NDRU said:

It is equal to one if viewed as a limit (which is really the only way it makes sense). It is like the infinite sum of 1/(2^k) where k goes from 1 to infinity. The sum expanded out is 1/2 + 1/4 + 1/8 + 1/16 etc. The sum is marching halfway to one with each increment, and reaches one as a limit.

coolcat said:

So...You can't really get closer to 1 than .999 repeating so it might as well be 1

but any number lower can be added to infinately as long as it doesn't equal .999 repeating?

The first time somebody told me that, I got really mad. I was just like, "no, it doesn't."

Somebody explain to death what you mean by a limit in this case. K? Because something still isn't clicking for me.

Moonbeam said:

I suppose, I don't know limits. But that doesn't acuually seem much different than adding another decimal (I assume the fractions go on forever, too 1/32...).

They both get closer & closer and when they reach infinity they're there.
[Edited 12/27/06 18:35pm]

HereToRockYourWorld said:

retina was right in a way, when he said it was smoke and mirrors... it is smoke and mirrors, until you really see if a repeating decimal makes sense as a number.

In the proofs, we multiplied 0.9999rep * 10 = 9.9999rep, but unless 0.9999rep makes sense as a number, this calculation is meaningless, and could very well be a trick...

So first, we've got to be convinced that 0.9999rep is actually a real number...

So what is 0.9999rep? If we try to define it like this:

0.9 + 0.09 + 0.009 + ....

it is actually undefined, because it is impossible to finish the calculation, and unless we finish the calculation the sum doesn't exist...

So how else can we try to define 0.9999rep. We have to introduce the idea of a limit...

Limits let you deal with infinite sums and infinite sequences... without them, I don't think infinite sums and sequences make sense at all...

You can create a sequence of numbers... start with any number greater than 1. Let's say 4. Then to get the next number in the sequence, take its square root.

4, 2, 1.414... , etc... this sequence will never actually hit 1. But the limit of the sequence is 1. It approaches 1.

For example (taking Moonbeam's example):

1/2 + 1/4 + 1/8.... is undefined, but if you start calculating it out you notice that it approaches 1... so intuitively, you see that it approaches 1. And you can prove that it has this limit mathematically (the actual precise mathematical definition of a limit is kind of tricky), so I'm not going to do this proof. But the intuitive definition is the number being approached.

We can say that 1/2 + 1/4 + 1/8.... has the limit of 1, so even though the sum of 1/2 + 1/4 + 1/8... is undefined, the limit is clearly defined.

So a lot of times when you see a question like 1/2 + 1/4 + 1/8.... it is being presumed that they're asking for the limit...

Not all infinite sums have a limit... For example:

1 - 1 + 1 - 1 + 1 - 1..... doesn't approach a number it alternates between 0 and 1.

It can be proven that 0.9 + 0.09 + 0.009 +.... has a limit, and that limit is 1. And you can see this intuitively...

So if we defined 0.9999rep as the limit of 0.9 + 0.09 + 0.009 +...., then 0.999rep makes sense as a number, and that number is 1.
[Edited 12/27/06 19:55pm]

coolcat said:

Makes sense. Still somewhat unsatisfying.

Thanks.

discuss!

SlamGlam said:

So if i keep saying "0.99999.....", i'll eventually stumble and say 1?

thesexofit said:

SlamGlam said:

That av

I got that video off the vh1 special he did in the mid 90's. Good times

thesexofit said:

yeah it is from the "rock and Roll is alive ..." video...

NERDS!

also:

1/3 = 0.333 repeating

1/3 x 3 = 3/3 = 1

as 0.3333 x 3 = 0.9999

It's controversial because it doesn't make intuitive sense.

The equasion makes mathematical sense, which is interesting, but logically, anyone can see that it's infinitesimally smaller than 1 (infinitesimal=so small that it can't be measured, or infinitely small)

so .999... is virtually 1, but conceptually smaller.

anyway, it's a very interesting concept and I'd never seen that equasion before
[Edited 12/24/06 23:05pm]

coolcat said:

No I wouldn't. It gets closer and closer for every decimal that you're adding but it never ever quite gets there. That's why you speak of it as an approximation even in the world of mathematics. I know the whole 0.999 repeating = 1 concept fits really nicely into your equation and that you're really happy about that, but it just doesn't apply to the real world, I'm sorry. Only 1 equals 1, that's why it's called 1 and not 0.999 repeating. 0.999 repeating is a traveller (forever), 1 is already there.

repeated/repeating edit
[Edited 12/25/06 18:52pm]

great thread! i've been perplexed over this for decades!

And i like to take issue with it!
i've argued for provacative shits and giggles, that if .99999 (ad infinitum) = 1.00000 (ad infinitum) then why couldn't you say that the 'last' digit of the repeating number defined as .33333 is a 4, just as the 'last digit' of .99999 is essentially a 0 (that tips the scales)?

i would say that if you can have an infinite number of digits following the decimal point then you can also have

isn't .99999..... just short of 1 by the infinitesimal? a 'something' that in some ways can be said to be that ever so crucial 'amount' that grounds calculus, the non-zero number that is smaller than any definable real number?
(Calulus now defines its terms in limits, rather than upon the infinitesimal, but the calculus infinitesimal still has its adherents)

Couldn't you say that .99999 (ad infinitum) + .00000(ad infinitum)1 = 1?

i don't know. something about their (supposed!) equivalence just rubs me the wrong way.
does that make me crazy?

coolcat said:

I'm starting to understand this. It's all about infinity.

When you look at .999... it's hard to appreciate that the 9's literally go infinitely--not a hundered decimal places, and not a million, but forever.

That's the only way the equasions work.

And even if (as I believe) the number is infinitely smaller than 1, that really still equals one for all intents & purposes. Once you actually get to infinity (which never happens, but if you could) you'd have 1.

NDRU said:

yes, BUT: counting sequentially one 9 at a time, does an infinite amount of time of adding 9s get you any closer to an infinite number of 9s? i say no, it gets you no closer at all to .99999 ad infinitum, than does the simple ol' number .999

The first time somebody told me that, I got really mad. I was just like, "no, it doesn't."

Somebody explain to death what you mean by a limit in this case. K? Because something still isn't clicking for me.

thesexofit said:

yeah it is from the "rock and Roll is alive ..." video...

Working with infintesimals, and expanding the numbering system, seems a much more complicated task to me... how exactly do infintesimals work... can you add, divide multiply them like the reals... it just introduces so many questions...

HiinEnkelte said:

Yes, you are absolutely right... Without the concept of a limit, .9999rep makes no sense as a real number. When a repeated decimal is used, the limit is implied. .9999 is the limit of the infinite sum .9 + .09 + .009 etc... Suppose, we don't use limits, how can we make .9999rep make sense?

coolcat said:

i know! and that's what i like about em!

hey, after what Cantor did the mathematics with the transfinite numbers, infinities of different sizes, ...why the heck not!???

When contemplating Dedekind's cut of the number line, which preserves its continuity, it all seems reasonable to me to say that there is a real (no pun intended) place for the reality of the infinitesimal.

who knows? all this stuff is so boggling in such a wonderous way, but sometimes it is just does this to my head and eyes

coolcat said:

so the limit works, and enables mathematics to do wonderful things with the real world, and in mapping, or representing, the 'real' world, but.....is it a legit concept? it's like it cheats, but how? if we only really understood what how it cheats....

well if it works, then of course it is legit!
but who cares about what works?!
i want the truth, dammit!! -even if it doesn't 'work'!

we all know work sucks anyway.

HiinEnkelte said:

That's cool! I was actually wiki'ing after seeing your post, and found this:http://en.wikipedia.org/w...d_analysis which seems to be a rigorous theory of calculus using infintesimals. You might know about it already, but I thought you'd be interested.

HiinEnkelte said:

Unfortunately, the truth usually works...

just as the 'last digit' of .99999 is essentially a 0 (that tips the scales)?

SlamGlam said:

doesn't this in a way relocate part of the problem into the fraction 1/3, in asserting that it equivalent to .33333 repeating?
it's like the bar over the three can 'define' the number for us conceptually supposedly), but it never really is done, fixed, or can be completed, or is finally determined.

can any thing ever really have this property?
can anything ever weigh .33333 repeating pounds?
or be .33333 repeating ft. tall?