I drunk what: i believe 0.999... is an asymptotic function whose limit is less than 1

A "function" is a specific mapping from an independent variable ("x"), or variables, to a dependent variable "y". What is the input and output of your "function" 0.9.... ?

0.9... is not a function. It's a number. And that number is equal to 1.

Prost!

I drunk what: the current popular theory is that 0.999... is precisely equal to 1 and anyone who disagrees is a big stupid dumb head janitor and probably flunked high school

What he means is the mathematically proven theory -- hence its popularity. It's akin to saying that the "current popular theory" is that a²+b²=c².

I drunk what: i think the difference between 1.0 and 0.999... is so tiny that even the smartest big brained mathematician...

...realizes that the attempt to add a difference to something infinitesimally small is that doing so would mean it was no longer infinitesimal, but just really, really, really, really farking long.

I drunk what: i have suggested the problem is more about logic and terminology than being able to derive algebra proofs to support one's position

Which is otherwise known as "special pleading".

I drunk what: my view doesn't require any leaps of faith (or miracles), concerning logic

It requires not understanding logic, which is the driving force behind all mathematics.

I drunk what: their view completely relies on assumptions, presumptions, optimistic confidence in obtuse math and the kool aid they were fed by their professors in order to receive a passing grade for that class

And rigorous mathematical proofs that can be replicated any number of times in any number of different ways to all arrive at the same result. There is that aspect, as well. I suppose if you consider just inventing math to be ok, then yes, I can see the hostility to hard proofs.

I drunk what: however i have yet to hear a single rebuttal to my logical proposition, that doesn't involve any algebraic rationalizations of trying to justify their wish-beliefs...

So now you're angry that your "imagine this!" scenarios are being rebutted by incontrovertible mathematical proof?

Alright, imagine this: a²+b²!=c². Prove me wrong -- without math! And every time you say anything, I'll just say "imagine this!" and repeat it.

evaned: Let me state my question again for the thick-headed: give me a real number which is larger than 0.9999... and smaller than 1.0.

likewise allow me to state my answer for the thick headed

let's use integers to possibly increase your chances of understanding my english

what number is higher than 9 but less than 10? there isn't one

does that mean that 9 = 10, because there is no number higher than 9 and less than 10? no, because the DIFFERENCE is still 1

the fact that a DIFFERENCE exists means BY DEFINITION that 9 =/= 10, even though you cannot produce a number (integer) that is higher than the lower bound and less than the higher bound

so my answer was that you are right there is no number real or imagined that could be BETWEEN 1.0 and 0.999... but there still is a DIFFERENCE between those two entities, therefore they are NOT EQUAL

capice??

I drunk what: what number is higher than 9 but less than 10? there isn't one

does that mean that 9 = 10, because there is no number higher than 9 and less than 10? no, because the DIFFERENCE is still 1

Too bad the integer numbers cannot, by definition, have infinitesimal length as the rational, irrational, and real numbers can. Otherwise you might really have a point there.

There is a mild spectacle in the idea that in an attempt to disprove basic high school math, you're regressing to kindergarten-level math. 9+1=10 sure was a fun flash card, wasn't it?

I must have seen this nine times.

Dr. Mojo PhD: There is a mild spectacle in the idea that in an attempt to disprove basic high school math, you're regressing to kindergarten-level math. 9+1=10 sure was a fun flash card, wasn't it?

it does sorely disappoint me when even the piled high deepers refer to simple kindergarden-level exercises as "imagine this!"

if attempting to complete a series such as 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. ?? what comes next, requires your imagination, then perhaps i better quit while i'm ahead?

otherwise some others in here will think i'm trying to be "condescending" or something?

but that's the nature of logic, either you get it or you don't

and on that note i'll leave you chaps with a lil quote:

"Insanity: doing the same thing over and over again and expecting different results." ~ Albert Einstein

alla salute

step 1: 0.99
step 2: 0.999
step 3: ????
step 4: profit = 1.0!!

///you should be a lil more explicit in step 3

cin cin

mamoru: /teach 7th grade math in Thailand

*sheds a tear*

Lsherm: I'm weeping for the kids these days.

me too

poor 7th graders don't even stand a chance...

/last word
//thanks for posting trollolololoru

I drunk what: step 1: 0.99
step 2: 0.999
step 3: ????
step 4: profit = 1.0!!

///you should be a lil more explicit in step 3

Oh sure. Step 3 is easy. Step 3 explains that steps 1 and 2 are finite and 0.999... is not finite. Seems to be what you're having extreme difficulty with.

I drunk what: i believe 0.999... is an asymptotic function whose limit is less than 1

Believe what you want. It doesn't make it true.

Is "1" a function?

I drunk what: let's use integers to possibly increase your chances of understanding my english. what number is higher than 9 but less than 10? there isn't one

Correct. That's because integers are not dense.

However, real numbers are. Between any two real numbers is another real number. This is provable. In fact, you don't even have to go to real numbers; rational numbers are also dense. Given two natural numbers a and b, the result of (a+b)/2 is also rational. (Rationals are closed under addition and division.) And if a < b, then a < (a+b)/2 < b is guaranteed to hold. Great, we've found a new rational number between a and b!

The same thing works for reals, though it's less obvious to show why reals are closed under each of the given operations; you have to pick a definition of the reals and prove it. (For rationals, it's easy.) It doesn't work for integers because integers are not closed under division.

(Incidentally, though it doesn't matter for this, rationals are dense in the reals. In other words, given any two different real numbers, it's possible to find a rational number between them. The same holds true in reverse: reals are dense in the rationals.)

I drunk what: so my answer was that you are right there is no number real or imagined that could be BETWEEN 1.0 and 0.999... but there still is a DIFFERENCE between those two entities, therefore they are NOT EQUAL

Correct on the first count, incorrect on the second.

If we let a=0.9... and b=1, and you cannot give a real number between them, that can only mean one of two things: either a=b or one of a or b is not a real number.

The fact that "there is not always another integer between two integers" is a false statement has no bearing on this.

By the way, the most interesting thing about 0.99...=1 is that there are nonstandard models where it doesn't hold, because people have invented ways to consistently work with infinities and infinitesimals. (The fact that she touches on this is why I like Vi Hart's video on this subject so much.)

However, the key word there is nonstandard. In this context, it means "unusual" or "atypical." That you're working in the standard model is taken as an assumption unless you state otherwise. I can't just come up and say "5=12!" and, when challenged, say I'm working in a different model.

("Model" in this sense is a technical term. It means a set (like the reals or hyperreals) and definition of operations (like * and +) that meets a set of axioms set out. The standard model is standard because it's the smallest -- that is, any potential model for the axioms, e.g. the hyperreals, contains the real numbers.)

I drunk what: Kahabut: Will both of you please state your case in clear language?

i believe 0.999... is an asymptotic function whose limit is less than 1

not less than or equal to

the current popular theory is that 0.999... is precisely equal to 1 and anyone who disagrees is a big stupid dumb head janitor and probably flunked high school

i think the difference between 1.0 and 0.999... is so tiny that even the smartest big brained mathematician cannot devise a proper way to accurately describe it and therefore feel confident to ascribe its insignificance to be the same as nothing-not existing

i have suggested the problem is more about logic and terminology than being able to derive algebra proofs to support one's position, but the old dogs and piled high deepers are appalled that i would ever question their traditional beliefs and demand an apology from me

but they aren't going to get it

i've clearly stated my reasoning multiple times in this thread if you'd like to continue the discussion

my view doesn't require any leaps of faith (or miracles), concerning logic

their view completely relies on assumptions, presumptions, optimistic confidence in obtuse math and the kool aid they were fed by their professors in order to receive a passing grade for that class

however i have yet to hear a single rebuttal to my logical proposition, that doesn't involve any algebraic rationalizations of trying to justify their wish-beliefs...

/not going to hold my breath either
//also not interested in the childish games of name calling and condescension

ready when you are

That is more or less what I thought.

And you 100% correct. You even managed to point to actual problem here, the difference between logical math and engineering math (physics).

Carry on then.

Kahabut: the difference between logical math and engineering math (physics).

Hahahaha, what?

wjllope: Kahabut: You aren't seriously arguing if .999999999 is = to 1... right?

They are not.

The confusion has to do with whether 0.9... is equal to 1 (the three periods there are key), not whether 0.999999999 (i.e. some finite number of trailing 9's) is equal to 1.
The trailing periods in "0.9..." imply an infinite series of 9's.

0.9... is equal to 1, and multiple posters have provided numerous well-known reasons why this is so.

The basic problem, i think, is that I drunk what does not, or chooses not to, understand these replies. Sadly, once (s)he learned (s)he was not capable of defending his(her) original claim, (s)he reverted to the sort of name-calling and passive-agressive deflections that one commonly sees in climate threads.

cheers

So you believe that 0.9... is the same thing as 1?

Then exactly what is the point of expressing 0.9... at all? Why not just put 1?

Kahabut: So you believe that 0.9... is the same thing as 1?

Then exactly what is the point of expressing 0.9... at all? Why not just put 1?

Why put ²⁸⁶/₂₈₆ instead of 1? Why not just put 1?

By that convention, do you believe ²⁸⁶/₂₈₆isn't 1?

Kahabut: So you believe that 0.9... is the same thing as 1?Then exactly what is the point of expressing 0.9... at all? Why not just put 1?

"So you believe that 2 + 2 is the same thing as 4?

When exactly what is the point of expressing 2 + 2 at all? Why not just put 4?"

(BTW, I'm not sure my previous post about nonstandard analyses is exactly on point. It's... at least sort of true, though I'm not sure it's completely true.)

Dr. Mojo PhD: Kahabut: So you believe that 0.9... is the same thing as 1?

Then exactly what is the point of expressing 0.9... at all? Why not just put 1?

Why put 286/286 instead of 1? Why not just put 1?

By that convention, do you believe 286/286isn't 1?

First you say .9/1=1 then you say 286/286=1 You do realize the fallacy of your logic right?

evaned: Kahabut: So you believe that 0.9... is the same thing as 1?Then exactly what is the point of expressing 0.9... at all? Why not just put 1?

"So you believe that 2 + 2 is the same thing as 4?

When exactly what is the point of expressing 2 + 2 at all? Why not just put 4?"

(BTW, I'm not sure my previous post about nonstandard analyses is exactly on point. It's... at least sort of true, though I'm not sure it's completely true.)

No, I believe that 2+2=4 it is not 4 but equal to it. However, .9...=1 is simply false. Not sure how hard this is really.

Kahabut: First you say .9/1=1

Is that a thing I said? I don't remember saying that.

Kahabut: No, I believe that 2+2=4 it is not 4 but equal to it. However, .9...=1 is simply false. Not sure how hard this is really.

Apparently too hard.

0.9... = 1 is true in the standard reals as assuredly as . This is just a fact. It's provable using several different methods; saying it's false is as untrue a statement as you'll ever make.

If you're actually interested in carrying on a more intelligent conversation as opposed to me just saying you're wrong, you can start by explaining what your objection is to any of the standard proofs: whether they be

let x=0.9..
10x = 9.9...
10x-x = 9.9... - 0.9...
9x = 9
9 = 1

or

1/3 = 0.3...
3*1/3 = 3*0.3...
1 = 0.9...

or my proof via the density of the reals ("give me a number greater than 0.999... and less than 1).

evaned: 0.9... = 1 is true in the standard reals as assuredly as .

Wheee, yay me messing up html.

"0.9.. = 1 is true in the standard reals as assuredly as 2 + 2 = 4."

evaned: let x=0.9..
10x = 9.9...
10x-x = 9.9... - 0.9...
9x = 9
9 = 1

Umm, messed up again. Last line should be x = 1.

Wow, this is a fantastic thread. I cannot stop reading it.

And, for the record, if you're arguing that 0.999... is not equal to 1, YOU ARE WRONG.

This whole thing smells strangely like evolution vs. creationism - people that just disregard scientific proof, create an alternative, then stick their fingers in their ears and just "la la la" through life.

OK, so I've done some more reading about the hyperreals and how 0.9...=1 looks in other nonstandard systems, particularly the part at here in the Wikipedia article.

From what I can tell, what happens in them is it depends on how you associate 0.9... with an actual number. After all, 1 isn't in itself a number; it's just a vertical-line-thingy on your computer screen. However, it gets associated with the abstract notion of one. If the Arabs had decided that 1's meaning is the number two, then 1+1 = 4 would hold.

For integers, the association between a numeral and the actual number is pretty easy. We define what 0-9 mean, and then it's the old 21=2*10+1 thing. For reals it becomes a bit more complicated, as you have to resort to the limit of an infinite series.

Apparently it becomes much more flaky in non-standard models. For instance, with the hyperreals, there are a couple options. The first one invented(?) expresses hyperreal numbers in the form ab.cd... ; ..ef... The expansion of 1/3 is 0.33...;...333..., and if you multiply by 3 you get 0.99...;...999... which can be proven to equal 1 in that formulation. If you say "I'll make the hyperreal part all 0s", as in 0.99...;...000..., then what you get isn't a valid number in that construction. (It's like how "xhtaoerc" is a series of letters but doesn't actually correspond to any word.) However, a different definition does set 0.99... to be infinitesimally far from 1; and in the hyperreals, that means strictly less than 1.

So in other words, not only should you say that you're using a nonstandard model if that's what you want to do, but you really need to say how you're translating 0.9... to an actual number.

Kahabut: And many of us farkers are amazed at the number of people that think they can come up with multiple answers to the same equation.

This statement is amusing in light of this:

x = 0.9...
10x = 9.9...
10x - x = 9.9... - 0.9...
9x = 9
x = 1

and

1/3 = 0.3...
3(1/3) = 3(0.3...)
1 = 0.9...

If they're the same equation -- and they are, they can be repeated with as many variations that all lead to any real, non-zero terminating decimal having an equal representation as decremented by 1 in the prior least significant bit and trailing 9s into infinity, regardless of base -- then no other answer can be arrived it. Since they, by Kahabut's own admission, cannot be different answers, they therefore must be the same number represented two different ways. Yet he (she?) refuses to accept it.

Most commented threads from Friday's main page




Not sure what to make of this.

Lsherm: Sure, mathematical theories change, but unless you're willing to put up a proof, they aren't a matter of opinion.

I'm sorry, but my right to believe 1 = 2 is protected by the 2nd amendment.

evaned: Kahabut: No, I believe that 2+2=4 it is not 4 but equal to it. However, .9...=1 is simply false. Not sure how hard this is really.

Apparently too hard.

0.9... = 1 is true in the standard reals as assuredly as . This is just a fact. It's provable using several different methods; saying it's false is as untrue a statement as you'll ever make.

If you're actually interested in carrying on a more intelligent conversation as opposed to me just saying you're wrong, you can start by explaining what your objection is to any of the standard proofs: whether they be

let x=0.9..
10x = 9.9...
10x-x = 9.9... - 0.9...
9x = 9
9 = 1

or

1/3 = 0.3...
3*1/3 = 3*0.3...
1 = 0.9...

or my proof via the density of the reals ("give me a number greater than 0.999... and less than 1).

I'll add:

.99999....... = summation(n=0,inf) .9(1/10)^n = .9/(1-(1/10)) = 1

and

1 - .9999..... = lim x -> inf 1/10^x = 0

i'm sorry. i'm so very sorry...

/subby.

Honestly, this is the ideal thread for testing what the April Fool's filter will do this year.

This thread could achieve orbit. Asymptotically. Boobies.

RickTheVote
This whole thing smells strangely like evolution vs. creationism - people that just disregard scientific proof, create an alternative, then stick their fingers in their ears and just "la la la" through life.

Yeah, I picked up the same scent yesterday.
I wondered if I drunk what is what bevets would be like should he ever actually do something else than dump some more or less random quotes and leave.

pelzo63: i'm sorry. i'm so very sorry...

/subby.

THIS IS ALL YOUR FAULT

_(j/k)

Truth is not the sum of that which you cannot disprove, it is the sum of that which you can.

where we create a number devised of a zero, a decimal point and x repetitions of 9 after the decimal point and call it y.

(1-y)*10^x=1

People have proposed a state where the answer to the above function becomes zero, indicating no difference. Basically show mathematical proof that this function is not true for all values of x, which is being claimed.

The thing iTake from this thread is that people care more about arguing the toss over unprovable maths than the do about bashing apple products.

FARK, what has become of you?

Kahabut: You even managed to point to actual problem here, the difference between logical math and engineering math (physics).

O.k. - I'm pretty sure I know what you mean here. But you're making a distinction that doesn't actually exist.

You're saying, I think, that there's this world of "pure math" - theorems & lemmas, manifolds, Riemann sheets, and all sorts of abstract stuff, and another world of "i need to calculate how much weight this beam will hold or the building will collapse". Both areas involve "math."

And indeed, the fact that 0.9... is equal to 1 is not likely ever something a "practical" calculation is ever going to need. In practical/experimental applications, a number is useful only up to its uncertainty, and i have never seen any numbers that need more than 10-15 digits. The field of Quantum Electrodynamics (QED) in physics is an excellent example of the place where numbers with 10+ significant digits can be experimentally measured and calculated from the theory. And they match!
Accelerator physicists regularly carry out perturbation theory to like 16th order to set up the optics for focussing and colliding particle beams. Very small numbers can indeed be very important in these cases, because they're being multiplied by very large numbers.

In undergraduate physics, students studying the radioactive decay law need to use a specific constant, called e, which is roughly 2.718. There's a boatload of digits after the the 2.718 - an infinite number of digits in fact. How many of these digits one needs to use to calculate the decay time constant depends on how many significant digits there are on the experimental measurements. That is, what are the experimental uncertainties on how accurately you've measured the number of atoms that decayed and the time interval within which they decayed.
In physics/engineering - an "infinite" precision on a number is never needed, practically, to my knowledge. Maybe someday we'll have theories and experiments that require more than QED's 10+ significant digits.

All that said, it is still true that 0.9... is equal to 1.

I admit, the first time I heard this i was [keanu] Whoa! [/keanu]. But the more I thought about it, the more I realized that it must be correct. Infinities are always hard to fathom, as are more than 4 dimensions, wave-particle duality, etc etc..

If you would have said "0.9... might be equal to 1, but what practical use is that?," I wouldn't have had a response to that - I don't know one - maybe someone else here does.
But to say that there are different kinds of math, or to claim that 0.9... is not equal to 1, is definitely not correct...

cheers

ryant123: To the math guys who say that .99... = 1, does the same sort of reasoning imply that 0.000...1 = 0 ? Cause that would be cool...

What you are really trying to say with 0.000...1 is the lim x-> inf of 1/10^x = 0 and the answer is yes. I used that as another mathematical proof to the .999.... = 1 proofs.

ryant123: To the math guys who say that .99... = 1, does the same sort of reasoning imply that 0.000...1 = 0 ? Cause that would be cool...

Assuming you mean that there are an infinite number of 0s, then a 1... there are two logical conclusions. The normal answer is that such a number doesn't exist, as you can't put stuff after a repeating decimal.... where would it go? The repeating decimal goes on forever! The second answer is that yes, it equals 0.

pelzo63: i'm sorry. i'm so very sorry...

/subby.

I started the 0.99...=1 thing so I am right with you.

Been reading & laughing at this whole thread. But since it seems to be dying down, I'll just leave this here:

http://www.youtube.com/watch?v=wsOXvQn3JuE

evaned: as you can't put stuff after a repeating decimal.... where would it go? The repeating decimal goes on forever!

I should point out that it goes on forever without going anywhere. I think that this is people's problem with a concept of infinity, that it's a number that just keeps adding more digits no matter how far you go, but it's really not. It's not adding anything, or anything like that. It's just sitting there, at any given moment in time, going on forever without moving. There's a kind of psychological horror at such an idea (which is perfectly natural) which you can overcome when you remind yourself that infinity is not a number.

The Voice of Doom: RickTheVote
This whole thing smells strangely like evolution vs. creationism - people that just disregard scientific proof, create an alternative, then stick their fingers in their ears and just "la la la" through life.

Yeah, I picked up the same scent yesterday.
I wondered if I drunk what is what bevets would be like should he ever actually do something else than dump some more or less random quotes and leave.

well i am a creationist, so your theory does hold water

Lsherm: I honestly think he's just funning with anyone

well like i've said, i've accepted most of it (aka kool aid drinker) to get shiat done, but yeah i like to play advocate for certain discussions to see where they go

at the end of the day i really don't care about that stuff enough to invest an appropriate amount of thought into it, though i do mean what i'm saying even if i can't back up some of the stuff, that i have a gut feeling about

it's all harmless thought exercises

but just so we're clear accepting the standard beliefs and practices of the day =/= 100% completely agreeing with them

but like dude said up a few posts, since we deal with 5-10%, i really don't think this 0.9999.... issue will EVER come up in real life anyway so i just skip that nonsense and focus on real life for the time being

Lsherm: He's also most likely a civil engineer

that's a bingo, like i said, a poop engineer, (we also do roads and bridges n stuff)

Lsherm: since the PE license is most important for that field in the US,

incorrect, you're thinking of Mech. E's and Chem E's since their career usually depend on it, our field it's more like icing on the cake, or in this economy a little thing i like to call, "backup insurance"

my boss doesn't have one, and he's runnin' the frickin firm

evaned: Believe what you want. It doesn't make it true.

what if i kept repeating it? would that make it true?

0.999... =/= 1
0.999... =/= 1
0.999... =/= 1
0.999... =/= 1
0.999... =/= 1
0.999... =/= 1

there now it's true :P

but they are really really close, you say tohmaytoh i say tahmahtoh

dlp211: ryant123: To the math guys who say that .99... = 1, does the same sort of reasoning imply that 0.000...1 = 0 ? Cause that would be cool...

What you are really trying to say with 0.000...1 is the lim x-> inf of 1/10^x = 0 and the answer is yes. I used that as another mathematical proof to the .999.... = 1 proofs.

for finite systems sure, i agree, but for infinite systems...

not so much

/this is the whole point

and since we currently inhabit a finite system (aka the physical universe) then for most intents and purposes (especially dealing with architects and contractors) i can agree that we are close enough, and let bygones be bygones

but if you run into a guy named abbey here, we won't be as forgiving

///that's why i like him

I drunk what: though i do mean what i'm saying even if i can't back up some of the stuff, that i have a gut feeling about

So in your magical, stupid world, your "gut feeling" is equal to and in fact surpasses mathematical proofs. Awesome. Wish I lived in a world as absurd and arbitrary.

I drunk what: but they are really really close

They are different ways of representing the same value. We have multiple mathematical proofs -- at least five discrete proofs were posted in this thread -- and so far you have your gut feeling based around a vague notion that if you subtract 0.9... from 1 you will, somewhere, get 0.0...1 -- you will not, because that involves a final nine. All I see is concrete evidence your mind is still like that of a teenager, where "infinity" is some other concept that doesn't mean infinity to the rest of us, but "really, really long".

I drunk what: and since we currently inhabit a finite system (aka the physical universe)

Ah, and there's that "I CAN'T ACTUALLY EVER COUNT THAT HIGH!" special pleading again.

I drunk what: then for most intents and purposes (especially dealing with architects and contractors) i can agree that we are close enough, and let bygones be bygones

No, that's just you realizing you're wrong, being unable to find any actual fault with the math, being unable to post a mathematical proof of your own, and trying to appear magnanimous where you have no right to magnanimity (you aren't, after all, the victor, we are) while trying desperately to bow out, save face, and end the argument.

evaned: Is "1.000..." a function?

I drunk what: evaned: Is "1.000..." a function?

i0.kym-cdn.com

Most amusing in that you felt the need to a) quote mine evaned, b) attempt to frame his rhetorical question as a literal one and, c) most importantly, in fact altered what he said.

That's right, a quick copy/paste and then ctrl-F job reveals the words 'Is "1.000..." a function?' to be nowhere in the thread save your post. Humourously, you, for no reason at all, not only quote-mined him to play contextomy games, but actually went and altered what he said, adding nine whole characters -- two quotation marks, a decimal point, three zeros, and an ellipses composed of three periods.

To what end, though? Rabid stupidity, perhaps, or grandiose intellectual dishonesty which, given that you not only quote mined him, you directly interfered with what you were quoting, reveals to us that not only are you intellectually dishonest, you are aware of it and choose to do it anyway.