The Speculist

Before one of our geek (and I use that word with the utmost love and respect) readers chimes in with how I'm an idiot either for not already knowing this to be true or for not immediately recognizing the flaw, I want to explain what I see as the problem. It looks to me like you can only get to

9A = 9

through the steps that Greg takes us through. If you try to multiply it out the hard way, logic insists that the answer must be

8.999999999...999991

with that final 1 being the mythical, infinite digit. I assume this because if you multiply 9 by any other string of .99, that's the structure you get. Thus,

9 X .99 = 8.91

9 X .999 = 8.991

9 X .99999 = 8.99991

and so on.

I can't find a flaw with the way Greg does the math, however. But my position is that approaching infinity and infinity ought to be two different things. But then maybe that's what he means when he says it's counter-intuitive.

Sorry, any math on a number with ... at the end cannot be called "precise". There's the root of the fallacy.

Phil,

I'm inclined to agree with you that the ultimate, infinite, decimal expansion of 0.999... is infinitesimally less than 1, and that such a difference is (or would be) significant to mathematicians and philosophers.

Whether it is practically significant is a matter for discussion and disagreement.

If my understanding of mathematics is accurate (and that, too, is a matter for debate), just such infinitesimals are the conceptual launching point from Algebra into Calculus.

To paraphrase Fermat: "I have a truly marvelous proof of this proposition which this comment is too narrow to contain."

"Cuius rei demonstrationem mirabilem sane detexi. Hanc commentis exiguitas non caperet."

Didn't we come up with the concept of a limit because it was useful for making this distinction?

I think this is one of those immovable object/unstoppable force problems. You're trying to get a discrete number out of an infinite series, and I think the rules don't allow that.

Okay, in C it'd be a cast, and you'd be aware you're rounding, but that's neither here nor there. :)

By the by, the unstoppable force/immovable object problem is easy. To exist in the same universe, unstoppable forces have to go around immovable objects, or immovable objects have to be permeable by unstoppable forces. :)

Please bear in mind that I'm neither a philosopher or a mathematician. So I reserve the right to be completely full of bantha poodoo. :)

-Jim

The difference between 0.999.... and 1.0000... is infinitely small.

This is perfectly defined in math. The bounds of a set of numbers on the real line can be inclusive of exclusive. I've seen it denoted > for exclusive, and [] for inclusive.

The biggest number in the set [0,1> is the number we're talking about. By definition, it isn't 1. End of story.

should read "inclusive or exclusive." above.

The problem is that real numbers and decimal sequences aren't identical (cannot be paired up one-to-one). There are a few numbers, namely the ones that can be represented with a finite decimal sequence, that can also be represented with a sequence that has an infinite number of nines in it.

The actual definition of a real number is the "limit" of any sequence of rational numbers that converges in the "Cauchy" sense. Given an infinite sequence of rational numbers a_1, a_2, a_3, ..., Cauchy convergence is the property that given any small number "e" greater than zero, there exists some lower bound L such that for any m,n greater than L, the absolute difference of the mth and nth elements of the above sequence, |a_m-a_n| is less than e. In other words, for any small number, there's some point along the tail (ie, everything past the Lth place) in the sequence so that everything in the tail is closer to each other than this number e. Pick a new smaller but positive e, then you just need to go further along the tail before you get this property.

It turns out that Cauchy sequences can have the same limit. That occurs when you can pick any small positive number and have past points on each sequence that any two elements, one picked from the tail of one sequence and the second picked from the tail of the second, are always closer that that little number to each other.

This is important in tha there's infinitely many Cauchy sequences for each possible limit.

The interesting part is that this precisely matches real numbers to limits of these rational Cauchy sequences. Rational numbers turn out to be a subset of the reals. And the reals also inherit addition, multiplication, and the greater than or less than comparisons too.

Decimal expansions turn out to be a special case of a Cauchy convergent sequence. They can be written as:

s_0 = d_0

s_1 = d_0 + d_1/10

s_2 = d_0 + d_1/10 + d_2/100

...

s_k = d_0 + d_1/10 + ... + d_k/(10^k).

And so on. In other words, these are the finite decimal representations of the number to k places. For the math constant pi (the ratio of the circumference of a circle to its diameter), the decimal sequence would generate the rational number sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, etc. For CS types out there, we're truncating the decimal sequence rather than rounding off at any point.

Since these just different in decimal places to the right, the difference between the elements of the sequence is just some power of 1/10. That is, if k L, |s_k-s_l| 0 was arbitrary and the sequence is Cauchy and by definition it has a limit which is a real number.

The final remark is just to note that the sequences:

1,1,1,1,1,...

and

0, 0.9, 0.99, 0.999, 0.9999, 0.99999, ...

have the same limit (as described above). The first one has "1" as it's limit. That means that the second sequence also has "1" as its limit.

This is essentially the same as the paradox of the arrow which supposedly can never reach its target, because first it must travel half the distance, then half the remainder, then half of that etc. the implication being that since an infinite number of such steps are required, it will "never reach" its target. One could substitute "nine tenths of" for "half" and arrive at the same quandry. But there is no real paradox here. The notation 0.999... indicates a limit sum, which is the sum of an infinite number of values which themselves approach a size of zero. Not all such sums actually converge to a specific finite value, but this one does. Maybe the difficulty lies in the fact that 0.999... indicates only the value this limit reaches, not any point along the way. There is no final digit in this series, so there is no "final digit 1" when you multiply this value by 9. If an ellipsis (three dots at the end) implies an imprecise value, then 1.000... must also be an imprecise value. As far as mathematicians are concerned (and hey, this is their game in the first place) 0.999... is exactly equal to 1. You can say that "the difference is infinitesmally small" if it will make you happy, but last I checked the only real value which meets that criterion is 0.

David --

Thanks. Your answer resolves my basic objection.

Great discussion, all. For more of the same, Rand Simberg and friends have picked up the thread and are running with it.