Simple mathematical problem (?)

[Skeptoid]

After doing some research on surreal numbers, I've come to the conclusion that attempting to convince Suggestologist that surreal numbers do not apply to this discussion is like trying to convince fractal/chaos cultists that the mind of god cannot be found in the Mandelbrot set. It is a hopeless cause.

FWIW, here's a .pdf document I found on the subject: http://www.tondering.dk/claus/sur14.pdf

It appears to be one of the more technical treatises available on the web with references to Conway, Knuth, and Gonshor.

 

[slimshady2357]

400th post. Wow, this is craziness. (edit: make that 401, too slow making the link to a specific post)

Suggestoligist,

This post by Cabbage is exactly what I would have wanted to say.

It was wondefully said, and I feel the same frustration I read in that post.

As the saying goes 'you know just enough to be dangerous'.

Adam

 

[DickK]

I'd just like to thank everyone who has contributed to this thread. I'm not a mathematician but have found a large part of the debate here comprehensible. Many thanks to xouper for the link to the geometric series proof and to others for proofs of 0.9~=1, some of them remarkably simple.

I'd also like to thank people for the explanation of sequence, series and partial sequences.

I go with xouper and co. on this one, but I have a nagging feeling that I'm missing an awful lot of context by having no working acquaintance with surreals, imaginaries and so forth.

Finally, the idea, stampeded over many times in this thread, of manipulating infinity in such a variety of ways, is also very compelling. Very many years ago, a friend and I, having got summarily stoned one Friday night, were discussing infinity (as you do). He said, "Imagine the set of infinite positive integers.", so I did (very briefly); he then said, "imagine the set of infinite negative integers...", I did this too (not so briefly, but with a distinct *sigh* in my demeanour). His last question, which caused a vodka shandy I was drinking to spurt out through my nose, was, "Are you going to have that last bhaji?", which, in the circumstances was probably wise.

 

[xouper]

I've already forgotten whether anyone has posted this link earlier in this thread, but here it is again, the official FAQ of the sci.math newsgroup. This FAQ was created and maintained by mathematicians.

http://db.uwaterloo.ca/~alopez-o/math-faq/math-faq.html

On that intro page are links to several formats of the FAQ, including a new pdf.

Below is the link directly to the discussion of 0.999... = 1.

http://db.uwaterloo.ca/~alopez-o/math-faq/node41.html

I thought for sure I had posted this link elswhere on this forum, but I can't find it, and the forum search tool doesn't allow searches on the word "FAQ".

That FAQ also contains a discussion of the construction of the real numbers, for those who are interested.

For those who insist that this math FAQ is wrong about 0.999... = 1, please go to the sci.math newsgroup and give them a piece of your mind. They will thank you for it.

 

[BillyJoe]

Therefore, 0.999... is NOTATION for the NUMBER that is equal to the limit of the SERIES:

0.9 + 0.99 + 0.999 + 0.9999 ....

Which.....is equal to 1.

BillyJoe
(Looks like no one else reads your posts slim, otherwise SURELY they would have picked up this glaring error )

 

[slimshady2357]

BillyJoe
(Looks like no one else reads your posts slim, otherwise SURELY they would have picked up this glaring error )

I knew it! Just proof of what I knew all along, nobody (but possibly BillyJoe) reads my posts

And I don't proof-read them enough

Um, I suppose I might have meant 0.9 + 0.09 + 0.009 + ...

Adam

 

[xouper]

slimshady2357: Therefore, 0.999... is NOTATION for the NUMBER that is equal to the limit of the SERIES:

0.9 + 0.99 + 0.999 + 0.9999 ....

Which.....is equal to 1.

BillyJoe: (Looks like no one else reads your posts slim, otherwise SURELY they would have picked up this glaring error )

slimshady2357: Um, I suppose I might have meant 0.9 + 0.09 + 0.009 + ...

I am guilty of not having proofread your post, either, slim. However, your oversight included more than just changing some nines to zeros. I would rephrase your comment as follows:

Therefore, 0.999... is NOTATION for the NUMBER that is equal to the SERIES:

0.9 + 0.09 + 0.009 + 0.0009 ....

Which.....is equal to 1.

I took out the words "the limit of". If you want to use "limit of" in your comment, then you might say it this way:

Therefore, 0.999... is NOTATION for the NUMBER that is equal to the limit of the SEQUENCE:

0.9, 0.99, 0.999, 0.9999, ....

Which.....is equal to 1.

Does this help?

 

[slimshady2357]

I am guilty of not having proofread your post, either, slim. However, your oversight included more than just changing some nines to zeros. I would rephrase your comment as follows:

I took out the words "the limit of". If you want to use "limit of" in your comment, then you might say it this way:

Does this help?

Yes, it helps remind me that no one reads my posts (including me, too much effort on the fancy colours, not enough effort on detail)

Well, not the first time around anyway

Adam

 

[xouper]

slimshady2357: Yes, it helps remind me that no one reads my posts

Well, maybe it could be some small comfort to know that I had read enough of that particular post to see that you were essentially saying something I agreed with and then I just skimmed the rest, not feeling the need to scrutinize it for errors.

 

[BillyJoe]

slim,

....and then I just skimmed the rest, not feeling the need to scrutinize it for errors.

Well, at least now everyone will read your post.
Even if for the wrong reason.

BillyJoe

 

[TillEulenspiegel]

I would add the clarification that mathematicians and other sufficiently math literate people seem to understand those notions rather well (or well enough) and that the confusion is on the part of those who do not sufficiently understand "the notions of sequences, limits, infinities and other basic math premises".

Edited to add - maybe that wasn't much of a clarification. The short version is, some do and some don't. The people who do understand those notions are not confused.

Well I'm not sure that I like Alberts visage without the mask as much as with.......Hmmm kinda like the pope on rollerskates...oop sorry thought this was a PM =)

 

[xouper]

TillEulenspiegel: Well I'm not sure that I like Alberts visage without the mask as much as with...

The Halloween mask kinda grew on me too. I was reluctant to part with it, but I also didn't want to be like those people who leave their Christmas decorations up until Valentine's Day.

 

[TillEulenspiegel]

You you..Take your christmas decorations down...?

 

[LW]

But sometimes I'd like to be a math teacher there. With a big red pencil.

Last winter I bought myself a big red stamp with the text "Tämä arpa ei voita" (translates to "This is not the winning raffle ticket") for exam grading work.

 

[AmateurScientist]

I just saw this thread for the first time, having been referred to it by the xouper appreciation thread.

I'm amazed that this age-old mathematical proof never fails to inspire much debate, even among some mathematicians.

As so many others with math degrees in this thread, I can state quite emphatically that .9999 (repeating infinitely) is exactly, not approximately, equal to 1. .9999 (repeating) does not merely approach one. It is one.

The proof does not contain any tricks, as has so often been claimed to me by anyone I show it to.

The proof more often used uses nothing more than simple algebra learned by most 12-14 year olds.

It's astounding how many well-educated adults have such difficulty accepting it. I think it has to do with the concept of infinity and its often counterintuitive implications.

AS

 

[xouper]

AmateurScientist: It's astounding how many well-educated adults have such difficulty accepting it. I think it has to do with the concept of infinity and its often counterintuitive implications.

I blame Zeno.

 

[whitefork]

Zeno-phobia!

 

[69dodge]

Let x = . . .9999.
10x = . . .9990.
x - 10x = 9.
-9x = 9.
x = -1.

So, . . .9999 = -1.

 

[xouper]

69dodge:
Let x = . . .9999.
10x = . . .9990.
x - 10x = 9.
-9x = 9.
x = -1.

So, . . .9999 = -1.

Interestingly, that very thing was published in The College Mathematics Journal, Volume 26, Number 1, January 1995, page 11-15, "The Repeating Integer Paradox," by Paul Fjelstad.

Although that notation does not represent traditional integers, there is something consistent about ...999 = -1. This can be shown using the notion of congruences.

Also, consider two's complement arithmetic as used by computers. In a 16 bit word, FFFF = -1. If the wordsize is infinite then ...FFFF is still equal to -1.

Two's complement can also be written in decimal. For example, in an 4 digit representation, 9999 = -1. It doesn't take much imagination to see that the limit condition is ...999 = -1.

The numbers represented by the notation ...PPP are called p-adics. I don't really know much about them, except that p-adics are not real numbers.

http://mathworld.wolfram.com/p-adicNumber.html

 

[clk]

Let x = . . .9999.
10x = . . .9990.
x - 10x = 9.
-9x = 9.
x = -1.

So, . . .9999 = -1.

S = 1 + 2 + 4 + 8 + 16 + 32 + 64 +...
so
2S = 2 + 4 + 8 + 16 + 32 + 64 +...
so
2S = S-1
solve for S to get:
S = -1
so...
1 + 2 + 4 + 8 + 16 + 32 + 64 +... = -1 !
How do you like them apples?

Obviously, this problem can be easily solved correctly by making S equal to infinity, which is the proper sum. Then, subtracting 1 from infinity still gives you infinity, so in the end, you get infinity equals infinity, which is a true statement.