## 0.999...=1 on wikipedia

Things that don't belong anywhere else. (Check first).

Moderators: Moderators General, Prelates, Magistrates

SpitValve
Not a mod.
Posts: 5130
Joined: Tue Sep 26, 2006 9:51 am UTC
Location: Lower pork village

### 0.999...=1 on wikipedia

Wikipedia's featured article for today:

http://en.wikipedia.org/wiki/0.999...

It is a hard concept to explain to people... read through the talk pages, lots of confusion there

fjafjan
THE fjafjan
Posts: 4766
Joined: Fri Oct 06, 2006 12:22 pm UTC
Location: Down south up north in the west of eastern west.
Contact:
but why not just point out htat 0.99999999... = 3/3

or, 0.3333..... = 1/3

0.33333 * 3 = 3/3

and also one could do say "where should we round this number?" and no matter where you chose you end up with 1...

It is a strange concept at first though, true
//Yepp, THE fjafjan (who's THE fjafjan?)
Liza wrote:Fjafjan, your hair is so lovely that I want to go to Sweden, collect the bit you cut off in your latest haircut and keep it in my room, and smell it. And eventually use it to complete my shrine dedicated to you.

ulnevets
Posts: 186
Joined: Wed Aug 09, 2006 1:45 am UTC
Contact:
what is the largest number that is less than 1, but not 1?

kira
I hate bananas.
Posts: 904
Joined: Fri Apr 14, 2006 4:21 am UTC
Location: school
Contact:
ulnevets wrote:what is the largest number that is less than 1, but not 1?

That's pretty much the equivalent of saying "what's the largest number that is not infinity?"

Gelsamel
Lame and emo
Posts: 8237
Joined: Thu Oct 05, 2006 10:49 am UTC
Location: Melbourne, Victoria, Australia
fjafjan wrote:but why not just point out htat 0.99999999... = 3/3

or, 0.3333..... = 1/3

0.33333 * 3 = 3/3

and also one could do say "where should we round this number?" and no matter where you chose you end up with 1...

It is a strange concept at first though, true

Because someone with an objection to .999... = 1 may easily have an objection to .333... = 1/3.

fjafjan
THE fjafjan
Posts: 4766
Joined: Fri Oct 06, 2006 12:22 pm UTC
Location: Down south up north in the west of eastern west.
Contact:
Gelsamel wrote:
fjafjan wrote:but why not just point out htat 0.99999999... = 3/3

or, 0.3333..... = 1/3

0.33333 * 3 = 3/3

and also one could do say "where should we round this number?" and no matter where you chose you end up with 1...

It is a strange concept at first though, true

Because someone with an objection to .999... = 1 may easily have an objection to .333... = 1/3.

But that is just the way one writes it, ultimatly, if ... = infinite 3s, then 0.33... is 1/3 .... Or are these people saying you cannot write 1/3 with decimals?
//Yepp, THE fjafjan (who's THE fjafjan?)
Liza wrote:Fjafjan, your hair is so lovely that I want to go to Sweden, collect the bit you cut off in your latest haircut and keep it in my room, and smell it. And eventually use it to complete my shrine dedicated to you.

mezz
Posts: 22
Joined: Fri Oct 06, 2006 4:55 am UTC
Location: CA, USA
Contact:
The largest number that is not 1 ?

<1 ?

lim 1 - [1/(n)] ??
n->infinity

lim n
n->1-

How do you say it mathematically?

Gelsamel
Lame and emo
Posts: 8237
Joined: Thu Oct 05, 2006 10:49 am UTC
Location: Melbourne, Victoria, Australia
fjafjan wrote:Or are these people saying you cannot write 1/3 with decimals?

Yes.

svk1325
Posts: 139
Joined: Mon Jul 17, 2006 9:10 am UTC
Location: Take a guess...
Contact:
Gelsamel wrote:
fjafjan wrote:Or are these people saying you cannot write 1/3 with decimals?

Yes.

Not in base 10 anyways

You may be able to explain that 0.999... = 1. You may even be able to mathematically prove it. Some people still won't believe you and just say "but they're written differently" or something to that effect.

Wasn't there some thread like this under "Logic Puzzles", or am I just imagining things?

Peshmerga
Posts: 2061
Joined: Wed Oct 04, 2006 1:56 am UTC
Contact:
I'm fairly convinced that that "fact" .999... = 1 is a Jewish conspiracy started to cripple world governments and online forums.
i hurd u liek mudkips???

Gelsamel
Lame and emo
Posts: 8237
Joined: Thu Oct 05, 2006 10:49 am UTC
Location: Melbourne, Victoria, Australia
I prefer...

Let x = .999....
x=.999....
10x = 9.999....
10x - x = 9.999.... - .999....
9x = 9
x = 1

Therefore 1 = .999...

wisnij
Posts: 426
Joined: Mon Jun 26, 2006 5:03 pm UTC
Location: a planet called Erp
Contact:
ulnevets wrote:what is the largest number that is less than 1, but not 1?

No such real number exists. The real numbers are dense, meaning that for any number n < 1, there will always be a number m such that n < m < 1.
I burn the cheese. It does not burn me.

SpitValve
Not a mod.
Posts: 5130
Joined: Tue Sep 26, 2006 9:51 am UTC
Location: Lower pork village
Wasn't there some thread like this under "Logic Puzzles", or am I just imagining things?

true...

fjafjan
THE fjafjan
Posts: 4766
Joined: Fri Oct 06, 2006 12:22 pm UTC
Location: Down south up north in the west of eastern west.
Contact:
svk1325 wrote:
Gelsamel wrote:
fjafjan wrote:Or are these people saying you cannot write 1/3 with decimals?

Yes.

Not in base 10 anyways

decimal

so it was sort of implied
//Yepp, THE fjafjan (who's THE fjafjan?)
Liza wrote:Fjafjan, your hair is so lovely that I want to go to Sweden, collect the bit you cut off in your latest haircut and keep it in my room, and smell it. And eventually use it to complete my shrine dedicated to you.

Narsil
Posts: 2995
Joined: Thu Oct 26, 2006 6:59 pm UTC
Location: Columbus.
This can be mathematically proven false by the simple question of "what happens if I show up at a dollar store with 99 and .99999999.... pennies?" Obviously I'll be .11111111..... penny short, so I won't be able to buy anything. Of course, a "Take a penny, leave a penny" jar could in theory solve that.

sandy
Posts: 12
Joined: Wed Oct 11, 2006 10:15 pm UTC
Narsil wrote:This can be mathematically proven false by the simple question of "what happens if I show up at a dollar store with 99 and .99999999.... pennies?" Obviously I'll be .11111111..... penny short, so I won't be able to buy anything. Of course, a "Take a penny, leave a penny" jar could in theory solve that.

In what universe does .999... + .111... = 1.0? Remember to carry the 1 when you add. I think what you mean to say is .999... + .000...1 = 1.0, but you can't do that -- you can't just say I have infinite zeros and then a 1 at the end. You should check out the logic puzzles thread, the various proofs of .999... = 1 are fairly straight-forward.

Posts: 105
Joined: Fri Sep 29, 2006 7:14 pm UTC
A few ways of putting it that I like:

0.999... = 0.9 + 0.09 + 0.009 + 0.0009 ...
= 9*0.1 + 9*0.01 + 9*0.001 + 9*0.0001 ...
= 9*10^-1 + 9*10^-2 + 9*10^-3 + 9*10^-4 ...
n=N
= lim ∑ 9*10^-n
n=1
= lim (9*10^-1 + 9*10^-2 + ... + 9*10^-N)
N->infinity
= lim (9*0.1 + 9*0.01 + ... + 9*0.00...01)
N->infinity
= lim (0.9 + 0.09 + ... + 0.00...09)
N->infinity
= lim (.999....99)
N->infinity
= lim (1 - 0.000...001)
N->infinity
= lim (1 - 10^-N)
N->infinity
= lim (1) - lim (10^-N)
N->infinity N->infinity
= 1 - 0
= 1

Another way:

For any two unique real numbers, it is possible to pick another real number between them.
In the case of 0.999... and 1, it is impossible to pick another number in between them. Therefore they must be the same number.

thomasjmaccoll
Posts: 541
Joined: Thu Oct 19, 2006 11:27 pm UTC
Location: cupar, fife, scotland
Contact:
this thread has got my head working about maths again for the first time in a year : )
slow down, you move too fast

Pathway
Leon Sumbitches...?
Posts: 647
Joined: Sun Oct 15, 2006 5:59 pm UTC
We take .999... and 1 to be real numbers.

But what are real numbers, exactly?

Well, most of the axioms defining them are pretty familiar, such as the additive identity axiom (for real x and n, there exists one and only one number n such that x + n = x) or the multiplicative identity axiom (for real x and n, there exists one and only one number n such that n*x = x).

But the weird one is the Least Upper Bound axiom. "Every nonempty subset S of the real numbers, if S is bounded above, has a least upper bound."

In other words, there exists a number M for every set of real numbers such that if there is no number in S greater than some number N, then M is less than or equal to N.

So: if we take the set of numbers from some arbitrary number less than 1 to 1, not including the right endpoint, S = (n<1, 1) , then S has a least upper bound M = 1.

But we also know that there is no number x in S = (n<1, 1) such that x > .999......, because any real number added to .999... would be at least equal to 1. That means, by the definition of least upper bound, that .999... is also the least upper bound M of S, so since M = M, .999... = 1.

Narsil
Posts: 2995
Joined: Thu Oct 26, 2006 6:59 pm UTC
Location: Columbus.
sandy wrote:
Narsil wrote:This can be mathematically proven false by the simple question of "what happens if I show up at a dollar store with 99 and .99999999.... pennies?" Obviously I'll be .11111111..... penny short, so I won't be able to buy anything. Of course, a "Take a penny, leave a penny" jar could in theory solve that.

In what universe does .999... + .111... = 1.0? Remember to carry the 1 when you add. I think what you mean to say is .999... + .000...1 = 1.0, but you can't do that -- you can't just say I have infinite zeros and then a 1 at the end. You should check out the logic puzzles thread, the various proofs of .999... = 1 are fairly straight-forward.

I feel stupid now. Anyway, if you change ".11111..." to ".000.....01", then doesn't that raise a good point?

SpitValve
Not a mod.
Posts: 5130
Joined: Tue Sep 26, 2006 9:51 am UTC
Location: Lower pork village
Narsil wrote:
sandy wrote:
Narsil wrote:This can be mathematically proven false by the simple question of "what happens if I show up at a dollar store with 99 and .99999999.... pennies?" Obviously I'll be .11111111..... penny short, so I won't be able to buy anything. Of course, a "Take a penny, leave a penny" jar could in theory solve that.

In what universe does .999... + .111... = 1.0? Remember to carry the 1 when you add. I think what you mean to say is .999... + .000...1 = 1.0, but you can't do that -- you can't just say I have infinite zeros and then a 1 at the end. You should check out the logic puzzles thread, the various proofs of .999... = 1 are fairly straight-forward.

I feel stupid now. Anyway, if you change ".11111..." to ".000.....01", then doesn't that raise a good point?

0.1 = 10^-1
0.01 = 10^-2
0.00...001 = limit as (n->infinity) of 10^-n

agreed?

Let's say that 0.00...001 is not zero. Then we can multiply it by a _finite_ number to get 1.

a * 0.00...001 = 1

so

a = 1/(0.00...001)

a = 1/(limit as (n->infinity) of 10^-n)
a = limit as (n->infinity) of 1/(10^-n)
a = limit as (n->infinity) of 10^n
a = infinity

which is not finite... so the initial assumption is wrong: 0.00...001 must be equal to zero.

In terms of pence - if you have 99.99... pence, you have 100p. If you have less than 100p, then you have less than 99.99.... pence.

Practially speaking, how on earth are you going to remove an infitesmal part of a penny anyway?

mezz
Posts: 22
Joined: Fri Oct 06, 2006 4:55 am UTC
Location: CA, USA
Contact:
1 - .898989... = .101010...

Very interesting, but I can't think of any time I will need to use it.

moopanda
Posts: 339
Joined: Tue Aug 22, 2006 12:35 am UTC
Location: Brisbane, Australia
Contact:
mezz wrote:1 - .898989... = .101010...

Very interesting, but I can't think of any time I will need to use it.

Whenever you need to calculate 99/99 - 89/99 = 10/99, obviously!

Posts: 4
Joined: Fri Sep 01, 2006 10:17 pm UTC

phlip
Restorer of Worlds
Posts: 7572
Joined: Sat Sep 23, 2006 3:56 am UTC
Location: Australia
Contact:
There is a rule with the real numbers:

If:
(1) x ≥ 0
(2) x < ε for all ε > 0
Then:
x = 0

It's closely related to limits, and all that fun stuff. It basically says there's no infinitesimals in the real numbers. If you can say that a non-negative number is less than every positive number, then it must be zero.

You can try to debate this through philosophy, but you'll be debating the definition of a "real number"... it can be proven that the real numbers have no infinitesimals.

Now, 1 - 0.999... < ε. There is no real number that is less than 1 - 0.999... but greater than 0. Therefore, 1 - 0.999... = 0; 0.999... = 1.

The thing to remember is that the decimal representations of numbers are just that. Representations. The number 16 isn't defined somewhere as "the number represented in decimal by a 1 followed by a 6"... A 1 followed by a 6 is just a simple way of representing the number. 0.999... and 1 are simply two ways of representing the same number in decimal, the same way that 1/1 and 2/2 are two ways of representing the same number as fractions.

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

Gemini25RB
Posts: 39
Joined: Fri Oct 27, 2006 9:38 pm UTC
I'll just take the simple(?) route and say that since the Real number system is complete, so between any two distinct numbers there exists a third. Now consider 0.999999... (note, the ellipsis designates that the series of nines repeats forever). Now consider 1. Can you name a number between the two? No, because there are an infinite number of nines, so we cannot tack on another number.

Pathway
Leon Sumbitches...?
Posts: 647
Joined: Sun Oct 15, 2006 5:59 pm UTC
Wow. That qntm site is absolute gold. His fiction reminds me of Asimov. No joke. He's good.

henre
Posts: 470
Joined: Sun Oct 29, 2006 10:04 pm UTC
Location: A healthy soul clings to a healthy spirit and a healthy body.
Hahahahahahahaha

qntm wrote:"I didn't understand that proof."

Try an earlier one.

hahahahahahaha

Air Gear
Posts: 227
Joined: Sat Oct 07, 2006 6:36 pm UTC
Since a math-related thread has recently been posted in and I don't feel like making this a new thread, I'll post something totally off-topic here...

...the worst math joke ever.

Yup, somebody today, in my analysis class, made the worst math joke ever and now you will hear it.

Basically, my prof brought pumpkin pie for the class...one remark she made, "I thought, 'There are twenty people in the class? Should I have gotten MORE than two?'"

And the response, almost instantly, by a guy in the back...

"We know by radian measure that two pie is enough to go around."

Though surely people have said this about that guy before, he seriously needs to be stopped.

fjafjan
THE fjafjan
Posts: 4766
Joined: Fri Oct 06, 2006 12:22 pm UTC
Location: Down south up north in the west of eastern west.
Contact:
that was brilliant, I am fjafjan, and i support this wittyness ^_^
//Yepp, THE fjafjan (who's THE fjafjan?)
Liza wrote:Fjafjan, your hair is so lovely that I want to go to Sweden, collect the bit you cut off in your latest haircut and keep it in my room, and smell it. And eventually use it to complete my shrine dedicated to you.

Hawknc
Oompa Loompa of SCIENCE!
Posts: 6986
Joined: Mon Oct 02, 2006 5:14 am UTC
Location: Melbourne, Australia
Contact:
I lol'd.

Gelsamel
Lame and emo
Posts: 8237
Joined: Thu Oct 05, 2006 10:49 am UTC
Location: Melbourne, Victoria, Australia
Air Gear wrote:Joke

Ahahah that was freakin' awesome. I can't believe he said that in response so quickly, I've had a few moments like that, but I actually took a few seconds to make up the joke.

moopanda
Posts: 339
Joined: Tue Aug 22, 2006 12:35 am UTC
Location: Brisbane, Australia
Contact:
Air Gear wrote:...the worst math joke ever.

I'm not so sure, check out the Nerdy jokes thread on this forum. And the paper I linked from it. It has many many contenders for that coveted title.

Air Gear
Posts: 227
Joined: Sat Oct 07, 2006 6:36 pm UTC
moopanda wrote:
Air Gear wrote:...the worst math joke ever.

I'm not so sure, check out the Nerdy jokes thread on this forum. And the paper I linked from it. It has many many contenders for that coveted title.

I dunno. The ones on that paper weren't quite as bad; I REALLY like the puns along the lines of "Zorn's lemon", "the real lime", and "simple pole in the complex plane"...though I will admit that the one with the constant, e^x, and the derivative operator walking around is a lot worse, so ok, it's not the worst, but it's really bad.