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

[packbat]

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. ^

Comments

[active-apathy.livejournal.com]

Ah, you're using that proof. My favourite for this is:

Let x = 0.9
  10x = 9.9
   9x = 9.9 - 0.9
      = 9
    x = 1
∴ 0.9 = 1

[packbat.livejournal.com]

That's a good one - I mainly used the one-third proof because it was dead easy to transition to. Edit: Also, because it emphasized my point - which is that allowing recurring decimals at all implies one equals point nine recurring.

Another one for your viewing pleasure:

Let x = 0.999...
      =  0.9  +   0.09   +   0.009    + ...
      = 1-0.1 + 0.1-0.01 + 0.01-0.001 + ...
      = 1

[chanlemur.livejournal.com]

I had the same negative visceral reaction to this idea when it was first shown to me in high school, but it's hard (if not impossible) to counter the proofs.

[packbat.livejournal.com]

Actually, curiously enough, I didn't have that reaction - not exactly sure why. I think I even thought it was really cool, as in, "Hah! The carry is infinitely far away!" (I even remember coming up with some trick for calculating fractions based on using 0.999... instead of 1.) I think it's some sort of brain-wiring thing.

...but it's hard (if not impossible) to counter the proofs.

Some proofs more than others, of course - although it's a good reference, I count Sam Hughes's "Quickest Proof" (http://qntm.org/?pointnine) (and, in fact, his Preliminary Note) as dubious.

[chanlemur.livejournal.com]

Hm. Yeah, I don't like how that one almost seems to dip its toes into linguistics in the formation of its argument. I prefer the more strictly numerical proofs.

which is bigger, e**pi or pi**e ?

[(Anonymous)]

without using a calculator, please: which is larger, e^π or π^e? (from a Russian math problem collection, seen a few decades ago ...)

- ^z - http://zhurnaly.com

Re: which is bigger, e**pi or pi**e ?

[packbat.livejournal.com]

e^π - smaller to the larger is bigger than larger to the smaller with 2 and 10, so it should continue. What's the right answer?

Edit: Merle did 2 and 3, which are the other way 'round, so he guessed π^e.

Edit 2: If you take the log, you get π and e ln π - taking the Taylor series, that's π and e * (π-1), and the latter's larger. So I guess I should switch too, to π^e?

[zwol.livejournal.com]

But if you deny that 0.999... = 1, and instead define ε = 1 - 0.999..., and work out the consequences from there, you get the hyperreals, which are fascinating and useful (for instance, they make a way of motivating differential calculus that is IMO better than the standard).

[packbat.livejournal.com]

I do like infinitesimals - that's a good point. Making the conscious choice to define it that way is the important bit, though - as it will imply, for example, 1/3 = 0.333... + ε/3. As long as you know what you're doing, it's good.

Totally irrelevant to your point

[baxil.livejournal.com]

I like your footnote notation - the superscript numbers and the return-up-arrow. Be aware, though, that if you're going to continue using it, your links will break on your journal page (and possibly on other people's friends pages). When I've linked footnotes, I've tried to insert dates in the links to prevent that; for full compatibility, links not buried beneath a cut tag may require the anchor tag be something like username_date_footnotenum.

Re: Totally irrelevant to your point

[packbat.livejournal.com]

Ah, excellent point! I'll be sure to correct it appropriately. (What about [month][abbreviated subject line] as a salt?)

Edit: Make that [yymm][keyword] between [foot/ref] and the number - that way, digits and numbers alternate.

(Also: I cannot take credit for the footnote notation - I've seen it many places, I merely borrowed it as good.)

[roaminrob.livejournal.com]

There is a subtle, but important, error in the proof you gave.

Where 1/3 = .999..., it must be assumed that there is still a fractional remainder. 1/3 doesn't exactly equal .999... any more than pi exactly equals 3.14159.

For example, I could use your proof to make some pretty wild conclusions:

1/6 = .16666...; 6 * .16666 = .9996; thus, .9996 = 1.
1/36 = ..02777...; 36 * .02777 = .99972; thus, .99972 also equals one!

Hmm, but wait.

If .9996 = 1, and .99972 = 1, then .9996 = .99972. Ooo. Hey, this could get fun!

[roaminrob.livejournal.com]

Or, to put it another way:

A mathematician walks in to Euclid's Machining, Inc., and speaks to the head machinist. "Sir," he says, "I'm working on a project, and I need a cylinder and a donut, CNC machined out of stock aluminum. The donut must have an inside diameter of 1.000 inches, and the cylinder must have a diameter of .999 inches."

"No problem," says the machinist, "but it'll cost you a fair bit for precision to thousandths of an inch." The mathematician agrees to the cost, and decides to return the following week for the completed pieces.

When he gets returns, he tries desperately to fit the two pieces together, and furiously approaches the machinist. "I want my money back! These parts aren't within tolerances!"

"Hey," shrugs the machinist, "you're the one that thinks .9999 is the same as 1 inch."

[packbat.livejournal.com]

The ellipsis is critical here. 0.999... ≠ 0.999, because the former decimal is nonterminating.

Look again at the one-third proof. I've numbered the lines this time.

[1] 1/3 = 10/30
[2]     = 9/30 + 1/30
[3]     = 9/3 * 1/10 + 1/3 * 1/10
[4]     = 3 * 1/10 + 1/3 * 1/10
[5]     = 3 * 1/10 + 3 * 1/10 * 1/10 + 1/3 * 1/10 * 1/10
[6]     = 3 * 1/10 + 3 * 1/102 + 3 * 1/103 + ...
[7]     = .3 + .03 + .003 + ...
[8]     = .333...

What you are pointing out is essentially what is stated on [4]: that if you terminate the decimal at any point, you need a 1/3 * 1/10^-n to represent the remaining part of the fraction. However, the notation does not represent the termination of the decimal at any point - rather, the decimal never terminates. The roundoff error you point out never comes. Instead, you get the infinite series (copied from Sam Hughe's page (http://qntm.org/?pointnine)):

0.9999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...

            n=∞
          =  Σ 0.9 * 0.1n
            n=0

            n=∞
          =  Σ a * rn where a=0.9 and r=0.1
            n=0

          = a / (1-r)
          = 0.9 / (1-0.1)
          = 0.9 / 0.9
          = 1

If you require that all decimals terminate, 0.999... is not a decimal.

[roaminrob.livejournal.com]

(Apologies for the repeated comments. Free LJ accounts apparently aren't cool enough to edit their comments, and LJ's comment preview feature bites hard, so I have to try a couple times to find the appropriate tags. Sorry.)



Exactly. If, at any point in working with a non-terminating decimal, you choose to operate on the number as though it did terminate, then you're working on an approximation.



Sorry, let me back up for a second.



I think we can agree that 1/3 != .333 (i.e., without the ... notation), right? We can represent 1/3 as .333... repeating infinitely. But, then what happens if you perform an operation on such a number?



You might say that 3 * .333... must be equal to .999... infinitely, but you've just induced a rounding error; to get that result, you had to behave as though .333... terminated at some point.



Or, let me try yet another tack:

[1] 1/3     = 10/30
[2]         = 9/30 + 1/30
[3]         = 9/3 * 1/10 + 1/3 * 1/10
[4]         = 3 * 1/10 + 1/3 * 1/10
[5]         = 3 * 1/10 + 3 * 1/10 * 1/10 + 1/3 * 1/10 * 1/10
[6]         = 3 * 1/10 + 3 * 1/102 + 3 * 1/103 + ...
[7]         = .3 + .03 + .003 + 1/(3 * 104)
[8]         = .333 + 1/(3 * 104)
[9] 3 * 1/3 = (3 * .333) + (3 * 1/(3 * 104))
[10]        = 1

The math I'm presenting here is as algebraically correct as what anybody else has presented, yet it arrives at a different conclusion.



I think you have to be really careful if you're dealing with infinitely repeating decimals and you're concerned about accuracy.

[packbat.livejournal.com]

(Apologies for the repeated comments. Free LJ accounts apparently aren't cool enough to edit their comments, and LJ's comment preview feature bites hard, so I have to try a couple times to find the appropriate tags. Sorry.)

No prob - it's cool.

You might say that 3 * .333... must be equal to .999... infinitely, but you've just induced a rounding error; to get that result, you had to behave as though .333... terminated at some point.

I think I see the problem.

Fundamentally, my argument is not, contrary to what I actually said, "point nines recurring equals one". My argument is "if one wishes to use the recurring-decimal notation, equating point nines recurring and one is useful." Why? Because then you can multiply 0.333... by 3 as if it terminated and still get the right answer. In fact, you can multiply 0.090909... = 0.09 by 11, or 0.142857 by 7, or any infinitely-repeating decimal representation of a fraction by its denominator, as if it terminates, and if you get the infinite string of nines in the decimal, know that you can replace it with a carry.

If you want to deny it, fine. But if you do, it's really inconvenient dealing with recurring decimals, not to mention mathematically inconsistent. I don't see that it's worthwhile.

[roaminrob.livejournal.com]

OK, I think I can go along with that, oddly enough.

The recurring nines problem is something that I'd thought about before, and I've spent a lot of time working it over in my head today. 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. It does make a certain sense to accept that, for all practical purposes, .999... = 1. I just balk at the notion that they're exactly equal.

[packbat.livejournal.com]

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.)

[roaminrob.livejournal.com]

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.

[packbat.livejournal.com]

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.)

[roaminrob.livejournal.com]

Incidentally, if you choose to just throw your hands up at me on this at some point, that's fine. I fully realize that I'm stubbornly refusing to accept something regarded as rigorously proven canon by a very large number of people who are both smarter and better trained in arcane mathematics than I am.

[packbat.livejournal.com]

Thanks - I'll probably take you up on that sometime. ;)

I fully realize that I'm stubbornly refusing to accept something regarded as rigorously proven canon by a very large number of people who are both smarter and better trained in arcane mathematics than I am.

Y'know, having written this proof, I'm not so sure the proofs are that rigorous. It's sorta definitional, in some ways. (Then again, all math is. :D )

[roaminrob.livejournal.com]

It's sorta definitional, in some ways.

Yep. I was very carefully going over the content of Sam Hughes' page that you linked to earlier. In any case where somebody is presenting a large volume of well-structured evidence supporting an idea that I have trouble accepting, I try to find the first point in their argument that I disagree with.

What I found was right near the top, where he says, "Different sets of numbers have different properties." (Which I don't disagree with, it's just from there I begin to follow a different line of reasoning.) I think -- without the necessary background in mathematics to give this notion any kind of legitimacy whatsoever -- that real numbers which necessarily have an infinitely repeating decimal value must be treated as a separate set of numbers, with some of their own properties. So, I see the case of 1 being represented as 1.000000... not any different from saying, 1 = 1 + 0 + 0 + ...; none of the decimals add any value at all to the number being operated on. However, in the case of something like .333..., each subsequent decimal adds some small value to the number in question.

Things begin to make some sense from there, although it's inconvenient. For example, that makes the precise value of the number somewhat difficult to deal with. It reminds me of the Koch snowflake; you have this thing with finite bounds but an infinitely long boundary.

So, let's say you wanted to add one Koch snowflake to another Koch snowflake. How would you do that? Well ... you sort of can't. You can get a very close approximation, close enough to do whatever it is you're trying to do. But you can't get the exact result, unless you first return the Koch snowflake you're working with to the original equation which described it. So, if I wanted to operate on .333..., the first thing I would do is return it to the equation that described it: 1/3. Then I could do useful things with it, like add another 1/3, and get the correct answer.

I think I'm making a distinction between the idea of a particular number, and its value, or its representation. I realize that that's something that would make a lot of mathematicians cringe. :-) But, again, there is a small but not insignificant difference between saying, "Koch snowflake of unit side four", and asking somebody to compare the drawing of such a snowflake to the drawing of another Koch snowflake. It's also better to say, "add a Koch snowflake of unit side 4 to a Koch snowflake of unit side 3 and tell me what you get", than it is to hand a person a drawing -- a representation -- of two Koch snowflakes, and say, "add these up for me".

[packbat.livejournal.com]

That's a good argument. I think I'm beginning to see where you're coming from - it's something that most of the people on the nines-recurring-equals-one side brush over, but one really shouldn't be so blasé about throwing infinities around. Recurring fractions don't seem to cause a lot of problems, but an amateur mathematician should be aware of the potential for mayhem in the original problem statement, whether or not they are reassured by any of the arguments.

[roaminrob.livejournal.com]

Right. Otherwise, they begin to go around saying silly things like, ".999... equals 1".

grins, ducks, and runs

[packbat.livejournal.com]

Y'know, this actually raised a interesting question of netiquette for me: is the correct response to a line like this (a) leave it as the last word, (b) a one-line acknowledgment, e.g. "Bah-dum-pssh!", or (c) something else entirely?

Incidentally: in a way, you're right, except that professional mathematicians are well-known to say ".999... equals 1". So amateurs should probably treat it the way they treat Gödel's incompleteness theorem (http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems) - with a humility born of acknowledging the (heh!) incompleteness of their understanding. :D