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XaJin
27-10-2004, 06:34
.999 repeating = 1

Discuss.

BiggestHobbit
27-10-2004, 06:51
.999 repeating = 1

Discuss.
Let x = .9999...
10x = 9.9999....
10x - x = 9.9999... - .9999...
9x = 9
x = 1
therefore .9999... = 1

BH

Lord Gargoyle
27-10-2004, 09:18
not again. :) look up the thread in d2 offtopic. there are all the explanations. ;)

dantose
29-10-2004, 14:51
in short, it does equal 1.

IamTheTest
30-10-2004, 20:43
(1/3)=.3 rep.
(1/3)*3=1
.3 rep. *3=.9 rep.
1=.9 rep.
So what if the system is perfect...it is good enough.

toader
01-11-2004, 15:20
In shorter (or maybe longer)...our current numerical system is flawed, and it does not equal one. The only reason we can "prove" it equals one is because there is a flaw in the nomenclature of our numerical system that we "made up" a long time ago.

SuggestiveName
01-11-2004, 16:36
In shorter (or maybe longer)...our current numerical system is flawed, and it does not equal one. The only reason we can "prove" it equals one is because there is a flaw in the nomenclature of our numerical system that we "made up" a long time ago.
It's not so much a flaw, because a flaw would imply a logical contradiction and in math contradictions simply cannot happen. Also if .999... did not equal one, we could not have calculus because we would not have convergent sequences, and we would not have continuous functions.

Maybe the easiest way to think about this is to use a different definition of equals then we are accustomed to. Simply put, when the distance between two points is arbitrarily small, we say that they are equal. That is very roughly put, and slightly inaccurate, but should help with the philosophy of this problem.

Another formulation that makes sense of this is a theorem about the real number system. We can represent any real number as a non-terminating decimal expansion, and if the decimal part of the number is finite, we simply subract one from the last decimal place and append an infinite string of 9's. So 2.45 = 2.4499999999.... and 1.0 = 0.999999.

ZiyiWang
02-11-2004, 05:20
SuggestiveName, I remember seeing you post last year from D2 forums during the First season of d2 1.10 patch.
Does GloryField sound familiar to you? I was the first US East ladder with perfect annihilius, also I had a nearly perfect (off by 1) Superior Archon Plate with that 4 socket runeword, I forgot what it was, it wasn't enigma, it was something else.

th5418
02-11-2004, 05:51
You people are slow in Guild Wars. We at Diablo2 have proved this over and over!

Tarrant Rahl
02-11-2004, 07:40
You people are slow in Guild Wars. We at Diablo2 have proved this over and over!

Actually, the topic came up from people that have frequented the D2 boards. In fact, many of the members that participate in the GW boards also participate at the D2 boards. Your comment really was unnecessary and completely irrelevant.

toader
02-11-2004, 17:58
Haha, th6726 = pwned by Mr. Rahl.

But really, go easy on him Mr. Rahl, he was joking with us, a common personality on OTF boards, especially the Diablo one. He is just a poor little asian kid who just lost his best forum friend due to a Delete+Bannange. :(


And Suggy....stop stalking me!!!! Somethings wrong with our system, I dunno what, if I knew, I would be a cabillionaire. But if two things arent the same, then they arent equal. The fact that our system allows this (IMO) means its flawed. I had that argument so many times at college in all my higher level math classes. I know it may not seem so, but I took a lot of higher level math classes in my day. I know most people that take those classes conform to the thinking of the teacher, but not special ole me. :thumbsup:

Silenoz
04-11-2004, 03:22
.999 (with about 30 9's) x 10 on a TI-83 will give you 10. So I suppose, for one reason or another, Texas Instruments agrees =)

1 = 2. I'll get back to you on that one soon enough.

Silenoz
04-11-2004, 03:48
(1/3)=.3 rep.
(1/3)*3=1
.3 rep. *3=.9 rep.
1=.9 rep.
So what if the system is perfect...it is good enough.

the real problem here is that .3 repeating cannot EXACTLY amount to 1/3- it is infinite. It can come infinitely close, but never be exact as the fraction.

locallyunscene
05-11-2004, 18:32
the real problem here is that .3 repeating cannot EXACTLY amount to 1/3- it is infinite. It can come infinitely close, but never be exact as the fraction.

the real problem is that this proves .333 repeating times 3 is equal to one not that .333 repeating equals 1.

.333 repeating in base four IS equal to one.

And .3 repeating = 1/3. What you just said makes baby mathmaticians cry.

Tarrant Rahl
05-11-2004, 18:35
the real problem is that this proves .333 repeating times 3 is equal to one not that .333 repeating equals 1.

.333 repeating in base four IS equal to one.

And .3 repeating = 1/3. What you just said makes baby mathmaticians cry.

Sure, it does technically equal 1/3, but what I can question is the validity of any equations utilizing .333 instead of 1/3, since 1/3 can't technically be written properly in decimal form for an equation. If you can't truly write that number in decimal form, how can you use it in equations in decimal form?

locallyunscene
05-11-2004, 18:42
numbers are abstract. They aren't real, so when we try to make them into the real world we have to put finite constraints on them. So technically, in this world .333 repeating cannot equal one becuase we cannot represent it as it actually is.

But when ever we are dealing with abstracts it is true. Suggestivename explained it fairly well.

toader
05-11-2004, 21:56
in this world .333 repeating cannot equal one


0.3333 cant equal 1 in any world :)

locallyunscene
06-11-2004, 01:13
0.3333 cant equal 1 in any world :)

that was a typo but if you're talking about a world that uses a base 4 number system then .3 repeating does equal 1 :winner:

trafalgar-zero
11-11-2004, 05:34
We use the flaws to all sorts of difficult equations

physical
calculus (calc 4 sucks, DON'T TAKE)

just think next time that Budwiser smelling, toothless, carney straps you into that centrafuse ride, that you are riding in a machine that is built on flawed mathmatics.

It's all gonna come back 10 fold when we need mathmatics the most

makes me feel warm and fuzzy

Tsume
11-11-2004, 06:42
I undestand easily how it is proven to be 1. But the term limit keeps popping up in my head when i look at that decimal number....increasing ever so slightly with every 9 you look at, never quite reaching 1.

trafalgar-zero
11-11-2004, 18:02
We assume that it is 1 because it makes calculations way easier, and the error it creates in reality is so little we can't notice it.

Maybe in the future it will cause problems.

We know .9999... doesn't =1
but in math we say this to cut down the paper work.
There's many other things in math that we assume or bend to our advantage.

so literaly .9999999 doesn't = 1
but for the sake of saving time we say it does.

Dreamsmith
16-11-2004, 00:25
Sure, it does technically equal 1/3, but what I can question is the validity of any equations utilizing .333 instead of 1/3, since 1/3 can't technically be written properly in decimal form for an equation. If you can't truly write that number in decimal form, how can you use it in equations in decimal form?

But you can write it in decimal form, it's ".3" with a bar over the three.

This is irrelevant, however. How you write a number has no bearing on the number itself. It's the same number whether you write it "1/3" or ".3..." or whatever, just as "3", "three", "tres", and "III" are the same number, regardless of the word, symbol, or notation used to represent it. Any equation written in demical notation could easily be rewritten in fractional notation using roman numerals if you wanted -- the validity of the equation has nothing at all whatsoever to do with what notational system you decide to use.

One third times three equals one. There's no magic associated with notational systems that would make this true if you write it one way but false if you write it another. Don't confuse the actual math with the way you write it out. "One third times three equals one" was true long before humans came along and invented writing...

Mortaj Delvas
16-11-2004, 00:33
I've been hesitant to reply so far but...


1/3 can be represented by 0.3 repeating. Technically they are not equal. Remember back to 8th grade, 0.3 repeating is found by dividing 1 by 3. If you were to actually do this calculation, you could never could finish.

zabadoohp
05-12-2007, 05:34
Hello everyone hahahaaa

Tucks
05-12-2007, 05:42
hahaha, epic thread necromancy.

The Avatar
05-12-2007, 05:53
We know .9999... doesn't =1
but in math we say this to cut down the paper work.
There's many other things in math that we assume or bend to our advantage.


.9999... (repeating) does euqals to 1. yes, blame math that u can manipulate any ways u want as long as both side equal and ur not breaking the rules.



so literaly .9999999 doesn't = 1
but for the sake of saving time we say it does.

.9999999 does not equal to 1. do the method below u wont come up with 1 but exactly .9999999.


Let x = .9999...
10x = 9.9999....
10x - x = 9.9999... - .9999...
9x = 9
x = 1
therefore .9999... = 1

BH

that one is the correct answer & proof method; learn in calculus about limits.

zabadoohp
05-12-2007, 06:05
Hello everyone hahahaaa

The Avatar
05-12-2007, 06:06
Your "equation" is false. If it was true you could do this.
Let x = .8888...
10x = 8.8888....
10x - x = 8.8888... - .8888...
8x = 8
x = 1
therefore .8888... = 1

no redo it & check ur work.

let x = .8888...
10x = 8.8888...
10x - x = 8.8888...- .8888...
9x = 8
x = 8/9 = .8888888889

Drec Sutal
05-12-2007, 06:13
and any other infinitely repeaping
number

Go away, unless you care to actually say something.

MoonUnit
05-12-2007, 06:50
Useless mathematics

You registered today to revive a 3 year old thread? Why would a "math whiz" be posting on a gaming forum?

You might want to check this (http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html) out.

Kahlan
05-12-2007, 07:14
hahaha, epic thread necromancy.

Yeah, that's what I was thinking.

When I read the "I saw this in the D2 off topic forums last year when first ladder season came to be" (ok, not exact quotes, but I think people will get the picture) is when I looked at the date...2004.

Man, I've seen some threads necro'd, but I do think this one takes the cake!

And zabadoohp, you do need to look at your math again. 10x-x (or 10x-1x) = 9x, not 8x.

Drec Sutal
05-12-2007, 07:21
(...edit time limit...)

My guess is you figured out what I figured out in like fifth grade....


better and more accurate description, how to get the EXACT numerical value of 1/3

If you use a base different then 10 you get nice even numbers for (some) repeating fractions!

Lets try base 9!

First lesson: count to 20(base9) (or 18 base 10) in base 9...

1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,18,20

Next lesson: 3 times 3 in base 9...

3+3+3 is (refer to your counting above now...) 10(base9)

What's 1/3 of a circle in base 9?

_.3
3|1.0
-1.0
0.0

Wow! An exact decimal (actually not decimal because that refers to base 10...) representation of one third! It's just .3 base 9!

Fact is, .333... is the exact representation of 1/3 base 10. Exact. If I did the same math as above but in base 10, I'd get .33333333333(and so on forever - a decimal point followed by infinite threes or written more simply, .333...)

We only use base 10 because of our 10 fingers.

Oh, and this doesn't work for pi, though as a little kid my initial impression was that it might. Of course, I didn't go writing to prestigious math institution and yelling this out to everyone I saw so when I realized how obvious and not useful this was I just ignored it. Anyway... bases are whole numbers so math (like what is 2+2) is easy. You could make a "base pi" but then you would find yourself with a non-repeating-non-terminating value for 1. And you can already do that - define a circle as having a circumference of one and find its diameter. It isn't useful. Pi is a "non-repeating non-terminating" decimal. Changing bases only fixes repeating "non-terminating" decimals. Don't worry, they probably won't even read your email.

And just for the record if changing base was your idea, I figured that one out in 5th grade when a teacher was trying to explain how pi went on forever. I figured out you could use "base pi" and "base 3" and such to get rid of long numbers, and within the course of an hour just figured out that it *moved* the long numbers. In base pi the circumference is easy, but you'll find the diameter is the 1/pi base10 - another repeating number. And in base 9 1/3 is simply .3, but 1/2 will go on forever. That took 11-year-old me an hour to figure out, not even knowing other base numbering systems existed.

Hints that I shouldn't have even bothered reading his post...


overexaggerating

Isn't a word. And it's redundant. Exaggerating already means too much.


disovered, descripitve, repeaping
I think you meant discovered, descriptive, and repeating.

You can find a flaw in basic math but you can't figure out how to work a spell checker before trying to publish your results. The next reason nobody should bother listening to you is that you also obviously don't understand the publishing process. Lets assume you actually found something here. You write a paper (a *real* paper complete with APA or similar citations for every single assumption you make in your complete proofs and pages of discussion about implications). You submit it to ONE journal. Only one at a time, and you sign a statement that this is the only journal you're currently submitting it to. They take it and reject it right away for improper formating and obvious spelling mistakes. But lets say you fix those and try again. Now it goes off for peer review and a month or so later a decision gets back to you. The *next* month at best you 'get published.'


The 6th Numerical System of Numbers has been disovered!

This from the Redundant Department of Redundancy, of course?


2000+ years of mathematics

Wait, math has only been around about 2k years? Try 4k (http://en.wikipedia.org/wiki/History_of_mathematics) buddy, since the first proof of the Pythagorean theorem.

MixedVariety
05-12-2007, 13:37
Isn't a word. And it's redundant. Exaggerating already means too much.


Definition: to overemphasize completely; to go beyond anticipated exaggeration.

You can find a definition for overexaggeration. Perhaps it might actually be considered as slang, but it does exist.

Just sayin'.

Lord Dragon
05-12-2007, 20:26
Let x = .9999...
10x = 9.9999....
10x - x = 9.9999... - .9999...


9x = 9 (sorry but this does not equal 9 it equals 8.9991 (9*0.9999)



x = 1
therefore .9999... = 1

BH

MoonUnit
05-12-2007, 20:28
9x = 9 (sorry but this does not equal 9 it equals 8.9991 (9*0.9999)

It's 0.999999999 repeating. So yes, it equals 1.

Lord Dragon
05-12-2007, 20:31
It's 0.999999999 repeating. So yes, it equals 1.

nope, sorry. jut did it long hand and by calculator 9*0.999 equals 8.9991 it's finite.

9*.009 = .0081
9*.09 = .081
9*.9 = 8.1

thus 8.9991

Drec Sutal
05-12-2007, 20:33
9x = 9 (sorry but this does not equal 9 it equals 8.9991 (9*0.9999)

you seem convinced there are 4 9's in the starting equation (x=.999...). There aren't. There are infinite. So what you probably mean is 8.999...1. Which of course means nothing (an endless list of nines with a one at the end?)

More importantly, they aren't asserting that 9x = 9. They are proving it. If you can find a problem in the proof, show it...

x= .999... (given)
10x = 9.999... (multiply by 10)
10x-x = 9.999... - .999... (subtract .999...)
9x = 9
x = 1 (divide by 9)
.999... = 1 (substitution)

So which step do you have a problem with?

Lord Dragon
05-12-2007, 20:36
you seem convinced there are 4 9's in the starting equation (x=.999...). There aren't. There are infinite. So what you probably mean is 8.999...1. Which of course means nothing (an endless list of nines with a one at the end?)

More importantly, they aren't asserting that 9x = 9. They are proving it. If you can find a problem in the proof, show it...

x= .999... (given)
10x = 9.999... (multiply by 10)
10x-x = 9.999... - .999... (subtract .999...)
9x = 9
x = 1 (divide by 9)
.999... = 1 (substitution)

So which step do you have a problem with?

Ah ok infinite nines..

yes I see what you mean now. It always makes me wonder when do you see the 1 at the end.... it has to be there but it is infinitely far away.

Well I do see a problem with the proof as 8.9999... never equals 9.0 You can agree that it's close enough to count but it will never equal 9
Although you can approach infinitely close to 9.0 you can never get there. the same as the old proof about "you must cross half the distance every time"

Drec Sutal
05-12-2007, 20:57
Ah ok infinite nines..

yes I see what you mean now. It always makes me wonder when do you see the 1 at the end.... it has to be there but it is infinitely far away.

No, it doesn't have to be there. Nines forever. Never did I multiply .999... by 9. I multiplied it by 10, then subtracted .999...


Well I do see a problem with the proof as 8.9999... never equals 9.0 You can agree that it's close enough to count but it will never equal 9 Although you can approach infinitely close to 9.0 you can never get there. the same as the old proof about "you must cross half the distance every time"

This is not a problem with the proof, it's a problem with you. A problem with the proof is pointing out something I did wrong, not saying the conclusion isn't true. If I didn't do anything wrong in the method the conclusion *is* true.

Now, you claim that 8.999... is smaller then 9. Presumably that .999... is smaller then one as well. If that's true, then by the density of numbers there are an infinite number of numbers between .999... and one. If that's true it shouldn't be hard to come up with exactly one such number.

The traditional method is an average. So tell me, what's an average of .999... and 1? Is it an infinite number of nines followed by an eight? Because I'm pretty sure that's smaller then an infinite number of nines. There are no numbers between them, so they are the same number.

Please stop trying to fight this, it's a mathematical fact that people far smarter then you have pondered and found to be true. It's a bit counter-intuitive but true.

David Holtzman
05-12-2007, 21:16
I am particularly bad at math (I barely made it through calc in High School) so forgive me if this question appears nonsensical. I understand that 0.999... is infinitely close to 1. What number would be infinitely close to 0.9999...? And, could a similar proof be given to show that this infinitely close number was identical to 0.999...?

Drec Sutal
05-12-2007, 21:28
I am particularly bad at math (I barely made it through calc in High School) so forgive me if this question appears nonsensical. I understand that 0.999... is infinitely close to 1. What number would be infinitely close to 0.9999...? And, could a similar proof be given to show that this infinitely close number was identical to 0.999...?

.999... IS one. So no. For exactly that reason, among others.

David Holtzman
05-12-2007, 21:37
Alright, then what number would be infinitely close to but not 1?

Drec Sutal
05-12-2007, 21:43
Alright, then what number would be infinitely close to but not 1?

There's no such thing. There's either something (infinite things) closer or it *is* one.

Lord Dragon
05-12-2007, 21:45
Please stop trying to fight this, it's a mathematical fact that people far smarter then you have pondered and found to be true. It's a bit counter-intuitive but true.

Thank you oh so much for saying it this way. You do realize that people Infinitely smarter than the ones YOU site have proven that 0.999...(infinite) does NOT equal 1. So yes I know the proof. I have taken calculus and I have even worked with string theory and quantum physics. I completely understand the way you are going about the proof. I just believe (and very rightfully so) that the premise is flawed.

Drec Sutal
05-12-2007, 21:52
Thank you oh so much for saying it this way. You do realize that people Infinitely smarter than the ones YOU site have proven that 0.999...(infinite) does NOT equal 1. So yes I know the proof. I have taken calculus and I have even worked with string theory and quantum physics. I completely understand the way you are going about the proof. I just believe (and very rightfully so) that the premise is flawed.

Then I'm sure you'd be happy to share the proof.

The Avatar
05-12-2007, 22:14
Well I do see a problem with the proof as 8.9999... never equals 9.0 You can agree that it's close enough to count but it will never equal 9
Although you can approach infinitely close to 9.0 you can never get there. the same as the old proof about "you must cross half the distance every time"

no, do the method u will find out that 8.9999... does equal to 9.

let x = 8.9999...
10x - x = 89.999...- 8.9999...
9x = 81
x = 9

line up the decimal places. as long as the decimal is repeating, which is .999..., all these 9 will be cancelled.

but if ur claiming 8.9999... doesnt equal to 9.0 or 9.00 then that's true cuz the sig figs are much smaller than 9, however, that only matters in scientific term. in mathematical term it has only a slight difference as the rule says 9.0 = 9, drops the decimal place.

in math as long as ur not breaking the rules then any method applies to be correct if it fits the problem.

Lord Dragon
05-12-2007, 22:28
that only matters in scientific term.

There is the exact crux of the problem. In scientific terms the proof is false. In mathematical terms the proof is true. So it depends on completely on the framework in which the problem is looked at.

Oh and as to the proofs... here's a good one for you... let me know if you need anymore.

---------------------------------------
There is no need for anyone to prove 0.999... < 1 because by definition of the decimal radix system, it is. The onus of proof rests on those who claim these two values are equal. All their proofs are false: starting with the most common (as in the above example) and moving to the most complex (as in the Archimedean property). If x = 0.999... then 10x is not well-defined. In fact it is not defined at all because our rules of arithmetic for multiplication apply to rational numbers only. We extend these rules to non-rational numbers by approximation. That is to say, we can never calculate the area of a circle exactly and can never find the exact length of the hypotenuse in a right angled triangle where the remaining sides are both of unit length. So, if multiplication is defined for rationals, we must therefore be able to express 0.999... as a rational number. This is impossible for there are no numbers a and b such that a/b = 0.999...

Rational numbers were at first invented to represent only quantities that are less than 1 but greater or equal to 0. Numbers were written in a finite representation format. Recurring numbers were not considered rational until centuries later when the limit concept was introduced. This definition was later reworked (or extended) so that we could perform mixed arithmetic a lot easier. For example arithmetic involving an originally defined rational number and a whole number, eg. 2 + 2/3 = 6/3 + 2/3 = 8/3 This is an exact calculation because it can be represented finitely.

Much later very stupid men decided that a number would be defined as a limit of a Cauchy sequence. The definition they invented is in fact circular because they use a limit to define a number. In fact, if they were even remotely intelligent, they would have stated that any number is a rational number or the limit of the sum of infinitely many rational numbers where a rational number is a/b with b not zero and a < b (Was part of original definition but later dropped).

So as long as we stay with the same rational representation of numbers, our result will generally be true. eg. 1/3 + 1/3 + 1/3 = 3/3 = 1. However, when we try to use other representations such as radix or mixed rational/radix, we end up with anomalies. eg. 1/3 + 0.333.. + 0.3333... = 1/3 + 0.666.... = ? Oops, 0.666... is not finitely represented and thus cannot be used in our arithmetic except as an approximation. e.g. 1/3 + 66/100. We cannot say 0.666... = 2/3 because it is not. Sorry, duplicate representation is not allowed in any radix system. And no radix system is capable of representing all numbers.

Let's summarize the rules: In order for arithmetic to be performed, numbers must be finitely represented using the same representation. 2pi/3 + pi/3 = pi. 1 + sqrt(5) = 1 + 2.23 = 3.23

In light of the above it is evident that even debating the 0.999... and 1 equality is completely idiotic

The Avatar
05-12-2007, 22:37
There is the exact crux of the problem. In scientific terms the proof is false. In mathematical terms the proof is true. So it depends on completely on the framework in which the problem is looked at.

science does not prove these things. a "theroem" in math is similar to the "law" in science. the question of .9999... = 1 is merely a mathematical problem, there is nothing scientific about it.

the smaller sig figs are used in chemistry & physic cuz u need the accuracy when dealing with inifinite large or inifinite small numbers

math is a tool to use in physic, but that doesnt mean math = physic even though they are related. if u know the estein's relativity u know wat i mean. everything work in the same framework, working on a different frame of reference completely makes problem meaningless.

Lord Dragon
05-12-2007, 22:42
science does not prove these things. a "theroem" in math is similar to the "law" in science. the question of .9999... = 1 is merely a mathematical problem, there is nothing scientific about it.

the smaller sig figs are used in chemistry & physic cuz u need the accuracy when dealing with inifinite large or inifinite small numbers

math is a tool to use in physic, but that doesnt mean math = physic even though they are related. if u know the estein's relativity u know wat i mean, everything work in the same framework.

Yes, Quite right. Physics does not work with "proofs" such as geometry and mathematics do. Oh, I just placed the "proof" before a Mathmatics Prof that I know. I will quote his answer here... I gave him this

Let n = 0.9999...
10n = 9.9999...
10n - n = 9.9999... - 0.9999...
9n = 9
n = 1 = 0.9999...

and he responded with...

"I consider this a demonstration, not a proof...."


Another portion of the proof that .999...~ does not equal 1 is this...

Graph the function y=1/x and tell me the point at which the line crosses either axis. You can't. There is no point at which the line crosses the axis because, infinitely, the line approaches zero but will never get there. Same holds true for .9999 repeating. No matter how many 9s you add, infinitely, it will never equal one.

Drec Sutal
05-12-2007, 22:55
There is the exact crux of the problem. In scientific terms the proof is false. In mathematical terms the proof is true. So it depends on completely on the framework in which the problem is looked at.

Oh and as to the proofs... here's a good one for you... let me know if you need anymore.

Undoubtedly I will, because the "proof" is flawed.


There is no need for anyone to prove 0.999... < 1 because by definition of the decimal radix system, it is. The onus of proof rests on those who claim these two values are equal. All their proofs are false: starting with the most common (as in the above example) and moving to the most complex (as in the Archimedean property). If x = 0.999... then 10x is not well-defined. In fact it is not defined at all because our rules of arithmetic for multiplication apply to rational numbers only.

It 10x is perfectly defined as 9.999...

In the decimal system, multiplying by ten simply shifts the decimal place by one to the right.


We extend these rules to non-rational numbers by approximation. That is to say, we can never calculate the area of a circle exactly and can never find the exact length of the hypotenuse in a right angled triangle where the remaining sides are both of unit length.

The two short sides of a right triangle are 3 and 4. The fourth side is exactly 5.


So, if multiplication is defined for rationals, we must therefore be able to express 0.999... as a rational number. This is impossible for there are no numbers a and b such that a/b = 0.999...

Of course there are. 3/3 for example. 1/3 is .333... and 3/3 is three times that. .999...


Rational numbers were at first invented to represent only quantities that are less than 1 but greater or equal to 0. Numbers were written in a finite representation format. Recurring numbers were not considered rational until centuries later when the limit concept was introduced. This definition was later reworked (or extended) so that we could perform mixed arithmetic a lot easier. For example arithmetic involving an originally defined rational number and a whole number, eg. 2 + 2/3 = 6/3 + 2/3 = 8/3 This is an exact calculation because it can be represented finitely.

Okay... congradats on the history of math...


Much later very stupid men

Horray insulting random mathmatitions.


Much later very stupid men decided that a number would be defined as a limit of a Cauchy sequence. The definition they invented is in fact circular because they use a limit to define a number. In fact, if they were even remotely intelligent, they would have stated that any number is a rational number or the limit of the sum of infinitely many rational numbers where a rational number is a/b with b not zero and a < b (Was part of original definition but later dropped).

I don't think you're qualified to make this statement.


So as long as we stay with the same rational representation of numbers, our result will generally be true. eg. 1/3 + 1/3 + 1/3 = 3/3 = 1. However, when we try to use other representations such as radix or mixed rational/radix, we end up with anomalies. eg. 1/3 + 0.333.. + 0.3333... = 1/3 + 0.666.... = ? Oops, 0.666... is not finitely represented and thus cannot be used in our arithmetic except as an approximation. e.g. 1/3 + 66/100. We cannot say 0.666... = 2/3 because it is not. Sorry, duplicate representation is not allowed in any radix system. And no radix system is capable of representing all numbers.

.666... is 2/3. A finite ratio.


Let's summarize the rules: In order for arithmetic to be performed, numbers must be finitely represented using the same representation. 2pi/3 + pi/3 = pi. 1 + sqrt(5) = 1 + 2.23 = 3.23

In light of the above it is evident that even debating the 0.999... and 1 equality is completely idiotic

No, I think it's just clear you don't understand the decimal system as well as you think you do. You've presented no evidence, no proof that .999... is less then one. You haven't invalidated any of the proofs that .999... is equal to one. You've declared that .999... can't exist in decimal form, and you aren't qualified to make that claim. Feel free to link me to someone who is.

Proof has been presented that .999... is equal to one. The burden is now on you to either find a flaw in the proofs, or present an alternate proof that .999... is less then one.

Lucis
05-12-2007, 23:01
Proof has been presented that .999... is equal to one. The burden is now on you to either find a flaw in the proofs, or present an alternate proof that .999... is less then one.

Sorry, I've got confused.

Firstly, I understand how to work out that 9.999etc. = 10, But isn't that just one of those cases in which the logic is simply flawed.

Secondly, where does this relation end. Is 9.88888888 = 9.9999999?

Sorry, never have been that good at maths.

draugaer
05-12-2007, 23:12
Let x = .9999...
10x = 9.9999....
10x - x = 9.9999... - .9999...
9x = 9
x = 1
therefore .9999... = 1


I may just be being stupid but somthing seems strange here:

Let x = .9999...
10x = 9.9999....
multiply both sides by 10, makes sense
10x - x = 9.9999... - .9999...
here you are subtracting x from both sides, but x=.9999... so wouldn't 10x-x be 9.0000.......9, not simply 9?

aren't you are subtracting 1 from one side of the equation and .9999 from the other? Therefore it isn't a balanced equation.

Lord Dragon
05-12-2007, 23:13
Sorry, I've got confused.

Firstly, I understand how to work out that 9.999etc. = 10, But isn't that just one of those cases in which the logic is simply flawed.

Secondly, where does this relation end. Is 9.88888888 = 9.9999999?

Sorry, never have been that good at maths.

It is a flaw in our system of mathematics. Yes. I will admit that the proof of .9999999... = 1 does exist and that most mathematicians even agree that it is true. But it is based upon a flawed system and we need to set limits to achieve any rationality at all. In truth (and understanding the mathematical definition of infinity) 0.9999... repeating can never truly = 1 although it can be shown with proofs that it does.

Really it all depends on what frame of scientific reference you wish to look at it from.

Lucis
05-12-2007, 23:14
EDIT: At Drauger btw. Not LD.
No.
X doesn't have to equal 1. In this case, x=0.999999...
I'm pretty sure at least.

The Avatar
05-12-2007, 23:15
Sorry, I've got confused.

Firstly, I understand how to work out that 9.999etc. = 10, But isn't that just one of those cases in which the logic is simply flawed.

the logic applies within the math rules. as i say, as long as ur not breaking the rule then u can use the correct method to prove a mathematical problem. the method provided above there is no way u can disprove it nor it breaks the rules.

like chemistry & physic, i can say a lot of theories in there are wrong if i judge from a common logical way. for relativity as example, u see a person inside a moving object is moving, but he sees ur moving as his persepctive so who's right? so the correct answer to this if u relate both of them in earth's frame of reference then u will be correct.



Secondly, where does this relation end. Is 9.88888888 = 9.9999999?

do the above method then u can figure out the answer.




I may just be being stupid but somthing seems strange here:

Let x = .9999...
10x = 9.9999....
multiply both sides by 10, makes sense
10x - x = 9.9999... - .9999...
here you are subtracting x from both sides, but x=.9999... so wouldn't 10x-x be 9.0000.......9, not simply 9?

aren't you are subtracting 1 from one side of the equation and .9999 from the other? Therefore it isn't a balanced equation.

no if u substitue at the start, x = .9999..., is 10(.9999...) - (.9999...) = 9.999... - .9999... = 9.

when dealing with infinite repeating numbers, it doesnt matter if there is an extra decimal place or not since there is no end. so .999... - .9999... = 0 even though .9999... has one more sig fig than .999...

if ur saying 9.00000....9, which has an end, then thats not an infinite repeated number, which would be bigger than 9.

Lucis
05-12-2007, 23:19
So x=0.8888888
10x= 8.888
10x-x=8.888888-.8888888
9x=8
So x = 0.8888888
Or around that.
Ok, definitely looks like my suspicion was wrong here.

Lord Dragon
05-12-2007, 23:23
like chemistry & physic, i can say a lot of theories in there are wrong if i judge from a common logical way. for relativity as example, u see a person inside a moving object is moving, but he sees ur moving as his persepctive so who's right? so the correct answer to this if u relate both of them in earth's frame of reference then u will be correct.



do the above method then u can figure out the answer.

I have heard the 0.999...= 1 argued by my Brother Mike (masters in mathematics) and my Uncle David (phd in physics) for over 3 decades and they still have never agreed on the correct answer.

Honestly I can agree with either of them although I lean more towards the physics side of the argument.

The Harlequin
05-12-2007, 23:25
No math major here, but I'm not sure how an irrational number can exactly equal a rational number. Doesn't that make all of mathematics explode in a firey ball of digits? :shocked:

The Avatar
05-12-2007, 23:30
No math major here, but I'm not sure how an irrational number can exactly equal a rational number. Doesn't that make all of mathematics explode in a firey ball of digits? :shocked:

.9999... (repeating) is a rational number.

an irrational number is like pie which is 3.141592654..., the fraction display is 22/7 but it comes out as 3.142857143....


I have heard the 0.999...= 1 argued by my Brother Mike (masters in mathematics) and my Uncle David (phd in physics) for over 3 decades and they still have never agreed on the correct answer.

Honestly I can agree with either of them although I lean more towards the physics side of the argument.

of course this type argument will go on forever if taken different references. the common ones in calculus are 0/0, x^x, sinx/x, inifinity +/- inifinity. physic u learn about light & the ether. (which does not exist) calculus deals with limits, physic deals with calculation of physical objects on earth. the higher level ur into either subjects, it becomes really abstract, so as english or any other subjects.

i can give u an example. when u go onto linear algebra level in math, one thing people like to argue about is the parallel. in math, the definition of two parallel lines will never meet or touch, but in linear algebra people argue that two parallel lines will meet & touch at a certain inifinity due to space & dimension, so wtf? in this case, u have to take a stance and try to prove either ones.

David Holtzman
05-12-2007, 23:46
There is the exact crux of the problem. In scientific terms the proof is false. In mathematical terms the proof is true. So it depends on completely on the framework in which the problem is looked at.

I'm no mathematician, but I am a philosopher. The above is false. Mathematics is an a priori study, science an a posteriori study. As such, a statement in one cannot even be expressed in the other. Science has precisely nothing to say about 0.999... = 1 just as mathematics qua mathematics has absolutely nothing to say about the results of some experiment being X.

************************************************** ********


There's no such thing. There's either something (infinite things) closer or it *is* one.

In this, I think, you cannot logically be correct. We can express numbers along a line. A given point in that line we can note as "1". There must either be a point directly next to "1" (that is, the number that is as close to 1 as possible without being 1) or there is a hole in the line which mathematics cannot express. However, even in the latter case there still would be an answer, it is merely that the language of mathematics would be insufficient to describe it.

Drec Sutal
06-12-2007, 00:01
I'll tell you what, I'll give up. Just go correct the relevant wikipedia (http://en.wikipedia.org/wiki/.999) article. And the 62 sources and more then 100 references. You'll want to contact the math professors at every major university as well.

David Holtzman
06-12-2007, 00:08
I'll tell you what, I'll give up. Just go correct the relevant wikipedia (http://en.wikipedia.org/wiki/.999) article. And the 62 sources and more then 100 references. You'll want to contact the math professors at every major university as well.

Forget those guys and answer me. They won't listen to deductive proofs, but I am more than happy to. I think my point about the number line is pretty solid though.

Whiskey River Seer
06-12-2007, 00:08
In this, I think, you cannot logically be correct. We can express numbers along a line. A given point in that line we can note as "1". There must either be a point directly next to "1" (that is, the number that is as close to 1 as possible without being 1) or there is a hole in the line which mathematics cannot express. However, even in the latter case there still would be an answer, it is merely that the language of mathematics would be insufficient to describe it.

No matter what number you can think of, I can name another one with another nine added onto the end of it that will be closer to one, like if you said .999, I could say .9999 and we could keep playing that game for a very long time (and that's how there aren't any holes; the method you describe, with discrete numbers, would create holes.) Incidentally, there are also an infinite number of numbers between .999 and .9999. The thing about .999... repeating is that you can't add another nine onto the end of it because there are already an infinite number of them.

The Harlequin
06-12-2007, 00:09
.9999... (repeating) is a rational number.
So what's the fractional equivalent?

David Holtzman
06-12-2007, 00:11
The thing about .999... repeating is that you can't add another nine onto the end of it because there are already an infinite number of them.

Yes, exactly. But as has been proven in this thread, 0.999... is identical with 1. So, there must be some version of 0.99999... that is not an infinite series (because that would be 1) but is as close to 1 as you can get. I want to know what that number is and what distinguishes it from 0.999... which is 1.

Drec Sutal
06-12-2007, 00:33
Yes, exactly. But as has been proven in this thread, 0.999... is identical with 1. So, there must be some version of 0.99999... that is not an infinite series (because that would be 1) but is as close to 1 as you can get. I want to know what that number is and what distinguishes it from 0.999... which is 1.

Infinity minus one is meaningless. .999... has infinite nines. Any smaller number would have a finite number of nines, to which you could add an additional nine making it larger but still smaller then 1(or .999...)

Infinity isn't a really big number. It isn't the biggest number. It means the nines go on literally FOREVER. You just can't wrap your mind around that concept, it seems to me.

.999... is equal to one. Because the number line is continuous, any number smaller then it has infinity numbers that are larger and still smaller then one.

There is no answer to your meaningless question. You would have a point if .999... was less then one.



.9999... (repeating) is a rational number.So what's the fractional equivalent?

1

The Avatar
06-12-2007, 00:34
So what's the fractional equivalent?

1, which is a fraction if u put it as 9/9, 3/3 or w/e. by definition, 1 is an interger (a real number) not a fraction. however, any interger can be broken down into fraction.

1/9 = .111111...
2/9 = .222222...
3/9 = .333333...
4/9 = .444444...
5/9 = .555555...
6/9 = .666666...
7/9 = .777777...
8/9 = .888888...
9/9 = 1

if u have a graphing calculator, 8/9 = .88888888888888888889 then it stops cuz it cannot go any further sig fig, thus it rounds up.

@David: an inifnity +/- any real number = undefine or infinite. since inifinity is not a number by mathematical definition.

Aiiane
06-12-2007, 00:36
The problem you're running into, David, is that you're attempting to examine an infinitely precise number in a finitely precise sense. You can't pick out a specific point on a line that is infinitely close to another number, because a distance on a line is by definition a finite object. "Infinitely close" is in actuality simply saying "equal to", because [lim(x-1/a) as a->infinity] is x. Limits are the only way to examine "infinite" concepts in a finite sense.

So yes, there is a hole in the gab between finite and infinite mathematics that cannot express what you're considering.

===========================================

The difference between .9(bar) and sqrt(2) or pi is that for .9(bar), I can define via a finite statement every digit in its decimal representation:

D(n) = 9 for any given decimal place n.

For pi, sqrt(2), or any other irrational number, I cannot construct a finite function to express the content of any given decimal place. I could write a 100-line function to express the first hundred digits of pi, but it'd break down before I got to the 1000th, and so on.

This is why 0.9(bar) is rational, while sqrt(2) and pi are not.


On a side note, there's actually a very handy trick:

If you have a decimal that repeats after N places, and you wish to find out its fractional equivalent, simply take the repeating digits and divide by N 9's.

1/7 = 0.142857142857142857....
142857/999999 = 3*3*11*13*37 / 3*3*7*11*13*37 = 1/7

2/13 = 0.153846153846....
153846/999999 = 2*3*3*7*11*37 / 3*3*7*11*13*37 = 2/13

This "trick" can be proven to be correct by the same method as the demonstration that .9(bar) is equivalent to 1.



Also, it has been proven that any rational fraction of form A/B will have its decimal representation repeat a sequence of no more than B digits.

Lucis
06-12-2007, 00:58
So, it's like when you draw a parabolic curve? The line gets closer and closer to being parallel with they y axis, but it never is. Or is that an incorrect analogy?

The Avatar
06-12-2007, 01:02
So, it's like when you draw a parabolic curve? The line gets closer and closer to being parallel with they y axis, but it never is. Or is that an incorrect analogy?

u mean like the graph y = 1/x? it approaches to 0 but never touches it, since 1/0 = inifinity.

but the analogy is incorrect because number line & graph have different uses & meaning.

number line generally uses as greater, less than or equal to a certain number. it does not define limits or show how a number approaches in infinite.

a graph shows the progression of a certain object or behavior. in calculus, u calculate the limit of a curve to see where it ends up to.

David Holtzman
06-12-2007, 01:46
Infinity minus one is meaningless. .999... has infinite nines. Any smaller number would have a finite number of nines, to which you could add an additional nine making it larger but still smaller then 1(or .999...)

So then, you agree that there is a hole that mathematics cannot fill? Namely. the number that is as close to 1 as is possible without being 1. It is not that I do not understand the concept of infinity, it is that you don't seem to understand what I am asking. I recognize that 0.999... is equal to one. I have said so many times now. Just because I am posting in a thread resurfaced by people who do not believe it does not mean I share their views. After all, you're doing the same and obviously you don't share them. I have a related but entirely distinct question. Namely, again, what is the number that is closest to 1 without being 1?

************************************************** ********


@David: an inifnity +/- any real number = undefine or infinite. since inifinity is not a number by mathematical definition.

I agree, however 1 is a number. 0 is a number. Adding to 0, we must eventually reach a place where to add any more is to become 1, but the X where we stand is not yet 1. It is the description of that place I am interested in. And, this place must exist because otherwise I can do a really fun syllogism that proves 1=0. Since this is by definition not true, the converse must be. To have the converse be true such a space as I am speaking of must necessarily exist.

************************************************** ********


So yes, there is a hole in the gab between finite and infinite mathematics that cannot express what you're considering.

If this is true, then the followup question comes into play. Is this hole merely a relic of the language of mathematics, or is it an actual metaphysical hole?

MasterNightfall
06-12-2007, 02:10
So then, you agree that there is a hole that mathematics cannot fill? Namely. the number that is as close to 1 as is possible without being 1.
Yes, there is a hole, if you want to think of it as such (Which you seem to be doing because you cannot quite grasp the difference between discrete and continious sets of numbers). It exists because it is completely useless, both for theoretical and applied mathematics. Also, because anything that would fill that hole would not be well-defined, which is useless to mathematics.

And lastly because no matter what number you choose, I can choose a number that's closer. When you define "Z" to be the closest number to 1 that is not 1, I can always point at (1-Z)/2 + Z, and say that it's closer. Hence, your initial definition is impossible. Hence, a contradiction forms. There is no way to specify a closest number to 1 that would not be self-contradictory, and hence, useless.


And, this place must exist because otherwise I can do a really fun syllogism that proves 1=0. Since this is by definition not true, the converse must be.

Wrong. You cannot do a "really fun syllogism that proves 1=0".

Trust me on that one. I strongly suggest that you don't waste your time trying. It can't be done.

The Harlequin
06-12-2007, 02:23
1
Yes, yes...I knew that would be the response if for no other reason than 3 x .3 repeating = .9999 and therefore 1.. The flaw in my line of questioning became obvious to me later: if .9 repeating were irrational, the answer would be ambiguous.

Still, I will cogitate on this puzzle in my copious OCD time...

Lord Dragon
06-12-2007, 02:26
If this is true, then the followup question comes into play. Is this hole merely a relic of the language of mathematics, or is it an actual metaphysical hole?

I just had a nice long talk with my Uncle David. Thanks for this thread people I haven't talked with him a a few years. (I know bad nephew!) Ok Drec Sutal You are correct from a mathematical perspective that 0.999r is technically equal to one. This is based upon following the rule that you cannot have anything after a repeating series (no 0.999....01 allowed). I will completely agree with the proof you presented as I have found many much more in depth ones and discussed a few with My Uncle.

Now on to Holtzman's "hole" It exists too. After talking with my Uncle we must come to the realization that to allow infinitely big we must allow for infinitely small. Thus 1-0.999r != 0 but some number that is as infinitely small as 0.999r is infinitely close to approaching 1. To discount this infinitely small number throws out the concept of infinity and as such it can no longer be applied to the infinite 0.999r

This is a hole in what we use to define numbers in the first place and is inherent to the system. We create and work with limits such as the one that gives us 0.999r=1 to get around these holes.

Drec Sutal
06-12-2007, 04:43
1-.999r is zero. Work it out as a geometric series. 1-9/10^1-9/10^2-9/10^3... The "leftovers" is 1/10^(infinity). Which is zero.


So then, you agree that there is a hole that mathematics cannot fill? Namely. the number that is as close to 1 as is possible without being 1. It is not that I do not understand the concept of infinity, it is that you don't seem to understand what I am asking. I recognize that 0.999... is equal to one. I have said so many times now. Just because I am posting in a thread resurfaced by people who do not believe it does not mean I share their views. After all, you're doing the same and obviously you don't share them. I have a related but entirely distinct question. Namely, again, what is the number that is closest to 1 without being 1?

You aren't listening. The number line is continuous. There *IS NO NUMBER* that is as close as possible to 1 while still being less. Take *any* number less then 1, average it with 1, and you'll find a bigger number that is less then one.


I agree, however 1 is a number. 0 is a number. Adding to 0, we must eventually reach a place where to add any more is to become 1, but the X where we stand is not yet 1.

No. This isn't true. A mathematical number can always be divided in half. You might add 1/(1 followed by 1 billion zeroes) but after that you can add 1/(1 followed by a trillion zeroes). This goes on literally forever. It isn't that hard a concept - the number line is continuous, not discrete.


It is the description of that place I am interested in. And, this place must exist because otherwise I can do a really fun syllogism that proves 1=0. Since this is by definition not true, the converse must be. To have the converse be true such a space as I am speaking of must necessarily exist.

Look, if it helps think of it this way. There are two ways to write any given number. Either the traditional way (like "1" and ".125156") or by taking the last number, decreasing it by one, and adding an infinite number of nines after it (like ".999..." or ".125155(999)..."). .999... is EQUAL TO one, it isn't "just a tiny bit less but functionally equal." And so there is nothing that is "just a tiny bit less but functionally equal" to .999... and so you can't do your silly 0=1 thing.


If this is true, then the followup question comes into play. Is this hole merely a relic of the language of mathematics, or is it an actual metaphysical hole?

Its an actual hole. The finite and the infinite are irrevocably separated. No matter how high you count you'll never reach infinity, and if you divide infinity into a trillion trillion trillion parts each one will still be infinity. The 13 billion year age of the universe isn't even a blink of the eye to "forever." The incredibly ten thousand years at the speed of light it takes to cross the galaxy, and is insignificant to the billions of light-years it would take to cross the universe, and both are effectively the same as zero compared to an infinite distance.

Now, taking that into consideration... there are infinite numbers between 0 and 1 on the number line. That's what continuous means. There are also infinite numbers between .99 and 1, because that's what infinite means. There are infinite numbers between .9 (followed by a billion nines) and 1. But .999... is equal to 1. Because the difference between "an unimaginably huge number" and infinity is is simply unreachable.

A billion isn't infinity (9 zeroes). A billion billion (1 billion zeroes) isn't infinity. A billion billion billion (10^81) isn't infinity. The number of stars in the sky isn't infinity. The number of atoms in the galaxy isn't infinity. It wouldn't be accurate to say that the number of fundamental particles in the universe is microscopic compared to the red giant star of infinity, because infinity isn't even a number.

zabadoohp
06-12-2007, 05:03
Hello Everyone hahahahaaa

Drec Sutal
06-12-2007, 05:09
Exactly. Until 4 days from now when I launch it. This is the closest I have seen to the answer. To all my "scepticists" (see how much I care about my spelling) when I finally show you how obvious this is, your going to go nuts on me. To be frank. I look forward to it. If I was completely off and .999......... forever really did = one; then why is the mere mention of a solution enough to spark this large of a debate. If it was real it would be widely accepted as real. It is a theory. Theories are believed to be real. They are BELIEVED to be real. Think about this. I did not make up any new numbers. It is still on a 10 number system. It is just more descriptive. This is a simple concept and should have been thought of before. It already exists in one of the most basic forms of math currently in use. We currently describe numbers everday with this basis. It is not like the controversy over complex numbers or otherwise known as "Imaginary numbers". It is quite simple and can be learned in less than 5 minutes. I am planning to publish, yes publish with JAMS or another Journal. I do use correct English on the final draft. Using slang and other speech is to announce it. I am a gamer / computer guru also. I own a Wii/Gaming PC and love World of Warcraft. This is one of the reasons why I am on here, and also the polymathematics forum.
I am just a normal kid with some, not an infinite, but some experience with high level math.
4 days til launch.

This launches a debate every time it is brought up because .999... = 1 is counter-intuitive. It's still true though. David appears to not understand the concept of a continuous number system or the concept of infinity. I suspect you don't either, which would make your closing statement mildly ironic. I look forward to laughing at you in four days. Even though you appear to still misunderstand the publishing process.

Manwithtwohands
06-12-2007, 05:38
You humans and your infinity nonsense.
Can your kind honestly not count that high.

/me floats into dimension X

David Holtzman
06-12-2007, 09:57
No. This isn't true. A mathematical number can always be divided in half. You might add 1/(1 followed by 1 billion zeroes) but after that you can add 1/(1 followed by a trillion zeroes). This goes on literally forever. It isn't that hard a concept - the number line is continuous, not discrete.


How do you reconcile this view with the geometric notion of the point? As I understand it, the point is extensionless in that it cannot be further divided. There is no such thing as "half a point." Lines, also according to what I understand of geometry, are made up of an infinite number of such points. The number line is a line. So, if there is a point on that line identical to the number 1, then there must either be a point immediately next to it (which I refer to here) or my definition of a line must be confused. What you seem to be telling me is that points do not exist, yet this seems to be in direct conflict with other branches of mathematics. I confess to not understanding how both these positions can be correct when they seem to be mutually exclusive.

Mister Smartypants
06-12-2007, 10:07
As far as I understand this (not very), there's no such thing as a point immediately next to any other point on a line of points infinitely long. There's always room for an infinite number of points between any two points on the line, even two points that to all intents and purposes from a human (as opposed to a mathematical) perspective are indistinguishable from each other.

Lensor
06-12-2007, 17:15
an irrational number is like pie

Mmm, pie...:azn:

Ok, ok, im sorry for intruding on all the serious math talk, but I just laughed out loud on that one.

Drec Sutal
06-12-2007, 18:20
How do you reconcile this view with the geometric notion of the point? As I understand it, the point is extensionless in that it cannot be further divided. There is no such thing as "half a point." Lines, also according to what I understand of geometry, are made up of an infinite number of such points. The number line is a line. So, if there is a point on that line identical to the number 1, then there must either be a point immediately next to it (which I refer to here) or my definition of a line must be confused. What you seem to be telling me is that points do not exist, yet this seems to be in direct conflict with other branches of mathematics. I confess to not understanding how both these positions can be correct when they seem to be mutually exclusive.

>.<

A point takes up no space, it is zero dimensional. As it doesn't take up any space, you can't divide it. You can also fit infinity of them in any given space, plane, or line. A point doesn't has a length, width, and depth of exactly zero.

Lucis
06-12-2007, 19:04
>.<

A point takes up no space, it is zero dimensional. As it doesn't take up any space, you can't divide it. You can also fit infinity of them in any given space, plane, or line. A point doesn't has a length, width, and depth of exactly zero.

I'm guessing that somes up tangents off circles too.

The Avatar
06-12-2007, 19:15
Mmm, pie...:azn:

Ok, ok, im sorry for intruding on all the serious math talk, but I just laughed out loud on that one.

okay is pi, yet pie still resemble to circle.

David Holtzman
06-12-2007, 21:26
A point takes up no space, it is zero dimensional. As it doesn't take up any space, you can't divide it. You can also fit infinity of them in any given space, plane, or line. A point doesn't has a length, width, and depth of exactly zero.

I know, that's what "extensionless" means. What I am asking is (framed differently but encompassing the same content): how can any line be made of any amount of unextended points? Talking about the number immediately next to 1 is the same thing as talking about the point immediately next to some specified other point. Because, if "1" is a point on the line, and lines are nothing more than an infinite series of points, then there must be a point next to that of the point we designate as "1."

Alternatively you could reject the notion that lines are made up of a series of points. I honestly have no idea what this would do to geometry, so maybe it is fully tenable. But, if you do reject this notion then it seems you also have to reject the number line. How can there be any place identical with "1" if there are no points to reference that concept?

Drec Sutal
07-12-2007, 00:01
You still don't understand the concept of continuous vs discreet. Let me explain it to you.

Here is the number line from zero to one, with 0, 1, and .99 marked...

http://i6.photobucket.com/albums/y247/drec/01.jpg

Here's a closeup of the very end. It's the number line from .99 to 1 with .9999 marked.

http://i6.photobucket.com/albums/y247/drec/99.jpg

Here's a closeup of the very end. It's the number line from .9999 to 1 with .999999 marked.

http://i6.photobucket.com/albums/y247/drec/9999.jpg

Are you starting to get the idea? I can keep doing that forever. FOREVER. No matter how many times I do it I can always just look at a smaller scale and find a closer number. *ALWAYS*. That's what continuous means... you can never get down to a set of "points" laid end to end to make your line. There are infinitely many points between any two numbers. A segment of a "line" is a collection of *all* the points between two given points. Yes, all infinity of them. Both points and lines are fictional constructions anyway so I can do that.

zabadoohp
07-12-2007, 00:07
Hello everyone hahahhaaaa

The Avatar
07-12-2007, 00:37
Even if the publishing process does take a month. I am still launching it in 3 days. This only means that I will be submitting it to a Journal, and sending a copy to anyone who has given me an email to send it to.
3 days til launch

i wonder how many people actually look into ur journal or even pay attention to ur email address.

mind saying wat are ur trying to prove here? because i have no idea wat ur previous post pointing at.

B Ephekt
07-12-2007, 01:41
I'm surprised this thread, on a stupid meme from like 5 years ago, actually made it to 3 pages. :shocked:

draugaer
07-12-2007, 04:18
well it really is just a problem of logic...

I mean .9999 is never as much as 1

take .99999.....% of something for example, there will always be something left even if it infinitely small.

problem is with the way our math works you can't really prove otherwise.

Drec Sutal
07-12-2007, 04:34
well it really is just a problem of logic...

I mean .9999 is never as much as 1

take .99999.....% of something for example, there will always be something left even if it infinitely small.

problem is with the way our math works you can't really prove otherwise.

It is not a problem of logic, it is a problem of math. Which is why you're wrong. Read the freaking thread for half a dozed proofs that .999... is equal to one. (counter to your claim that "you can't really prove otherwise")

B Ephekt
07-12-2007, 04:44
Wait, why am I even replying to this?

Delete this please.

Manwithtwohands
07-12-2007, 05:10
Decimal places are only used by species that can't even define yet what the lowest possible value of something is.
The space council is laughing at earth.
/me flies into dimension Y and hits a space wall

MasterNightfall
07-12-2007, 09:22
How do you reconcile this view with the geometric notion of the point?

A point can represent a number. When it does so, that point only represents that number. No other number. It cannot overflow into adjacent numbers. It takes up an infinitely small amount of space.


then there must either be a point immediately next to it
There's an infinite number of points immediately next to it.



(which I refer to here) or my definition of a line must be confused.
It is. A line isn't a discrete sequence of points. It is continious. Any space on it is occupied by an infinite number of points. There cannot be a closest point to 1, because there is an infinite amount of points between that point and 1. And so on.



What you seem to be telling me is that points do not exist,

No, what we are telling you is that points occupy an infinitely small, or zero space on the number line.



yet this seems to be in direct conflict with other branches of mathematics. I confess to not understanding how both these positions can be correct when they seem to be mutually exclusive.

Your understanding of points and the number line is confused.


I know, that's what "extensionless" means. What I am asking is (framed differently but encompassing the same content): how can any line be made of any amount of unextended points?

It can, if that 'amount' is infinity.

Here's an exercise for you, assuming you know what a limit is:

What is:

Lim(n-->infinity, n * (1/n))?

You can consider that to represent all the points between 0 and 1 on the number line.

1/n is the size of each point.
n is the number of points.

And by the way, the answer to that question is 1.

Even though n approaches infinity, and 1/n approaches zero.

You can fit an infinite number of points taking up an infinitely small space on a finite space.

There's nothing contradictory about it.



Talking about the number immediately next to 1 is the same thing as talking about the point immediately next to some specified other point.
Which cannot happen because I can always name a closer point.



Because, if "1" is a point on the line, and lines are nothing more than an infinite series of points, then there must be a point next to that of the point we designate as "1."

Not if each point is infinitely small.

You still do not understand the difference between discrete and continious math.



Alternatively you could reject the notion that lines are made up of a series of points.
No, alternatively, you could reject the notion that each point takes up a non-zero space.

Aiiane
07-12-2007, 09:25
Actually, David, the answer to your question is quite simple:

The term "closest" is not defined in infinite mathematics for non-countable* infinities. The concept of "close" is only viable for finite bounds - i.e. I say X is close to Y if and only if the distance between X and Y is less than Z, where Z is a discrete non-zero value. With non-countable infinities (such as the set of real numbers) however, there is no discrete minimum value for Z such that we could have a "closest".

Finite mathematics and infinite mathematics are two separate realms with what works in one not always working in the other.


* For those not familiar with infinite mathematics, "countable" infinities are those can be be mapped to the set of natural numbers, while "non-countable" are those which cannot be. For instance, the set of integers is a countable infinity, because we can simply map 0=0, 1=+1, 2=-1, 3=+2, 4=-2, and so on. The set of real numbers, on the other hand, is not a countable infinity as there is always a smaller interval with which we could start to map.

Katy K
07-12-2007, 23:18
So then, you agree that there is a hole that mathematics cannot fill? Namely. the number that is as close to 1 as is possible without being 1. It is not that I do not understand the concept of infinity, it is that you don't seem to understand what I am asking. I recognize that 0.999... is equal to one. I have said so many times now. Just because I am posting in a thread resurfaced by people who do not believe it does not mean I share their views. After all, you're doing the same and obviously you don't share them. I have a related but entirely distinct question. Namely, again, what is the number that is closest to 1 without being 1?


Rational numbers are dense so you can always find a rational number between two other rationals. This also applies to real numbers.

Suppose we have a real number x closest to 1 with x < 1. Real numbers are dense so we can find another real number z such that x < z < 1. This means z is closer to 1 and we have a contradiction, therefore no such number closest to 1 exists.

On the case of hole in our number line. When we worked wth rationals there certainly were "holes". The fact that there are sets that do not have a supremum was unacceptable.
The real numbers were made to "fill the holes" and so real numbers are complete.

Katy K
08-12-2007, 00:20
As for 0.99999... = 1

http://img363.imageshack.us/img363/3880/proofhv8.gif

Drec Sutal
17-12-2007, 21:01
Even if the publishing process does take a month. I am still launching it in 3 days. This only means that I will be submitting it to a Journal, and sending a copy to anyone who has given me an email to send it to.
3 days til launch

It has been 11 days. Math has yet to be turned on its head, and google still doesn't know this guy. I win by default! Also: because he didn't bother to post his silly theory here I maintain that my earlier deduction that it was using a different base then ten was correct and he didn't even bother trying to get published because he realized I was right and his idea was useless and far from new.

Akirai Annuvil
17-12-2007, 21:40
It has been 11 days. Math has yet to be turned on its head, and google still doesn't know this guy. I win by default! Also: because he didn't bother to post his silly theory here I maintain that my earlier deduction that it was using a different base then ten was correct and he didn't even bother trying to get published because he realized I was right and his idea was useless and far from new.

I have this strange feeling you counted down the number of days till it'd be published, then waited another week, and then decided to post and gloat.

Though I too am hoping to see his book. :fortuneteller:

Blastsniper
17-12-2007, 22:54
What about this one,

20/20= 1
2/2 = 1
1/1=1
0/0=1?

Fish On Fire
17-12-2007, 23:38
What about this one,

20/20= 1
2/2 = 1
1/1=1
0/0=1?

Can't divide by 0 :shocked:.

Anyway, check this out..
(assume that whenever I say .9 I mean .999repeating)

First, set x=.9
this being said, 10x=9.9

Subtract the original equation from the new equation:
10x=9.9
-
x=.9

9x=9, therefore x=1.. And .9

Further proof:
.3333....=1/3
.6666....=2/3
.9999....=3/3
3/3 = 1

sorudo
18-12-2007, 18:26
no mather how many .9 you use, you still are missing a 1.
so even if you make it .9999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999
it's still no 1, you are still missing one.

Drec Sutal
18-12-2007, 21:27
no mather[sic] how many .9 you use, you still are missing a 1.
so even if you make it .9999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999
it's still no 1, you are still missing one.

That's why you don't use a number of nines. You need infinity. Read the freaking thread, and try to learn what 'infinity' means.

Manwithtwohands
19-12-2007, 04:39
The universe doesn't like infinity.
It never allows it to happen.
To prove it, I show no proof, beyond theoretical math, that infinity even exists.
/me opens hands to reveal nothing
Ta-da!

I'm so silly. Infinity doesn't exist. Boop!

Feannag
19-12-2007, 15:46
The universe doesn't like infinity.
It never allows it to happen.
To prove it, I show no proof, beyond theoretical math, that infinity even exists.
/me opens hands to reveal nothing
Ta-da!

I'm so silly. Infinity doesn't exist. Boop!

http://img444.imageshack.us/img444/7581/thevillagersarerestlessbi3.gif (http://imageshack.us)
There's some mathematicians here who would like to have a word with you.

Azrael STX
19-12-2007, 16:30
I'm an ultrafinitist. This thread is totally baseless.

Manwithtwohands
18-04-2008, 10:38
http://img444.imageshack.us/img444/7581/thevillagersarerestlessbi3.gif (http://imageshack.us)
There's some mathematicians here who would like to have a word with you.

I just reread what I said, and have absolutely no idea what I meant back then.
I think I am too random for my own good.

Skyy High
18-04-2008, 14:41
The question is if you had any idea what you meant at the time.

Manwithtwohands
18-04-2008, 14:52
The question is if you had any idea what you meant at the time.

Ahhh. It was after playing Assassin's Creed. The quote "nothing is true, everything is permitted" got me to thinking about stuff. Apparently Hasaan I Sabah (http://en.wikipedia.org/wiki/Hassan-i-Sabah), uttered it before death claiming it as the greatest secret ever to be known by men.

After thinking about it some more, I believe there is an infinity.
Weird.

mocax
19-04-2008, 06:47
Are we going for the 4th year anniversary of this thread?

Manwithtwohands
19-04-2008, 07:11
*Eerie voice*

It is eternal. Wooooooo.
And now it has multiplied into 2. Even more woooooooo.

gervasium
19-04-2008, 09:08
Back in december I told my teacher about this, and she got really confused.
Then I showed it to my father and he told me that 0.999999x10 is 9.99999999999... but since you moved the coma to the right simply, or the numbers to the left, you still have the same number of nines (and yes, I know it's infinity). So, when you take x away, it would work like this (if it weren't an infinity number): 9.999-0.9999 which equals 8.9991, and not 9. That's because you have an extra nine on the right of the coma. Of course when dealing with an infinite number, you'd think this wouldn't apply, since there's an infinite number of nines in both cases right of the coma. However, on one of them, you have infinite and on the other you have infinite - 1 , since every nine moved to the left when multiplying by ten. It might be a hard concept to grasp, but I've learned it when, three years ago I made a fool out of myself when stating, in maths class that infinite+1 equaled to infinite (It is wrong).

Now, about this:
20/20=1
5/5=1
0/0=1
When you divide any number in 0 parts, you get 0 parts of that number, which means you get no part, which means 0. Any number dividing by 0 equals 0.

Aiiane
19-04-2008, 09:54
Back in december I told my teacher about this, and she got really confused.
Then I showed it to my father and he told me that 0.999999x10 is 9.99999999999... but since you moved the coma to the right simply, or the numbers to the left, you still have the same number of nines (and yes, I know it's infinity). So, when you take x away, it would work like this (if it weren't an infinity number): 9.999-0.9999 which equals 8.9991, and not 9. That's because you have an extra nine on the right of the coma. Of course when dealing with an infinite number, you'd think this wouldn't apply, since there's an infinite number of nines in both cases right of the coma. However, on one of them, you have infinite and on the other you have infinite - 1 , since every nine moved to the left when multiplying by ten. It might be a hard concept to grasp, but I've learned it when, three years ago I made a fool out of myself when stating, in maths class that infinite+1 equaled to infinite (It is wrong).

Unfortunately, you've managed to confuse yourself. For most standard definitions of infinity, "infinity + 1" is equivalent to "infinity" is equivalent to "infinity - 1". Thus, this:



Of course when dealing with an infinite number, you'd think this wouldn't apply, since there's an infinite number of nines in both cases right of the coma. However, on one of them, you have infinite and on the other you have infinite - 1 , since every nine moved to the left when multiplying by ten.


...is incorrect. Name me a decimal place, and I can tell you for sure that there is a 9 in it in both numbers.

There are two "types" of infinities: countable infinities and uncountable infinities. I'll refer to them as infC and infU respectively.

Countable infinities (infC) are infinities that can be uniquely mapped by a one-to-one function onto the counting numbers (1, 2, 3, 4, 5, ....). Examples of countable infinities are things such as the set of integers, the set of repeating decimals between 0 and 1, and the possible sequences of results from rolling a 6-sided die repeatedly.

Uncountable infinities (infU) are the opposite; they cannot be uniquely mapped to the counting numbers. The set of real numbers is a good example of this; for instance, if you tried to map the real numbers in increments of .1 to the counting numbers, I could simply show that .01 was not mapped to anything. No matter what interval you chose, there would still be a smaller possible one. Another example is the set of all improper fractions - whenever you chose to move onto the next denominator in your mapping, I could simply point out another numerator you missed.

Now that we have those definitions, we can take a look at how certain operations are defined in infinite math (notation: infC[x] means 'the element of a countability infinity mapped to the counting number x):

infC + finite = infC
We simply map 1,2,...N to the finite elements, where N is the length of the finite set, and then proceed to shift the mapping of the infC to start at N+1.

infC + infC = infC
Also easy to show: we map the first infC[1] to 1, then the second infC[1] to 2, then the first infC[2] to 3, and the second infC[2] to 4, and so on.

infC + infU = infU
Since we can't map the infU to the counting numbers, it stands to reason that we can't map all of the infU, plus even more, to the counting numbers either.

infU + infU = infU
By the same reasoning, neither can we map twice as many numbers to the counting numbers if we can't map the original set(s).

infU + finite = infU
Again, since we can't map one part, we can't map all of the parts put together.

We say that infU's are larger than infC's because any given infU contains an infinite number of infCs within it.

Note that all of this is addition. Another key concept of infinite mathematics is that subtraction of an infinity from something which is the same size not defined. That is to say:

infU - infU = undefined
infC - infC = undefined

However, we can subtract different size infinities, although it doesn't do much:

infU - infC = infU
infU - finite = infU
infC - finite = infC
infC - infU = -infU
finite - infC = -infC
finite - infU = -infU

This brings up another key point of infinite mathematics: Countable infinities are all the same size. The set of integers is exactly the same size as the set of all repeating numbers between 0 and 1. How do we know this? Because we have a one-to-one mapping between them! infC1[x] <-> infC2[x] for all x.

Thus, the number of 9's after the decimal point in .9(repeating) and 10*.9(repeating) is exactly the same, because they are both countable infinities.



Now, about this:
20/20=1
5/5=1
0/0=1
When you divide any number in 0 parts, you get 0 parts of that number, which means you get no part, which means 0. Any number dividing by 0 equals 0.
Division by zero is undefined in any sane mathematics system, for good reason.

Skyy High
19-04-2008, 13:40
Oh I'm so glad you did that Aiiane, much better than how I could have explained it.

Good news gervasium, whoever made a "fool" out of you 3 years ago was just making a fool out of themself.

Manwithtwohands
19-04-2008, 20:16
I heard we use multiples of 10 because we have 10 digits on our hands.
What if we had 8 digits on our hands?
Would .777...=1?
I think I understand a bit more now. It's only about the relationships between points and how infinity affects them.

For a creature with 6 digits that counts in multiplies of 6 .555...=1!

Skyy High
19-04-2008, 20:39
Pretty much. But as to why we generally use base 10 (not "multiples of 10") instead of base 2, 8, 16, or 60 (which was used by, iirc, the ancient Babylonians, and is the reason for our incrementation notation for units of time and angles)...no idea. The "10 finger" guess is as good as mine.

Manwithtwohands
19-04-2008, 20:47
Oh base. Not multiples.
lol. Now I understand what Aiiane was saying doing it using different base groups in Djinn's topic.

mocax
20-04-2008, 03:54
in 0.9999.... we use 10x to conveniently shift the number by 1 place and get the convenient position of 10-1=9

It's just a technique to proving. You can't apply it to arbitrary numbers.

if we use 10x on 0.777..... we'd get
10x - x = 7
x = 7/9 which can't prove anything.

Conversely, we can prove by contradiction that 0.7777.... will never equal one.

Manwithtwohands
20-04-2008, 04:27
The .777...=1 example I gave was me just be being silly.

It's believed we use base 10 systems because we have 10 digits on our hands. I was only pretending to be a creature with 8 digits on it's hand that counts 1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20 for random craziness fun.

Aiiane
20-04-2008, 05:30
in 0.9999.... we use 10x to conveniently shift the number by 1 place and get the convenient position of 10-1=9

It's just a technique to proving. You can't apply it to arbitrary numbers.

if we use 10x on 0.777..... we'd get
10x - x = 7
x = 7/9 which can't prove anything.


Actually, it proves that .777.... = 7/9. Which it does.

djacob
20-04-2008, 08:39
It's believed we use base 10 systems because we have 10 digits on our hands. I was only pretending to be a creature with 8 digits on it's hand that counts 1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20 for random craziness fun.

Now answer me this: How would something count with more than 10 numbers such as a hexadecimal system (note: they do not have the alphabet yet). Truly, I do not believe that they would use the symbols we use for any of the numbers. So taking that into account...

.{{{...=& yes, a crazy statement eh fellow 16 fingered friends?

Manwithtwohands
20-04-2008, 08:43
I bet 16 fingered aliens get more accurately paid for their work.

neoflame
20-04-2008, 12:49
Ugh, infinite decimals. Needs moar rigor!
http://img20.imageshack.us/img20/962/9eq1ou8.jpg

Monotheisticfreakizoid
20-04-2008, 13:30
I'll only add that 1+1=3
for unusually large cases of 1

djacob
20-04-2008, 16:43
I'll only add that 1+1=3
for unusually large cases of 1

Any mathematician worth their salt would have your head for rounding issues. :P

Drec Sutal
20-04-2008, 17:37
Now answer me this: How would something count with more than 10 numbers such as a hexadecimal system (note: they do not have the alphabet yet). Truly, I do not believe that they would use the symbols we use for any of the numbers. So taking that into account...

.{{{...=& yes, a crazy statement eh fellow 16 fingered friends?

I'm going to go ahead and assume a species with 16 fingers wouldn't independently develop arabic numerals. So they'd count:

1,2,3,4,5,6,7,8,9,a,b,c,d,e,f,10,11,12,13,14,15,16 ,17,18,19,1a,1b,1c,1d,,1e,1f,20

But with *all* the numerals and letters replaced with their own 16 unique numbers.

Celestial Kitsune
20-04-2008, 17:54
Well, since we talk about fingers... If you have trouble with basic multiplication, you can use your natural calculator ;)

http://www.youtube.com/watch?v=oamy8L2lDZM

Manwithtwohands
20-04-2008, 22:16
Well, since we talk about fingers... If you have trouble with basic multiplication, you can use your natural calculator ;)

http://www.youtube.com/watch?v=oamy8L2lDZM

That was fun to watch. :laugh:
I just like when he says to use your pre-installed calculator and then laughs.

Gaendaal
21-04-2008, 09:04
Is 9.1... = 9.2?

Manwithtwohands
21-04-2008, 09:07
I'm not good at math so I'm guessing it isn't.
Unless it was 9.199... then I'm guessing it is.

Aiiane
21-04-2008, 10:39
Is 9.1... = 9.2?

No.

9.1111..... = 9.0 + 0.1111..... = 9.0 + (1/9) = 81/9 + 1/9 = 82/9

9.2 = 92/10 = 46/5

82/9 != 46/5


But 9.1999999..... is indeed equal to 9.2, like so:

9.199999.... = 9.1 + (1/10)*(0.9999....) = 9.1 + (1/10)*(1) = 9.1 + 0.1 = 9.2

mocax
21-04-2008, 14:19
The trick only works wihth 9's

But most people try to apply it to any number.

Gaendaal
21-04-2008, 14:29
The trick only works wihth 9's

But most people try to apply it to any number.
Why? That seems counter-intuitive, to me. If the "trick" only works with 9s then it does seem like a "trick", a mathematical sleight of hand (and, yes, I know it can be proved mathematically, I'm not doubting that).

If 0.99999999... is equal to 1 why is, say, 0.88888888888... not equal to 0.9?

Aretelio
21-04-2008, 15:59
It has something to do with gas prices. (http://www.wisegeek.com/why-do-gas-prices-always-end-in-910-of-a-cent.htm)

Trust me, I'm an untrained amateur.

Alaris
21-04-2008, 16:19
Why? That seems counter-intuitive, to me. If the "trick" only works with 9s then it does seem like a "trick", a mathematical sleight of hand (and, yes, I know it can be proved mathematically, I'm not doubting that).

If 0.99999999... is equal to 1 why is, say, 0.88888888888... not equal to 0.9?

The trick works with any number, by substituting 0.11111 with 1/9 because the two are equal. It's not a rounding error, or magic trick. It's a substitution.

Let's see what 0.88888 is equal to.

0.88888 = 1/8 * 8 = 8/9.
0.9 = 0.9*9/9 = 8.1/9.

So 8/9 is not equal to 8.1/9

themagicmoedee
21-04-2008, 21:36
Why? That seems counter-intuitive, to me. If the "trick" only works with 9s then it does seem like a "trick", a mathematical sleight of hand (and, yes, I know it can be proved mathematically, I'm not doubting that).

If 0.99999999... is equal to 1 why is, say, 0.88888888888... not equal to 0.9?

It's not the represented value. Just like I can't call Bob Steve even though they kind of look the same.

In base 10 notation, any infinitely repeating single value (from 0.1 to 0.anything) is equal to X/9. If you translate 0.8... to base 9, it's just 0.8. It's not a trick - just an error of translation.

Lucis
21-04-2008, 23:10
If 0.99999999... is equal to 1 why is, say, 0.88888888888... not equal to 0.9?

Can you think of a number between them? Yes? Well then they are not the same number.
However it would work with 0.8999999999....

Celestial Kitsune
22-04-2008, 05:42
You people got it all wrong! 1 + 1 = 3 Tachibana Muneshige won battles with a smaller group than the enemies due to good strategy. Even if the soldiers have the same skill, you can't judge the result of the battle by simple addition. Thus, according to different strategies, 1+1 can be equal to 3 or 5 or even 10! :P

Lucis
22-04-2008, 23:00
I have a question.
Is 0.999.... recurring the same number as 1, or does it just behave in the same was as 1 in most situations?

Skyy High
22-04-2008, 23:10
It is equal to one. It will "behave" the same way as one in all situations, since they are, again, equal.

Aiiane
23-04-2008, 10:56
I have a question.
Is 0.999.... recurring the same number as 1, or does it just behave in the same was as 1 in most situations?

It's a different way of writing the same number.