Simple mathematical problem (?)

[Suggestologist]

For every conceivable intent and purpose, 0.999(recurring) = 1.

Do you know what every conceivable intent and purpose could be? I didn't think so.

Here's my argument from the surreal number perspective.

1. Does there exist a "real" number that is BOTH Less Than and closer to 1 than ANY other real number (which is less than 1)? The surreal number system says that such a number must exist.

2. If you were to represent this number in decimal form, what would it look like?

 

[Zep]

Everyone read Earthborn's posts again she(?) actually knows what she is talking about!

The original proof (in the first post) is perfectly convincing to me (ignoring the typo in it)

If you think 0.9999.... is not the same as 1 then I am assuming you believe it to be less than 1. In which case if it is less than 1 then you must be able to find another number between 0.9999... and 1. If anyone can provide such a number I would be interested to see it. If not we will have to accept they are one and the same.

Some-unknown-number = (1 + 0.9~) / 2

 

[Zep]

re: Zep's string cutting example. You cannot cut a string into 90%/10% pieces an infinite number of times. This is impossible. You cannot do it, you cannot construct a machine to do it or even simulate doing it. You must accept that infinity is not a number, but a concept. You must take the limit of the series for the use of infinity to have any meaning whatsoever in real numbers.

It was a conceptual example only! Now, your views on the Cantor Set??

On the surreal "iota" being an infinitely small number. This is dumb. An iota does not exist, except perhaps in some bizarre number system. If you think it does, I give you:

iota/2

Excellent! My point exactly. For any arbitrarily small variation away from a point on a number line, there is always another point between there and the original point.

Oh, and kiss my arse.

 

[Ziggurat]

Some-unknown-number = (1 + 0.9~) / 2

But (1+ 0.9~)/2 = 1.9~ /2 = 0.9~. Try it. That result only makes sense, of course, if 0.9~ = 1. Which it does.

 

[LFTKBS]

I can't believe this thread is five pages. Here's a solution for kids:

1/9 = 0.11111...
2/9 = 0.22222...
.
.
.
8/9 = 0.88888...
9/9 = 0.99999...= 1

 

[Cabbage]

"If sn [partial sum] converges to a limit S as n -> infinity: lim(n->inf) sn = S, then we say that the series converges and that its sum is S".

But remember, like I said earlier, an arbitrary real number is defined to be equal to the limit of it's "partial decimal expansions". In other words, by definition, .999... is equal to the limit of the sequence .9, .99, .999, .9999,....

It's irrelevant whether you want to pick nits over whether a series is "equal to" versus merely "converges to" its limit--When we're talking about a decimal expansion, it's defined to be equal to that limit.

Originall posted by Suggestologist
1. Does there exist a "real" number that is BOTH Less Than and closer to 1 than ANY other real number (which is less than 1)? The surreal number system says that such a number must exist.

2. If you were to represent this number in decimal form, what would it look like?

1. No. Additionallly, the surreal number system most certainly does not say that such a "real" number must exist--between any two distinct real numbers lie infinitely many more real numbers, and hence no real number can be both less than one and closer to 1 than any other real number.

2. I already mentioned that no such real number exists, but for the sake of argument, I'll assume you're asking here, "If you were to represent a surreal number in decimal form, what would it look like?"

And I would respond that the question is meaningless.

If you were to represent a negative number as a positive number, what would it look like?

If you were to represent an irrational number as a ratio of two integers, what would it look like?

If you were to represent an infinite ordinal as a finite ordinal, what would it look like?

If you were to represent a non-integral rational number as an integer, what would it look like?

All equally meaningless questions. You don't represent an apple as an orange.

 

[Suggestologist]

But remember, like I said earlier, an arbitrary real number is defined to be equal to the limit of it's "partial decimal expansions". In other words, by definition, .999... is equal to the limit of the sequence .9, .99, .999, .9999,....

It's irrelevant whether you want to pick nits over whether a series is "equal to" versus merely "converges to" its limit--When we're talking about a decimal expansion, it's defined to be equal to that limit.

1. No. Additionallly, the surreal number system most certainly does not say that such a "real" number must exist--between any two distinct real numbers lie infinitely many more real numbers, and hence no real number can be both less than one and closer to 1 than any other real number.

Um, no.

2. I already mentioned that no such real number exists, but for the sake of argument, I'll assume you're asking here, "If you were to represent a surreal number in decimal form, what would it look like?"

And I would respond that the question is meaningless.

If you were to represent a negative number as a positive number, what would it look like?

If you were to represent an irrational number as a ratio of two integers, what would it look like?

If you were to represent an infinite ordinal as a finite ordinal, what would it look like?

If you were to represent a non-integral rational number as an integer, what would it look like?

All equally meaningless questions. You don't represent an apple as an orange.

You don't represent a rational number as a decimal number?

 

[Cabbage]

Um, no.

What, exactly, do you disagree with?

You don't represent a rational number as a decimal number?

Where, exactly, did I say that?

 

[patoco12]

This discussion has gone on long enough.

Suggestologist, or anybody, can end this right now by doing something I've been requesting all along:


Please give me the fractional representation of .9~. It is a rational number (I established this pages ago), so it must have a fractional representation.


If you can do this, I will accept it as proof that .9~ != 1. (As long as your fractional representation doesn't equal 1.)

There have been plenty of proofs offered to show that .9~=1, but I have yet to see one that shows otherwise. Why is that?

 

[xouper]

Zep: I haven't confused them, but I will simply quote more of my textbook that says,

"If sn [partial sum] converges to a limit S as n -> infinity: lim(n->inf) sn = S, then we say that the series converges and that its sum is S".

Agreed. That's exactly what I said in my previous reply to you, but in different words. That quote from your book says the sum is S. In this context, the word "is" is synonymous with "equals". Referring back to the geometric series theorem you quoted from your book, if the infinite series is convergent, then it equals a/(1-r). Are we in agreement on this point yet?

 

[xouper]

patoco12: Please give me the fractional representation of .9~. It is a rational number (I established this pages ago), so it must have a fractional representation.

If you can do this, I will accept it as proof that .9~ != 1. (As long as your fractional representation doesn't equal 1.)

There have been plenty of proofs offered to show that .9~=1, but I have yet to see one that shows otherwise. Why is that?

OK, I'll jump on this bandwagon and repeat your challenge.

0.999.... is a rational number.

Zep, Lux, Suggest, or anyone else who thinks 0.999... does not equal 1, please show us two positive integers a and b such that

a / b = 0.999...

and also that a does not equal b.

If you cannot do that, then please have the honor to admit it cannot be done, and concede that 0.999.... exactly equals 1.

 

[xouper]

LFTKBS: I can't believe this thread is five pages.

I can't believe it either.

Here's a solution for kids:

1/9 = 0.11111...
2/9 = 0.22222...
.
.
.
8/9 = 0.88888...
9/9 = 0.99999...= 1

You would think that would be a sufficient hint, at least. BTW, that particular demonstration was posted last July, but apparently certain people still don't find it convincing. What's that saying about a horse and water?

 

[slimshady2357]

Also, could you please tell me what 1 - (0.999...) is? Can you please just write out that number in decimal notation for me?

Adam

 

[roger]

0

 

[T'ai Chi]

Also, could you please tell me what 1 - (0.999...) is? Can you please just write out that number in decimal notation for me?

Adam

Sure. The number is any e, where e is tiny and e>0.

 

[aerosolben]

Sure. The number is any e, where e is tiny and e>0.

Counterproof by contradiction:

e = 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

tiny, >0, and != 1 - 0.9~

Unles perhaps you have some alternate definition for tiny, like x is tiny => x = 0

 

[aerosolben]

Excellent! My point exactly. For any arbitrarily small variation away from a point on a number line, there is always another point between there and the original point.

Don't you see the flaw in your reasoning? This would mean that there's a number larger than infinity, which is impossible by definition. Therefore, you cannot treat infinity the way you have been treating it. You must treat it as a limit to think of it infinitely rather than finitely.

Example: If we define a > 0 such that

a = 1 - 0.9~ (infinity nines)

we can also define b such that:

b = a/10 = 1 - 0.9~ (infinity+1 nines)

But infinity+1 does not exist! This is completely impossible! Therefore, a cannot be greater than 0; the only possible value for a is 0, such that a = b.

 

[slimshady2357]

0

Exactly!

Adam

 

[slimshady2357]

Sure. The number is any e, where e is tiny and e>0.

That's decimal notation?

And can you now define tiny for me and give me an example?

Adam

 

[slimshady2357]

Unles perhaps you have some alternate definition for tiny, like x is tiny => x = 0

But then he would have another problem, since he says:

"where e is tiny and e>0"

So he would then have e = 0 and e > 0!

Adam