won’t work for pi. But “e” is probably a more useful number than pi, given that we don’t need pi if we use rational trigonometry. The exponential function, on the other hand, is defined by the special properties of its derivative, yet has interesting extensions all over Math. Seems a shame to get rid of that, since we already know Calculus can be done without limits. It is probably not “random” that e has a simple continued fraction representation. It is possible to do useful work with exponentials (even if you don’t use limits). IMO.

    1. Tom W.

      “The exponential function, […] has interesting extensions all over Math. Seems a shame to get rid of that,”
      If it turns out that it’s gotta go, then make it swift and painless – but it’s gotta go.

  2. njwildberger: tangential thoughts Post author

    Continued fractions are a nice approach, but ultimately they suffer the same kind of indeterminacy as the other approaches, and arithmetic with them is too hard generally. In my opinion there is no approach to real numbers in the full generality, or even the more limited generality of computable reals that works. It is not real mathematics.

    1. Tom Holroyd


      You might like that. I’d be interested in any comments.

      It talks about “very big” numbers. You can apparently compute not only all real numbers but also uncomputable functions, if you wait longer than infinity.

      This is like how in hyperbolic space, a Turing machine can solve NP complete problems in polynomial time, at the expense of exponential space, which you get for free in hyperbolic space (or on an appropriate graph).

      Also I’d like to hear your thoughts on SIA, SDG, Fermat Dual numbers, etc.; basically that just as we expanded the number system to include sqrt(-1) we can expand it to include infinitesimals as nilpotent elements of a ring, such as R[X]/X^2. Once you go there, you find that R is not “the sum of its parts”. You can’t express the continuous with the discrete. One needs to expand the notion of number to include microlinear quantities. Once that’s done, you can do all of high school calculus without using the word “limit” once. And you are forced to use intuitionistic logic, which I consider a win.

  3. Mark

    My favourite absurdity of so-called ‘real’ numbers is the claim that 0.999… equals 1.

    0.999… is simply an endless series just like 1+2+4+8+… is an endless series.

    The most famous proof that 0.999… equals 1 involves multiplying it by 10 and subtracting the original series, and this supposedly removes the endless part.

    But we can apply the same logic to 1+2+4+8+…; by simply multiplying it by 2 and subtracting the starting series we can supposedly remove the endless part giving the result 1+2+4+8+… = -1.

    This uses exactly the same logic. And so if 0.999… = 1 then 1+2+4+8+… = -1.

    In both cases the logic is flawed because we are not lining up the 1st terms of the series when we do the subtraction. In both cases we are comparing the first (n+1) terms of the multiplied series with the first n terms of the starting series in order to create the illusion that the trailing terms all cancel out.

    However, 1 is related to 0.999… in the same way that -1 is related to 1+2+4+8+…, it is the ‘fixed part’ of the expression for the sum to the n-th term (or the fixed part of the ‘partial sum expression’ if you prefer – but I don’t like the word ‘partial’ as it implies there may be a full sum).

    Prof. Wildberger, please can you do an article and/or video on 0.999…? If you claimed it was not equal to 1 it would cause a massive earthquake, not just in the world of mathematics but all the way to infinity and beyond!

    1. Peter Tobias

      I hope Prof. Wildberger will do no such thing, because it would set back any attempts of him to convince more mathematicians to work within a math world without infinity. He has some arguments for working without infinity, due to it not existing in computers, neither in hardware nor in software. But publicly defending your claim that 0.999999… is NOT 1.000000… would expose him to ridicule among his colleagues that wouldn’t stop before his death.

      Why is your argument wrong?
      First, the series 1+2+4+8+… is not a Cauchy sequence and does not converge against a real number. Any “calculations” with it would involve infinity, oo, as a value, and the expression oo-oo has no defined result. But 0.999999… is a Cauchy sequence, and, therefore, the equation for its real limit, x, in 10x-x=9 makes sense.
      Another way to look at it, is considering the two Cauchy sequences
      (1) 0.9, 0.99, 0.999, 0.9999, 0.99999, … and
      (2) 1.0, 1.0, 1.0, 1.0, 1.0, … (a constant series)
      and determine that their difference converges against zero. Therefore, the two series converge against the same real value. The two series are two DIFFERENT representations, but represent the SAME real number, 1.

      1. njwildberger: tangential thoughts Post author

        Sorry, but it is much more nuanced than that. Professional mathematicians are usually bemused by lay people’s distrust of 0.999…=1. But there is an argument here. For example, your argument is clearly wrong, as your statement ” But 0.999999… is a Cauchy sequence” is clearly wrong.

      2. Peter Tobias

        Prof. Wildberger replied to my comment: your statement ” But 0.999999… is a Cauchy sequence” is clearly wrong.
        Because there was no Reply button under his answer, I reply to my own original answer instead.
        “0.999999… is a Cauchy sequence” is not wrong, least of all clearly.
        Assume we have the decimal representation of a number, x out of [0;1], with a sequence a(n), with each element a(n) of the sequence out of the set {1;2;3;4;5;6;7;8;9} and with x written as 1.000000… or as 0.a(1)a(2)a(3)a(4)a(5)a(6)…
        This notation expresses that x = Sum(j=1 to oo, a(j)*10^(-j) )
        The Sum is indeed a Cauchy sequence of its partial sums: For any 1>epsilon>0, there is an N with N>(-ln(epsilon))/ln(10) that for all n and m larger than N, the partial sums of n and m differ by less than epsilon.

        Each decimal representation is a Cauchy sequence, and we assume it converges to a real number.
        But a real number can have two different decimal representations, if for one of its representations the sequence a(n) is zero above some n.

      3. Peter Tobias

        Correction: I write the description on the fly, and I made a mistake. I should have written:

      4. Peter Tobias

        CORRECTION (tried again)
        I wrote my comment on the fly and made a mistake. I left out the ZERO. I should have written:
        … sequence a(n), with each element a(n) of the sequence out of the set {0;1;2;3;4;5;6;7;8;9} …

  4. Mark

    By the way, just to clarify, 1/3 does not equal 0.333… because the algorithm to covert the fraction to a decimal has no defined end point. The endless terms are not zeros and so the algorithm must have some defined way of ending in order to produce a fixed value.

    0.333… is an endless series that cannot equate to any fixed value because the series does not end; it has no ‘last term’. It has an endless supply of positive non-zero terms.

    An endless series should be treated as an endless series, not a fixed value. We should not simply assert/define that a series must equal the fixed part of the expression for the n-th sum (known as the ‘limit’ for a series that is said to converge).

    1. njwildberger: tangential thoughts Post author

      I agree quite whole heartedly with what you are saying here. People naively suppose that there is a coherent theory of arithmetic with rationals and repeating decimals, but it is much more subtle than it seems. And I know of no place where it has been worked out clearly and carefully.

      1. Mark

        Maybe we should not try to achieve theories of arithmetic with rationals and repeating decimals?

        I like to think in terms of how a calculator or computer might attempt to perform operations. So for me, 1 divided by 3 is no different to the square root of 2 or finding the circumference for a circle of diameter 1 (i.e. pi), these are all just algorithms applied to finite values.

        If the algorithm has no defined way of completing, then you cannot have a fixed value result.

        In your MF89 video, you showed how to revert from a repeating decimal to a fraction by multiplying by different powers of 10 until you get the trailing parts to apparently match and cancel out. This uses the same trick as the 10-x proof, and so I believe the method is flawed.

        It is easy to be persuaded that a series might be able to equate to a fixed value when you see the terms getting smaller and smaller. But if the terms were getting bigger, like 1+2+4+8+… or 1+2+3+4+5+… then it is more obvious that an endless series cannot equal a fixed value.

        Professor, would you say that 0.999… (defined as an endless series) does not equal 1?

  5. njwildberger: tangential thoughts Post author

    I would say that if we want to proceed logically, that after we have completely understood arithmetic with finite decimals, we might turn our gaze to repeating decimals. The first thing to do is to define what these objects are. We are the ones obliged to say what the meaning is. Of course one approach is just to define 0.999… to be equal to 1, just at the symbolic level. That does not necessarily have to have anything to do with finite decimal arithmetic. It does not mean that we yet have a valid theory of arithmetic with “repeating decimals”, since we have not defined these: only one single instance.

    So overall there are considerable challenges, not to say that it could not, and should not be done. But I know of no place that does it correctly, so currently statements like 0.999…=1 ought to be taken with a large grain of salt.

  6. Mark

    Hi Norman,

    Thank you for another brilliant response. Your observation about definitions is what people like me find so frustrating.

    Since the sum of a so-called ‘infinite’ series is defined to be equal to the series limit (BUT only for converging series), then it is claimed that no proof is required. We have defined it, so it is true by definition.

    For me, the problem is with any object that is defined to consist of ‘infinitely many’ parts, because I have not found a reasonable (clear, precise, and useable) definition of what ‘infinitely many’ means.

    Mathematicians will never take the rants of non-mathematicians seriously, not least because of the loose, informal language we use and our lack of in-depth maths knowledge. But if nothing else, it is a measure of the frustration experienced by non-mathematicians when we resort to writing articles like this one (about 0.999…=1 being a con trick):

    Surely something is wrong when reasonably educated people complain so vigorously about fundamental school-level maths principles!

  7. Ross

    Firstly, please forgive any logic errors herein – I am mostly self taught in mathematics. Any correction to errors in my logic here would be greatly appreciated.

    That said, I too have a problem accepting strings of infinite decimals as a definition of a “real” object. I may be guilty of constructionism, but I am of the opinion that there are no exact values at all – only things that are approximately equal to such values.

    When I think of, let’s say, a number line I think of something that can be divided into sections which can be labeled as numbers. So for the Integers this works fine. Anywhere between 1 and 2 could be thought of as the value “1” (or “2” depending on your floor / ceiling definition of integer). But this breaks down with the Rationals, since you would have to be able to locate extremely small intervals such as those of length (10^10^100)^-1 or smaller, but this is a lot smaller than any mark we could measure to on said line. So what I’m trying to show is that there is an error which becomes larger as you try to measure to exactly a given value.

    Am I making any sense here?

  8. Mike

    If I understand correctly, you seem to be arguing that what cannot be accurately represented by our means of communication doesn’t exist. If a value cannot be accurately represented by any number system, it doesn’t exist. However, when I think of a diagonal of a 1 x 2 rectangle, my concept of that diagonal is not an approximation. If it were, I would also have to conceive of some inconsistency in my concept of the figure, such as a gap at the vertex, or the diagonal extending past the vertex. But that’s not the case. If you could enter my mind, you would find that the diagonal meets the vertex with infinite precision. I can even conceive of zooming in on it to the quadrovizillionth magnitude, and the diagonal still meets the vertex with utter perfection.

    The difficulty isn’t the concept, but our impoverished means of communicating concepts. You’re right as far as that goes; no matter how many decimals you include, it’s still just an approximation. However the problem isn’t just with mathematics. No matter what description I try to give of the taste of a mango, for example, it could never be more than just an approximation.

    By the way, I came here from YouTube, and I really think your videos are excellent. Thanks!

  9. extranosky

    Correct me if I am wrong Prof. Norman. From what I know about you a few years ago, you object to the notion of an infinite set, not to the concept of infinity. Has this changed now, in that you reject the concept of infinity altogether?


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