Math makes me happy

[Dr Unne]

Math is only fun once it becomes non-intuitive.

.999... is another way of expressing the infinite series:

.999... = 9/10 + 9/100 + 9/1000 + ...

Now even though the series is infinite, so long as it converges, it has a finite sum. So let's find it. (I hope. I haven't done this in a while.)

2) SUM = 9/10 + 9/100 + 9/1000 + 9/10000 ...
3) SUM / 10 = 9/100 + 9/1000 + 9/10000 + ...

Subtract 3) from 2):

(9 * SUM) / 10 = 9/10
9 * SUM = 9
SUM = 1

Put another way:

SUM = 9/10 + 9/100 + 9/1000 + 9/10000 ...
SUM = 9/10 * (1 + 1/10 + 1/100 + ...)
SUM = 9/10 * (1 + (1/10) + (1/10)^2 + (1/10)^3 + ...)

The equation for solving a geometric series of the form

s = 1 + q + q^2 + q^3 + ...

is

s = 1 / (1 - q)

In this case q is 1/10, so

s = (9/10) * ( 1 / (1 - 1/10) )
s = (9/10) * ( 1 / (9 / 10) )
s = (9/10) * (10/9)
s = 1

1.000... and .999... are just two ways of writing the same number.

Flying Mullet: Since we've already proven that .999... = 1, it follows that 1 - .999... = 0, yes. This is another way of saying "There is no real number between .999... and 1", like I said above. Indeed there is no real number between them. If 1 and .999... weren't equal, there would in fact be an INFINITE number of real numbers between them.

[Flying Mullet]

Flying Mullet: Since we've already proven that .999... = 1, it follows that 1 - .999... = 0, yes. This is another way of saying "There is no real number between .999... and 1", like I said above. Indeed there is no real number between them. If 1 and .999... weren't equal, there would in fact be an INFINITE number of real numbers between them.

Yeah, I know, I've been pretty much convinced for a while now. I'm just playing Devil's Advocate and trying to prove people wrong, without much luck. Somebody's gotta stick up for the ".999... < 1" team. *shrug*

And for my last log to throw on the flame, .999... will always appear behind 1 on the number line. Granted, the space/amount between .999... and 1 will get infinitely smaller, but that space is always there.

EDIT: Here's another thread in another forum that my fiance linked me to pretty much summing up that .999... = 1 in a few easy proofs: http://www.physicsforums.com/showthread.php?t=22866

[Loony BoB]

(SPOILER)I've actually been convinced for a while now, too, but arguing against Unne and Doomy is so much fun, and in this thread I get to argue with both at once. In fact, the thing that bennator/Unne posted (the long equation) has actually been shown to me by one of my teachers back in high school. xD Although I do still think that 0.000...001 should be a legitimate number on some level. Not to say that it'll have anything to do with this stuff, but yeah. Maybe I'm just too creative for my own good.

EDIT: That thread you linked to has some interesting thoughts on the 0.0...01 thing, too.

[frr_vegeta]

.999 = .999 *Nods*

[Flying Mullet]

Okay, I have another "problem" that I think is fun to think about:
2/3 = 0.6

2/3 + 2/3 + 2/3 = 6/3 = 2

0.6 + 0.6 + 0.6 = ???

I just find this interesting because the more sets of 3 2/3's(2/3 + 2/3 + 2/3) that we try and add, whole numbers in essence, the more we start to move away from 0.9 and 1 if we try to calculate it by means that we understand, i.e.:

2/3 + 2/3 + 2/3 = 6/3 = 2
0.6 + 0.6 + 0.6 = 0.666... with an 8 at the end

2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3 = 12/3 = 4
0.6 + 0.6 + 0.6 + 0.6 + 0.6 + 0.6 = 0.666... with a 6 at the end

and so forth. Yet by our proofs from earlier we should always equal one. Just something to ponder.

It's almost like the more we try to break down infinity, the more it deteriorates.

[Loony BoB]

How did you get the 8 at the end? x_x

[Logan]

I have a headache now from TRYING TO FIGURE IT OUT. *sux*

[Del Murder]

This is the greatest thread in the history of Eyes on Final Fantasy. I love you guys.

[Dragonflame]

Working by the rules of the decimal system, I am forced to say that .999...=1. However, this is actually an inherent flaw in the decimal system, because the concept of infinity is impossible.There is no way it can possibly be expressed in a way that humans can comprehend. Thus, the whole debate is meaningless because we are working with the rules humans have created for math, when at this point the whole concept is eluding us. We are trying to compress the workings of the universe into a system of numbers when such a system is inadequate for the task. When you really think about it, we have yet to define 1. What is 1? Can anyone explain that?

I have a better question than the one we are currently discussing. Does any of this matter. Is there anyone who will ever need to know whether or not .999...=1? Why don't we accept that there is no answer to the question and move on?

[Shlup]

That's it! I can't be on staff with this guy anymore! I resign as administrator!

...

...I hate math.

[Del Murder]

Unne already solved it, you silly goose. All he really needs to do now is prove the convergence of a geometric series is s = 1 / (1 - q), which isn't hard and can already be found in many textbooks.

[Dragonflame]

But are those textbooks right or not? They were written by people who could not possibly have a complete understanding of the subject, because it is beyond the scope of human experience and thus is impossible to comprehend fully.

[Yamaneko]

This is the greatest thread in the history of Eyes on Final Fantasy. I love you guys.

Worst, but yeah.

[SeeDRankLou]

It should be noted that an irrational number is any real number that cannot be expressed as a ratio between two integers. .999.... is a number divisible by 1, 3 and 9. If divided by 9, we get .111...., which is 1/9. So .999....=9*(1/9) (dividing by 3 will give the same results). But numbers must always be looked at in their simplest terms when using theory, so 9*(1/9)=1. The number 1 is not a ratio between two integers (in simplest terms), and therefore .999.... cannot be expressed as a ratio between two integers and is thusly an irrational number. A number cannot exist simultaneously as a rational and irrational number, thus it is not possible for .999.... to equal 1.

[HOOTERS]

I don't get how there has to be number in between two numbers for the first number to be less than the second number. Won't that mean that every number is equal to every other number? Sooner or later you have to run out of numbers to stick in between.