Is 0.999 … = 1? (wonkish)

29 Feb, 2016 at 12:52 | Posted in Statistics & Econometrics | 8 Comments

What is 0.999 …, really? Is it 1? Or is it some number infinitesimally less than 1?

The right answer is to unmask the question. What is 0.999 …, really? It appears to refer to a kind of sum:

.9 + + 0.09 + 0.009 + 0.0009 + …

9781594205224M1401819961But what does that mean? That pesky ellipsis is the real problem. There can be no controversy about what it means to add up two, or three, or a hundred numbers. But infinitely many? That’s a different story. In the real world, you can never have infinitely many heaps. What’s the numerical value of an infinite sum? It doesn’t have one — until we give it one. That was the great innovation of Augustin-Louis Cauchy, who introduced the notion of limit into calculus in the 1820s.

The British number theorist G. H. Hardy … explains it best: “It is broadly true to say that mathematicians before Cauchy asked not, ‘How shall we define 1 – 1 – 1 + 1 – 1 …’ but ‘What is 1 -1 + 1 – 1 + …?'”

No matter how tight a cordon we draw around the number 1, the sum will eventually, after some finite number of steps, penetrate it, and never leave. Under those circumstances, Cauchy said, we should simply define the value of the infinite sum to be 1.

I have no problem with solving problems in mathematics by ‘defining’ them away. But how about the real world? Maybe that ought to be a question to ponder even for economists all to fond of uncritically following the mathematical way when applying their mathematical models to the real world, where indeed “you can never have infinitely many heaps” …

In econometrics we often run into the ‘Cauchy logic’ —the data is treated as if it were from a larger population, a ‘superpopulation’ where repeated realizations of the data are imagined. Just imagine there could be more worlds than the one we live in and the problem is fixed …

Accepting Haavelmo’s domain of probability theory and sample space of infinite populations – just as Fisher’s “hypothetical infinite population, of which the actual data are regarded as constituting a random sample”, von Mises’s “collective” or Gibbs’s ”ensemble” – also implies that judgments are made on the basis of observations that are actually never made!

Infinitely repeated trials or samplings never take place in the real world. So that cannot be a sound inductive basis for a science with aspirations of explaining real-world socio-economic processes, structures or events. It’s — just as the Cauchy mathematical logic of ‘defining’ away problems — not tenable.

In social sciences — including economics — it’s always wise to ponder C. S. Peirce’s remark that universes are not as common as peanuts …


  1. I think a more forceful attack would aim directly at derivatives and integrals which rely on the same definitions of limits. The first argument to go would be Keynes’ marginal propensity to consume, I guess. Bummer!

  2. “The first argument to go….. ”

    And just about every equation in a New Classical model.


  3. Great article. You are right to be concerned that infinity cannot occur in the real world. More and more people are beginning to think it cannot occur in a mathematical sense either. These include mathematicians called ‘Finitists’.

    One of these mathematicians, Professor N J Wildberger, recently said “statements like 0.999…=1 ought to be taken with a large grain of salt.” (in a comment on his own website:

    A layman’s objections to 0.999…=1 are described in this article that includes two 10 minute videos:

    Further discussion of this article can be found here:

  4. There are many versions of the same problem. Is 1/3=0.333 … ? If so, isn’t 1/3+1/3+1/3=0.999 … ? But since 1/3+1/3+1/3=3/3=1, we have that 0.999 …=1.
    My favorite though is instead of asking is 0.999 …=1, to ask if 1-0.999 …=0.
    That is if 0.000…=0?

  5. “1/3+1/3+1/3=0.999….”

    Doesn’t necessarily follow.

    Until we know precisely what 1/3 equals (and we can’t) we can’t say what the sum equals.

    • It follows if 1/3=0.333 …, which is what I wrote.
      The same goes for 0.000 … which I suppose you would deny equals 0.

  6. Pontus,

    “It follows if 1/3=0.333 …, which is what I wrote.”

    Yes you did, but let’s face it, there’s just a little bit of pea and thimble gaming in your post. 🙂

    “The same goes for 0.000 … which I suppose you would deny equals 0.”

    I have no idea if it equals zero and I doubt if anyone else does either.

  7. If 0.999… stands for infinitely many commands

    Add 0.9 + 0.09
    Add 0.99 + 0.009
    Add 0.999 + 0.0009

    then following all of these infinitely many commands won’t get you to point 1. If you reached point 1 you have disobeyed those commands, because every single of those infinitely many commands tells you to get closer to 1 but NOT reach 1.

    Therefore, if you want to define the position of a “point” 0.999… on the number line, it cannot be at position 1 – and for the same reason (“disobeying those commands”) it cannot be short of 1 nor can it be past 1.

    So, if you want to measure the distance |1 – 0.999…| you know where to start the measurement (at point 1) but you don’t know where to stop the measurement, because the position of a “point” 0.999… is not defined on the number line.

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