Is 1 = 0.9999......

[bleeb]

Its ALIVE! ALIVE!

 

[jman19]

Ross,

The Rudin text you mention is currently causing me much pain... some of the exercises are the hardest problems I've ever had to solve. Don't know if it teaches the material in the best way possible. but definitely opens the reader up to many nuances of the world of mathematics.

 

[bleeb]

Mathematics can get a lot more complex...

 

[RossGr]

Originally posted by: bleeb
Mathematics can get a lot more complex...

Yes it can, strangly enough, there is an intire field known as Complex Analysis, it is that!

 

[silverpig]

I've heard horror stories from my about-to-graduate math major friend about Partial Differential Equations and Analysis II. Oddly enough, he said that Ordinary Differential Equations and Analysis I were fairly easy (I can agree with him on ODEs). Complex Analysis is pretty bad, but not quite in the same league as PDE or A II.

 

[jman19]

Originally posted by: bleeb
Mathematics can get a lot more complex...

Oh, I know it can get worse... but the current analysis course I'm taking is my first foray into analysis, which is probably why I find it to be difficult. A different mindset is required than is typically needed for lower level math courses.

 

[RossGr]

Originally posted by: jman19

Originally posted by: bleeb
Mathematics can get a lot more complex...

Oh, I know it can get worse... but the current analysis course I'm taking is my first foray into analysis, which is probably why I find it to be difficult. A different mindset is required than is typically needed for lower level math courses.

That is exactly right, the jump from Calculus to Real Analysis is as major a mind set change as the jump from Algebra to Calculus.

I think a telling sentence is the last one before the section on the Extented Real numbers

Since we will shall never use decimals. We do not enter into a detail discussion.

So a course in Real Analysis which states on page 10 that it will not be using decimal numbers. The fact is you never use ANY numeric representation. It is all symbolic and in general.

 

[PowerEngineer]

This thread still lives?!?

If each reply were another "9" after the decimal point, then we'd be very nearly to "1" by now.

 

[bleeb]

You're right. But we would never reach one because the nines would continue to infinity because 0.9999... != 1

 

[PowerEngineer]

Originally posted by: bleeb
You're right. But we would never reach one because the nines would continue to infinity because 0.9999... != 1

Oh yes it does!!!! (Chaulk up another "9")

 

[TuxDave]

dieeee..... omg... how can this thread still be here. Esp since the 0.999...=1 group has already won. Hehehe... but madrat's math does seem rather interesting. Kinda bends traditional thinking. I heard one idea that all parallel lines will intersect each other at infinity. And all lines will end up at infinity. This is probably where Madrat's getting his ideas from because if a point CAN exist at infinity, the 0.000...001 can exist... otherwise.. no.

 

[TuxDave]

Originally posted by: silverpig
Let's take 3 playing cards. Since I can't show them on here, we'll let 1 = a face up card, and 0 = a face down card. For 3 cards we have 8 possibilities:

000
100
010
001
110
101
011
111

There are 2^n ways to arrange the cards (where n is the number of cards).

Now, let's try that with an infinte deck of cards:

1000101001...
0110100010...
1000101010...
1001101011...
0001010011...
etc...

(note: I'm doing it in a random order because they would all look the same if I was to order them).

The number of columns indicate the number of cards, and the number of rows indicate the number of ways to arrange the cards. Obviously, there are an infinite number of each. If we have an infinite number of rows, then we must be able to represent EVERY possible combination of face-up and face-down cards right? WRONG.

Let's consider the following arrangement:


0000101001...
0010100010...
1010101010...
1000101011...
0001110011...

Notice that this set is exactly the same as the previous one, except for the bolded numbers. All I have done is gone down the diagonal and flipped every card. This diagonal must represent a way of ordering the cards right? After all, it consists of an infinite number of cards either face-up or face-down.

Now comes the cool part. This new diagonal is NOT a member of our original set. Why? Well, it can't be a copy of the first row, because the first card differs. It can't be a copy of the second row because the second card differs. Further, it can't be a copy of the nth row, because the nth card differs. We have just found a way of arranging the cards in such a way that was not included in our infinite set. We have produced a subset that cannot be in our previous list, despite the previous list being infinite.

This shows that such a set cannot be matched one to one with the integers, and is therefore of a different cardinality.


edit: fixed boldness

That's...... wow... mind boggling..... holy cow.

 

[Krakerjak]

hmmm, this thread is farrrrrrrrrrrrrrr too long for me to look for myself.......but is bleeb really that ignorant???

or are you just trying to drag this beast on to the ends of time.

 

[TuxDave]

Originally posted by: Krakerjak
hmmm, this thread is farrrrrrrrrrrrrrr too long for me to look for myself.......but is bleeb really that ignorant???

or are you just trying to drag this beast on to the ends of time.

I think he's just trying to mess with our heads.... I'll consider it a great victory if bleeb eventually puts 0.999... = 1 in his sig. Hehehehe

 

[JupiterJones]

Consider Skolem's Paradox. Take the theory of sets. In this theory you can distinguish between countably infinite sets and uncountably infinite sets (a countably infinite set is one which can be put into a one-to-one correspondence with the numbers {1,2,3,...}), and furthermore, you can prove that there are uncountably infinite sets (Cantor's construction can be carried out). This theory, if it is consistent, has a model. This model must be infinite (it contains elements for each of the numbers 1, 2, 3, at least!) so by the downward Lowenheim-Skolem theorem, it also has a countable model. But what of the sets that the theory takes to be uncountably infinite. From inside the model, they are uncountable. But from outside, we see that they are not. What is the story?

Well, that's a real problem. Skolem's enticing response was to say that the difference between the countable and the uncountable was relative and not absolute. Another response would be to say that the predicate logic theory of sets is incomplete, and it must be extended somehow.

 

[JonBarillari]

I didn't go through all the... gazillion ...posts to see if what I'am about to say has already been said, so I apologize if it has been said...

If the mathematical proofs given cannot be accepted for proofs, here is another way of thinking about it:

Number representations are not unique, I'm sure we would all agree that 1.0000 = 1/1 = 1.0 = 9/9 = 9.0/9.0 = 1 = 1.0000000...

so why is it so difficult to believe 1 = 0.999999.... ??

 

[xirtam]

...not that anybody really cares...

If there is no difference between two entities, they are the same. For if they do not differ in any way, they are the same entity.

Can someone take the mathematical difference between 1 and .999999... and tell me what they get?

That is to say, is there any number between .9999... and 1? Tell me what it is, and you have proven that 1 and .9999... are not the same number.

This is a question to be left to the philosophers. And philosophy should get a swift kick in the rear for not being profitable in today's society. Good day.

BTW I'm surprised that this thread attracted over 800 posts. Now I can clearly see how people can go on and on about nothing.

And on...

And on...

And on...

Ad nauseum...

Ad infinitum...

blah

 

[Kyteland]

Originally posted by: bleeb

Originally posted by: Kyteland

Originally posted by: bleeb
BUT, I'm still holding on to the belief that 0.9999... != 1. (Until the 1000th post)

So does the 1000th poster get to decide the final outcome of the debate?

well i'll finally and offically say what I believe regarding this topic.

I think you should wait until the 1000th vote instead of the 1000th post. That would make it even more interesting, don't you think?

 

[bleeb]

Originally posted by: Kyteland

Originally posted by: bleeb

Originally posted by: Kyteland

Originally posted by: bleeb
BUT, I'm still holding on to the belief that 0.9999... != 1. (Until the 1000th post)

So does the 1000th poster get to decide the final outcome of the debate?

well i'll finally and offically say what I believe regarding this topic.

I think you should wait until the 1000th vote instead of the 1000th post. That would make it even more interesting, don't you think?

that might take forever, but i'll continue until the 1000th post. Lets keep this thread alive guys!

 

[JupiterJones]

Originally posted by: xirtam
...not that anybody really cares...

If there is no difference between two entities, they are the same. For if they do not differ in any way, they are the same entity.

Can someone take the mathematical difference between 1 and .999999... and tell me what they get?

That is to say, is there any number between .9999... and 1? Tell me what it is, and you have proven that 1 and .9999... are not the same number.

This is a question to be left to the philosophers. And philosophy should get a swift kick in the rear for not being profitable in today's society. Good day.

BTW I'm surprised that this thread attracted over 800 posts. Now I can clearly see how people can go on and on about nothing.

And on...

And on...

And on...

Ad nauseum...

Ad infinitum...

blah

 

[bleeb]

The difference is 1 - 0.9999.... = 0.00000.....(infinity)....00001.

QED

0.9999 != 1

 

[juiio]

Originally posted by: bleeb
The difference is 1 - 0.9999.... = 0.00000.....(infinity)....00001.

QED

0.9999 != 1

There is no last digit. Hence infinity.

 

[bleeb]

haven't you been reading about the difference sizes of infinity???

In this particular solution, i'm using the aleph-NULL infinity set, instead of the aleph-ONE infinity set...

 

[juiio]

1/3 = .333 repeating
2/3 = .666 repeating

1/3 + 2/3 = 1
.333 repeating + .666 repeating = 1

It doesn't matter what you think of inifinity. There is not a trailing number. The second you stop putting 0s and add a 1, it ceases to be infinity.

 

[Kyteland]

Originally posted by: bleeb
haven't you been reading about the difference sizes of infinity???

In this particular solution, i'm using the aleph-NULL infinity set, instead of the aleph-ONE infinity set...

Um, I think you're using that wrong. Go back and reread those pages about cardinality.

*hint* just because your number ends in 1 and there was an impressive word that had a 1 in it (aleph-1) on that page doesn't imply any correlation between the two.