packbat | Because I haven't done it yet: 0.999... = ?

Point nines recurring^[1] equals one.

I understand that this claim is intuitively displeasing to many intelligent people. "Point nine recurring doesn't equal one!", they say, and give various reasonable accounts of why the two should be distinguished. However, as an engineer, I submit to the reader that we should equate them, however displeasing it should appear, on the following practical grounds.

Decimal notation is an incredibly effective way of describing numbers. (Or, more accurately, positional notation is. We use decimal specifically because we have ten fingers. I think senary or quarternary notation would be a superior substitute, but I'm not in charge.) Using a small set of symbols and a few characters, we can represent in a very close-grained fashion numbers of a huge span of orders of magnitude - and even larger if we permit exponential notation (e.g. 6.022 * 10²³). Further, in decimal notation, comparison of the magnitudes of numbers is dead simple. (Is 14/25 larger than 9/16? 14/25 = .56 and 9/16 = .5625, so no. Compare that with performing the fraction subtraction!) However, it pays for these advantages in the corresponding flaw: there are many^[2] numbers which cannot be written in standard decimal notation. Included among these are the majority of fractions. A good example of this is 1/3.

1/3 = 10/30
    = 9/30 + 1/30
    = 9/3 * 1/10 + 1/3 * 1/10
    = 3 * 1/10 + 1/3 * 1/10
    = 3 * 1/10 + 3 * 1/10 * 1/10 + 1/3 * 1/10 * 1/10
    = 3 * 1/10 + 3 * 1/10² + 3 * 1/10³ + ...

As is obvious, there's no way to write this completely out as a decimal, like you could with 14/25 and 9/16. So, what do you do? You make up a new notation. Instead of trying to hit the end, you go until you've got the pattern, and then...

1/3 = 3 * 1/10 + 3 * 1/10² + 3 * 1/10³ + ...
    = .3 + .03 + .003 + ...
    = .333...

Now, recall, there's no requirement that this notation exist. If, for example, you were programming a digital computer, you might choose not to bother with this notation, as you would never store an infinite decimal. However, it is convenient, defined this way. One especially convenient thing about it, for example, is that one can perform the regular arithmetic operations without trouble:

2/3 = 2 * 1/3
    = 2 * .333...
    = .666...

4/3 = 4 * 1/3
    = 4 * .333...
    = 1.2
     + .12
      +.012
       +...
    = 1.333...

Et cetera.

However, the other consequence is the following.

3/3 = 3 * 1/3
    = 3 * .333...
    = .999...
 and
3/3 = 1
 thus
 1  = .999...

So, one is left with the dilemma. Preserve this useful notation, or throw it out to avoid this (to some) distasteful equality? And, looking at it like that, is it really any sort of choice?

Thus, point nine recurring equals one. Q.E.D.

1. Some may say, for example, "repeating", rather than "recurring". (Google suggests the terms are approximately equal in popularity.) Naturally, the specific term is irrelevant. ^
2. "Many", in this case, meaning "an infinite number of" - specifically, 2 raised to the (infinite) number of counting numbers. What aleph infinity this is depends on whether you accept the continuum hypothesis. ^

Flat | Top-Level Comments Only

I didn't like the conclusion I was coming to, which was that math on a real number with a necessarily infinite decimal is effectively impossible.

Well, you don't have to do math on the decimal. You can opt to say, for example, that 1/3, √(5), π, e, etc. don't have exact decimal representations, and have to be represented by other means.

Apropos of nothing: are you familiar with Hilbert's hotel (http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel)? I know most people know Zeno's paradoxes, but that's another good infinity-screws-with-your-head one. (And actually, the cigar example they gave suggests another trick for thinking of the infinite nines: you can conjure a carry-at-infinity the same way they conjured the cigars, I suspect.)

Well, you don't have to do math on the decimal. You can opt to say, for example, that 1/3, √(5), π, e, etc. don't have exact decimal representations, and have to be represented by other means.

I think what I'm settling on is that any rational number with an infinitely repeating decimal value can be exactly represented using the ... notation, but it doesn't make sense to do any math on that representation. Irrational numbers don't have exact decimal representations at all.

Apropos of nothing: are you familiar with Hilbert's hotel?

I wasn't! That was cool. It didn't really mess with my head at all -- it's been a while, but I did once read George Gamow's "1 2 3 ... Infinity" (http://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642), and it does a great job of explaining countable infinities.

But, then they got to the cigars bit, and of course I wrinkled my nose.

(And actually, the cigar example they gave suggests another trick for thinking of the infinite nines: you can conjure a carry-at-infinity the same way they conjured the cigars, I suspect.)

Oh, eww, yuk, god, no. Please.

Oh, eww, yuk, god, no. Please.

Yeah, that's probably a new record for the worst proof, isn't it? :D That particular supertask is very dubious.

("1 2 3 ... Infinity" sounds like a good book - I'll watch out for it.)

Because I haven't done it yet: 0.999... = ?

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