2=1.999...infinity! What are your thoughts?

[sjjordan]

I'm having a great discussion with a friend about the following...

1. if x=1.999...infinity, then 10x=19.999...infinity. Subtracting 10x from x gives 9x=18. Dividing both sides results in x=2.

2. Using your calculators on this one. 1/9=0.111...?. 2/9=0.222....?. 3/9=0.333...?. 4,5,6,7,8 follow the same pattern. What, then is 9/9? It is 0.999...infinity.

Do you think those are valid? I think they are and thinking in infinite terms 1.999...infinity is equal to 2.

My friend thinks he can add any number to infinity because he just can.

Food for thought. Any you want to add?

[Stelliform]

I won't get into proofs here, but I am certain someone on this board may feel the urge.

Basically 2.00000000 isn't equal to 1.999999999 to infinity... they are very close, but not equal. I think your first argument has a flaw, but I think you are breaking some proof law I don't know about.

If your calculator is indicating anything different there might be a flaw in its logic.

[jxyama]

you have to be careful when doing arithmetics with infinite decimals...

that said, i think mathematically, 2 is equal to 1.99999.... depends on the definition of equal. i believe a little more technical way to say two numbers, say N1 and N2, are equal is to show that no matter how small of a number you choose, you can make the difference of N1 and N2 smaller than that number. and in this case, by extending the decimals far enough, you can make the difference between 2 and 1.9999... smaller than any possible number.

math is kinda convoluted... it's been a while since i've had to think about this stuff. (i majored in math in college, but that was more than 5 years ago.) any active mathematicians out there?

[zapp]

Originally posted by sjjordan
I'm having a great discussion with a friend about the following...

1. if x=1.999...infinity, then 10x=19.999...infinity. Subtracting 10x from x gives 9x=18. Dividing both sides results in x=2.


x=1.99... is an irrational number, a number that cannot be expressed as a fraction. Thus cannot be used as a true value for x.

On number 2 you 1/9 = .11.. ets, they are rational numbers. Of course 9/9 =1 The division you use shows the standard conversion that happens in division. It looks cool though and did get my rather slow brain working again. Thanks, Now to ponder the meaning of life.

[trashyspaceman]

You're saying that 2 == 1.999...
That is incorrect (but approximately true).
What is more correct is to define 1.999... as being (2 - 1/n), where n tends to infinity.

1/n does approach a limit (0), so
2 - 1/n approaches 2, for an arbitrarily large n.
(You could write it as:
2 - 1/n --> 2, n --> infinity)

You're discussing limits here, which is 1st year university mathematics.

The definition of a limit is:
"For any |1/n - 0| arbitrarily close to 0, there exists such an n"

(arbitrarily close ~= as close as you like)

-matt

[mangoduck]

Originally posted by jxyama
i think mathematically, 2 is equal to 1.99999.... depends on the definition of equal.
...
by extending the decimals far enough, you can make the difference between 2 and 1.9999... smaller than any possible number.
...
math is kinda convoluted...

definition of equal? smaller than any possible number? math is convoluted?

1, equal means equal.
2, impossible, there is always something smaller.
3, math is the language of pure and universal logic.

[Dippo]

You falsey assume that 10x - x = 9x.

While this may work for real/complex number, it won't for infinite numbers.

If you multiply both 10x and x by the same "size" infinity then you will get:

x times infinity = 199999...99999.0
10x time inifinity = 1999999...99990.0

You can't say that x infinity minus 10x infinity is 18 infinity because as you get to infinity the 10x will be just a bit smaller.

I guess this doesn't make any sense, but infinite numbers never do :)

[jxyama]

Originally posted by mangoduck
definition of equal? smaller than any possible number? math is convoluted?

1, equal means equal.
2, impossible, there is always something smaller.
3, math is the language of pure and universal logic.

1. what does "equal means equal" mean? that doesn't define anything!

2. many definitions within math use limits. the point is, the difference can be made smaller than any arbituary (but fixed) number. i didn't say that the difference is the smallest number.

3. you left out an obvious but important adjective: math is the language of pure and universal human logic.

[XnavxeMiyyep]

Math itself is not a problem. However, decimals are only measured in powers of 10, therefor fractions that do not divide into powers of 10 cannot always be displayed completely accurately in decimals. I'm pretty sure that there's no actual way to get to .99999_, as all multiplications, additions, etc. of decimals/fractions would come out to 1 instead of .99999_



Edited for grammar

[Dippo]

I am no longer going to work on this problem, it is making my head hurt really bad.

[isus]

1.99999999999 does not equal 2.

otherwise it would be 2.

calculators just do that crap because they can't handle big numbers.

imagine your calculator trying to do the sine of .4444444444444.

it would burst into flames.

[betta]

1.999... is the same number as 2, just written differently. Take any positive number and the absolute difference between 2 and 1.999... (that is, |2 - 1.999...|) will be less than that number, which can be true only if they are the same number. Some numbers have more than one decimal expansion.

Put it differently, The decimal expansion 1.999... represents a limit of a sequence of numbers that each can be represented as a finite decimal expansion. 2 is also a limit of that sequence, and since a convergent sequence can only have one limit, the two numbers are the same.

[Powerbook G5]

This is why I am studying law, it doesn't make my head want to implode like math does.

[Veldek]

Originally posted by betta
1.999... is the same number as 2, just written differently. Take any positive number and the absolute difference between 2 and 1.999... (that is, |2 - 1.999...|) will be less than that number, which can be true only if they are the same number. Some numbers have more than one decimal expansion.

Put it differently, The decimal expansion 1.999... represents a limit of a sequence of numbers that each can be represented as a finite decimal expansion. 2 is also a limit of that sequence, and since a convergent sequence can only have one limit, the two numbers are the same.

As I studied math, I will only say that all the guys who said the above are right.

[virividox]

all i know is i can have
1 car, 2 cars, 3 cars
1 computer, 2 computer, 3 computer
but no 1.99999999999, not 2-1.99999 n2, not y=x^-p(cos R2D2)

[grabberslasher]

Aah, but 1 = 2!
How?

Well using the rules for algebra do the same thing to both sides (divide by zero).

1 = 2

: 1/0 = 2/0

so therefore infintity = infinity

[kevin49093]

But you can't divide by zero!

[kevin49093]

Originally posted by jxyama

3. you left out an obvious but important adjective: math is the language of pure and universal human logic. [/B]

Have you guys ever read anythign into this http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html ?

[leo]

The notation 1.9999... is defined as the limit of the series. This limit is 2. Period.

Please google for "p-adic numbers" or "p-adic expansion".

[Roger1]

I can count to 10 on my fingers. Therefore 10=10. If 10=10, then 2=2.
:)

[jxyama]

i believe the explanation that those two numbers are the "same" just different representations, as posted by others. that's something i remember from my days in math.

all this discussion about 2 = 1.999... reminded me of one of the most important things i learned in physics...

i understand that some of us, not having studied math vigorously, would be uncomfortable that the number "two" can have more than one representation...

the story i remember is the question of "is light wave or particle?" people have problems because they consider wave and particle to be two distinct things and can't imagine things that are both... when in fact, light is both, in the sense it will display wave like properties under certain conditions and particle like properties under another.

we are most comfortable with the number "two" represented as 2. however, that doesn't mean the number "two" is restricted to being represented by the symbol "2"... no?

[Powerbook G5]

I did a science fair project on the particle-wave theory of light. I had a pretty nifty demonstration with flashlights, a cookie sheet, and a cardboard box with a few holes cut into it. Funny thing was, I just did it to be lazy since it was such an easy project and I hated doing science fair stuff and it somehow impressed my teacher and I got stuck going to the state science fair and wasting 5 days bored out of my mind.

[Dros]

Originally posted by jxyama
i believe the explanation that those two numbers are the "same" just different representations, as posted by others. that's something i remember from my days in math.

all this discussion about 2 = 1.999... reminded me of one of the most important things i learned in physics...

i understand that some of us, not having studied math vigorously, would be uncomfortable that the number "two" can have more than one representation...

the story i remember is the question of "is light wave or particle?" people have problems because they consider wave and particle to be two distinct things and can't imagine things that are both... when in fact, light is both, in the sense it will display wave like properties under certain conditions and particle like properties under another.

we are most comfortable with the number "two" represented as 2. however, that doesn't mean the number "two" is restricted to being represented by the symbol "2"... no?

But light acting as a particle or wave doesn't mean that it is a particle and a wave, but just that we are unable to describe light properly using these crude terms.

Similarly, I agree the limit of 1.999... is 2. But 1.999... is an irrational number, 2.0 is not. So under some systems of math, they may be considered equivalent. But other terminologies have been developed to describe the differences between these two numbers. Ancient humans probably counted 1, 2, many. So in their view, 10 deer and 12 deer are the same. But we have better systems for counting, and so they are different. Likewise with 1.999... and 2.

[Veldek]

Originally posted by Dros
But light acting as a particle or wave doesn't mean that it is a particle and a wave, but just that we are unable to describe light properly using these crude terms.

Similarly, I agree the limit of 1.999... is 2. But 1.999... is an irrational number, 2.0 is not. So under some systems of math, they may be considered equivalent. But other terminologies have been developed to describe the differences between these two numbers. Ancient humans probably counted 1, 2, many. So in their view, 10 deer and 12 deer are the same. But we have better systems for counting, and so they are different. Likewise with 1.999... and 2.

No. To make it easy, just look at 0,999... = 9/9 = 1. So it's the same, there's no difference, just forget about it.

If you want to know more, look here:
http://mathquest.com/dr.math/faq/faq.0.9999.html

[jxyama]

Originally posted by Dros
But light acting as a particle or wave doesn't mean that it is a particle and a wave, but just that we are unable to describe light properly using these crude terms.

exactly. so 1.999... may not look like "2" that we are used to but it's equal to two. the symbol "2" is perhaps too crude to adaquately describe all the properties of the number "two"?

math can't change its definitions... either 1.999... is equal to 2 or it is not. "equivalent" doesn't really make much sense. what does it mean? it's equal sometimes, but not all the time? that's too ambiguous for math...

who says 1.99999... is not a rational number? 0.1111... is a rational number because it's 1/9. so it's 1 + 9*0.1111..., sounds perfectly rational to me.

[Veldek]

Originally posted by jxyama
who says 1.99999... is not a rational number? 0.1111... is a rational number because it's 1/9. so it's 1 + 9*0.1111..., sounds perfectly rational to me.

Exactly.

[jefhatfield]

Originally posted by jxyama


math is kinda convoluted... it's been a while since i've had to think about this stuff. (i majored in math in college, but that was more than 5 years ago.) any active mathematicians out there?

i live in a retirement area in northern california near silicon valley and a lot of the locals i know were in computer science and data processing...and there are tons of them i can ask so i will see who i can find...one of them is bound to have stayed current with math to know the answer to this...SO WHY do i pick retired computer scientists you say??

these days i am a computer techie and EVERY software side computer techie i have ever met in my life over the age of 50 has a math degree or were math majors in college and may have dropped out for one reason or another

...however, i do have one friend who is exactly 50 and he is a programmer who does not have a math degree...but an actual computer science degree back when nobody offered the degree as a stand alone subject and when just about nobody in the world had a grasp of why programming may be important or if programs/software could be a commodity...when he studied for his phd at MIT back in 1974, the school yanked the specific program he was in thinking there was really no need for his brand of programming and specialty on the grad level since it was hard enough just to find any entry level computer scientists to fill a program and keep it going financially...my friend had to leave MIT after 7 years of intense computer science study (being awarded an MS but not a phd) and re-entered graduate school at stanford later on in the 70s and got his phd there in 1982...he was very ahead of his time studying programming and computer science for programming's sake without a strong math background

what is fascinating is that MIT and Stanford had access to intel processors that did not officially exist yet and my friend really thought nothing of it since he did not, and still does not follow, commercial and industrial trends of computers...he still uses a 486 at home if that is any indication of what he thinks about endless comsumerism and having to keep up with the jonses:p

[zapp]

Originally posted by jxyama
exactly. so 1.999... may not look like "2" that we are used to but it's equal to two. the symbol "2" is perhaps too crude to adaquately describe all the properties of the number "two"?

math can't change its definitions... either 1.999... is equal to 2 or it is not. "equivalent" doesn't really make much sense. what does it mean? it's equal sometimes, but not all the time? that's too ambiguous for math...

who says 1.99999... is not a rational number? 0.1111... is a rational number because it's 1/9. so it's 1 + 9*0.1111..., sounds perfectly rational to me.

You are right

1.99.. does equal 2

1.99.. = 1.99..

1.99.. = 1+ .99..

1.99.. = 1 + 9 * .11..

1.99.. = 1 + 9 * 1/9

1.99.. = 1 + 1

1.99.. = 2

[sjjordan]

Wow,

That's a great way to logically express it.

Thanks!

May we all be enlightened to a fact that has no significance in our lives whatsoever.

[MetallicPenguin]

Originally posted by zapp
It looks cool though and did get my rather slow brain working again. Thanks, Now to ponder the meaning of life.

I am sure someone has said this:

The answer to life is 42.

[amin]

Originally posted by zapp
You are right

1.99.. does equal 2

1.99.. = 1.99..

1.99.. = 1+ .99..

1.99.. = 1 + 9 * .11..

1.99.. = 1 + 9 * 1/9

1.99.. = 1 + 1

1.99.. = 2

for many, accepting that 1/9 truly = .11... is as difficult as accepting that 1.99... = 2; therefore, i'm not sure how the "proof" above is helpful. nevertheless 1/9 does = .11..., and 1.99... does = 2.

[Dros]

Originally posted by zapp
You are right

1.99.. does equal 2

1.99.. = 1.99..

1.99.. = 1+ .99..

1.99.. = 1 + 9 * .11..

1.99.. = 1 + 9 * 1/9

1.99.. = 1 + 1

1.99.. = 2

That is nicely put. I agree .111... = 1/9, but does .999... = 9 * .111...? Is that an appropriate operation on an infinitely long decimal?

[zapp]

Originally posted by Dros
That is nicely put. I agree .111... = 1/9, but does .999... = 9 * .111...? Is that an appropriate operation on an infinitely long decimal?


I would imagine it would be ok, just like simplifying variables, 9x divide by 9 gives you x. your not actually performing a math function with it. So its infinite characteristics remain unchanged just simplified.

[yosoyjay]

Originally posted by kevin49093
But you can't divide by zero!

But, if you take 1/n as n approaches zero 1/n becomes infinitely large.

But, the original post still dosen't hold true...

[johnnowak]

Originally posted by sjjordan
I'm having a great discussion with a friend about the following...

1. if x=1.999...infinity, then 10x=19.999...infinity. Subtracting 10x from x gives 9x=18. Dividing both sides results in x=2.

2. Using your calculators on this one. 1/9=0.111...?. 2/9=0.222....?. 3/9=0.333...?. 4,5,6,7,8 follow the same pattern. What, then is 9/9? It is 0.999...infinity.

Do you think those are valid? I think they are and thinking in infinite terms 1.999...infinity is equal to 2.

My friend thinks he can add any number to infinity because he just can.

Food for thought. Any you want to add?

I just want to add that this is some of the most brain damaged logic ever.

1. Christ.

2. Super christ. Since when did calculators get infinite precision you dolt? I sincerely hope you are still in middle school or life is going to be tough.

[johnnowak]

Originally posted by zapp
You are right

1.99.. does equal 2

1.99.. = 1.99..

1.99.. = 1+ .99..

1.99.. = 1 + 9 * .11..

1.99.. = 1 + 9 * 1/9

1.99.. = 1 + 1

1.99.. = 2

This math is insane.

You are saying .999 ... is the same is 9 * 1/9. That's not true. You are introducing some kinda laymen haze with this 1 + business.

9 * 1/9 = 1.
1 CLEARLY does not equal 0.99999...

This math is nothing short of ridiculous.

Let's take out the 1+ which does nothing.

.9999... = .9999...
.9999... = 9 * 1/9
.9999... = 1

That's what you are saying. Clearly that's nonsense.

You are replacing .9999... with 9 * .1111, which equals 1. You clearly cannot replace .9999... with 1.

I hope this ends the discussion.

[cddonline]

ahhh, but you CAN replace .9999... with 1. That's the whole point, johnnowak. Think of it this way:
If .9999... is different than 1, then 1-.9999... would have to be something other than zero. But because .9999... means a decimal with an infinite number of nines after it, the only possible value for 1-.9999... would be .0000000....1 in other words, an infinite number of zeroes..... followed by a one. Clearly, this is absurd, and therefore 1-.999999... must be zero.

P.S.- Do not take this as a mathematical proof! This is just a way to explain the issue in a way that people can understand.

Furthermore: BY DEFINITION, 1/3 is equal to .3333bar, 2/3 is equal to .6666bar, and 3/3, or 1, is equal to .9999bar. BY DEFINITION. This is simply the case. It cannot be helped. I am truly sorry.

[johnnowak]

You cannot replace .9999... with 1!

1 - .9999.. = 1/infinity

NOT ZERO.

You can have your own feelings about what is absurd, but I'm guessing you didn't exactly max out on math classes in college.

You thinking 1 minus a number less than 1 is 0 is what's absurd to me.

[cddonline]

Johnnowak,
I am currently in college, and I am going to MAJOR IN MATHEMATICS. I am really sorry to repeat myself, but BY DEFINITION: 1 is equal to .9999bar. As much as you would like this not to be the case, it still is. There is nothing you can do to change that FACT. Infinity is indeed confusing, but that is no reason to start throwing insults, especially if you are ENTIRELY WRONG. It will just make you look stupid. I'm sure you are a very intelligent human being, but at this point you need to cut your losses and realize that you are wrong. Sorry.

And yes, thinking 1 minus a number less than one is equal to zero WOULD be absurd. But .99999bar IS NOT LESS THAN 1, IT IS 1!!!!

[johnnowak]

It is true that 0.9 + 0.09 + 0.009 etc does equal 1. However the method in the original post for proving this is nonsense.

I do apologize though for assuming you were talking crazy talk. I'm a math major as well, and have often gotten into some interesting discussions when infinity came up. :-) I feel that the way they work conceptually and mathematically are different, at least going on my intuitive sense of what infinity is.

But yes, if there is an idiot in this discussion, it would be me. Got a bit carried away. :-)

[cddonline]

I must admit that when it comes to infinity, I have trouble wrapping my mind around things. I don't think infinity is an easy thing for the human mind to comprehend. I would also agree that there is a difference between mathematical and conceptual reasoning, and my initial post was indeed more conceptual than mathematical. I felt that I could explain it better if I pointed out conceptually why it makes sense instead of going into a mathematical proof. Also, sorry I got a bit preachy there at the end! :D

[johnnowak]

You're a fine human being. :-) I haven't dealt with infinity for awhile, so I instantly snapped into conceptual reasoning mode instead of recalling basic math facts.

Next time you're in New York we'll go out for coffee.

[cddonline]

haha, Good idea. But if we're going out for COFFEE, we should do it in my home city of Seattle! The single most coffee obsessed city in the entire world. : )

[joemueller]

alright... lets see if i cant explain this somewhat

x = 1.999...
x = 1 + .999...

now lets just work with .999...

in order to complete this problem properly we must understand that

1/(1-x) = 1 + x + x^2 + x^3 + ...

proof:

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

multiply both sides by x

sx = x + x^2 + x^3 + ...

now subtract the bottom line from the top line

s - sx = 1
s (1 - x) = 1
1 / (1 - x) = s

therefor

1 / (1 - x) = 1 + x + x^2 + x^3 + ...


alright, now that we have that we can apply it to this problem

lets break down the decimal expansion of .999...

.999... = (9/10) + (9/10^2) + (9/10^3) + ...

then we can factor out 9/10 from the right side

.999... = (9/10) (1 + (1/10) + (1/10^2) + (1/10^3) + ...)

we can now apply our understanding of the equation:
1 / (1 - x) = 1 + x + x^2 + x^3 + ...

1 / (1 - (1/10)) = (1 + (1/10) + (1/10^2) + (1/10^3) + ...)

and finally we can subsitute in to get:

.999... = (9/10)(1 / (1 - (1/10)))

if we simplify this we get

.999... = (9/10)(10/9)

therefor

.999... = 1

---------------

and in the case of x = 1.999...

it would seem that we could do the following now that we know
.999... = 1

x = 1 + .999...

x = 1 + 1

x = 2

1.999... = 2


-----------------

is this a bit more convincing?

[baby duck monge]

You clearly cannot replace .9999... with 1.

I hope this ends the discussion.

1/3 = .333333333333...
2/3 = .666666666666...

1/3 + 2/3 = .999999999999...
1/3 + 2/3 = 3/3 = 1

.999999999999... = 1

[Doctor Q]

I add my confirmation to those above. Yes, .999... is exactly the same number as 1. They are two notations for the same value, just as all of these are notations for the number 13:

13
15 (base 8)
XIII
9+4
http://www.smithit.com/number_systems/images/tally_marks.gif

[Don't panic]

I add my confirmation to those above. Yes, .999... is exactly the same number as 1. They are two notations for the same value, just as all of these are notations for the number 13:

13
15 (base 8)
XIII
9+4
http://www.smithit.com/number_systems/images/tally_marks.gif

and, of course, 12.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999 (to infinity and beyond!)

[Mr. Anderson]

I think people get bogged down in the semantics of the whole thing.

1/∞ approaches but never reaches 0 - its an infinitely small number...so, for standard math, you have to use the limit of 0 to solve the problem and have a usable answer.

If your solving proofs, you have forms for representing this, so I'd say 1.999...∞ approximately equals 2, etc. in most other cases.

D

[keysersoze]

I did a science fair project on the particle-wave theory of light. I had a pretty nifty demonstration with flashlights, a cookie sheet, and a cardboard box with a few holes cut into it. Funny thing was, I just did it to be lazy since it was such an easy project and I hated doing science fair stuff and it somehow impressed my teacher and I got stuck going to the state science fair and wasting 5 days bored out of my mind.


HAHAHAHAHA!!!!

That's Great!

[mpw]

Jeez the last 1.99999∞ pages just make me feel stupider than I thought I was!

It reminds me of a time i was surveying a property at work with a lazer distance measurer. All was going well until we got to a bath room with mirror on every wall the lazer went crazy and we recorded the rooms area as ∞. Needless to say the valuation was a little on the high side!

[Doctor Q]

1/? approaches but never reaches 0 - its an infinitely small number...so, for standard math, you have to use the limit of 0 to solve the problem and have a usable answer.

If your solving proofs, you have forms for representing this, so I'd say 1.999...? ? 2, etc. in most other cases.You are confusing two separate issues. The series 1.9, 1.99, 1.999, 1.9999, etc. does indeed approach 2 but never reach it. Its limit is 2 and its value isn't 2 because a series does not have a single value, just as the series 1/? approaches 0 but does not have a value.

But this is not the same thing as the number 1.999... (or written with .9 with a bar over the nine), which is one particular value that is indeed 2.

P.S. Typing & # 8 7 3 4 ; will get you a ∞ symbol.

[Loge]

Yes, 1.999999... and 2 are the same number. The difference between them is exactly zero. I also recall that every real number has a unique non-terminating decimal expansion; in this case the expansion of 2 is 1.999999.... Don't ask for proofs, it's too many years ago :o

[Hemingray]

Holy crap... I thought some of you guys were nuts until I saw the 1/3+2/3=3/3=1 so 1=0.999bar... now that's some crazy ****! :eek:

[adamcoop]

You may want to look at this, as it goes into some detail on the subject.
"Is 1 = 0.999....?" (http://www.math.fau.edu/Richman/html/999.htm)

[oldschool]

YOU'RE ALL NERDS!!!!!


(says he who just took a "break" from studying cell biology to read a macintosh computer rumors forum....)

[Roger1]

What this means is if I buy something for 1.99, you might as well say I bought something for 2.10. (allowing for 6% tax). But if I buy something for 2.00, then I might as well say I bought something for 2.12, allowing for tax. So if 1.99 =2.00, why does that extra .01 cost me an extra .02 when I buy something? After all, 1.99 equals 2.00, right???

:p

edit: first/third person b/s

[Doctor Q]

YOU'RE ALL NERDS!!!!!No, we all hate math and are in this thread because we all have the same homework problem and want somebody else to answer it for us. :rolleyes:

Math joke: Did you hear about the race between two infinite sets? Aleph won!

[Counterfit]

and, of course, 12.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999 (to infinity and beyond!) You both forgot 00001101!!

[Counterfit]

And of course, the answer to this problem is 2 + 2 = 5











For very large values of 2.

[Doctor Q]

I had a computer science professor who said his office hours were "3pm plus epsilon to 4pm minus epsilon". That let him claim his office hours approached an hour, but still let him come in a bit late and/or leave a bit early, and then say "don't say I didn't warn you!"

[Santiago]

I just want to chime in to say that the mathematicians are right, and 1.9-repeating is an alternate representation of 2. It is only the case that sums over a finite number of terms of the series represented by 1.9-repeating approach 2; the sum of the infinite series is precisely 2. (The failure to understand this, or even the principle that an infinite series can have a well-defined finite sum, is what lead to one of Zeno's famous "paradoxes" regarding Achilles and the tortoise.)

[benbondu]

(The failure to understand this, or even the principle that an infinite series can have a well-defined finite sum, is what lead to one of Zeno's famous "paradoxes" regarding Achilles and the tortoise.)

After reading this whole thread, I think that pretty much sums it up. (no pun intended)

However, something that wasn't brought up: (this is for people who believe 1.9(bar) equals 2)

If 1.9(bar) and 2 are the same number and 2 is a member of the integers, is 1.9(bar) an integer?

I guess this is the same question as "is 2/2 an integer?"

[Santiago]

However, something that wasn't brought up: (this is for people who believe 1.9(bar) equals 2)

If 1.9(bar) and 2 are the same number and 2 is a member of the integers, is 1.9(bar) an integer?

Yes, in the same way that "two" is an integer. They are alternate notations for the same numerical value.

[Hemingray]

Math joke: Did you hear about the race between two infinite sets? Aleph won!

Naught again... :p

[pianojoe]

No way I'm going to read thru all this!

Please note:

1/9 = 0,1111.... periodic

therefore

9/9 = 0,9999.... periodic = 1,0000

q.e.d.

[PlaceofDis]

this thread has helped me appreciate all the books and all the criticisms i have read, all the papers i have written and all the discussions and lectures that i have sat through to become an English major, i cannot stand math, i am incapable of large math problems, Literature, writing, creative writing, thats what im good at :D

[Counterfit]

No way I'm going to read thru all this! You should have because that was said at least three previous times. :rolleyes:

[AmigoMac]

2 = 1.9999999 , it's great, it's a hope that we can be tolerant. :rolleyes:

[billtanderson]

I look at this as a conversion problem.

Think of non-integer numbers as an analogue quantity. Our decimal number system is a digital representation of a number.

In some cases the representation can be exact, but it many it is just the nearest representation of the value. How close it is depends on the size of the quantization (I think thats the right word), or number of decimal places.


Take digital music, if we sample using 8 bits we get 256 discrete values to represent every possible value on the waveform, if we use 16 bits we get 65,636 etc. Thus the smaller the quantization (someone shout if this is the wrong word) the more accurately we can represent an analogue value with a digital number, but there are always infinitely more values in between each discrete one.

[King Cobra]

No way I'm going to read thru all this!
Heh, that's what I thought at first. (All right, who brought this thread back up from the dead anyways?)

Still, I think zapp's simple method/verification helps the most. (Though it also got the most controversy.)

Now, here's a riddle of a different sort: A mathematician and an engineer can't agree on the number of frames in a movie. If the movie is 2 hours long, then [A] how many frames are there according to both the mathematician and the engineer, [B] whose answer is correct, and why?

Another question for the math majors right above.

[twenties1234]

It's true 1.999... ≡ 2. The real numbers are complete and therefore have the betweenness property which states that between any two real numbers lies an infinite amount of real numbers. So, unless you can produce an infinite amount of numbers between 1.999... and 2, they must be the same number.

[Flying Llama]

Now i'm no pro in math, but from my simple logic, 1.999999999999999 is not the same thing as 2.0000000000000000. They are close, but not the same...

llama :o

[clayj]

I think it's pretty obvious that 2 != 1.99999999....9

If you assume the number of 9's in 1.99999999....9 is "x", then the difference between 2 and 1.99999999....9 is 0.["x-1" number of 0's]...1.

The difference maybe infinitely small, but it is a difference nonetheless.

[tobefirst]

Was this thread *really* just drug up from a year ago so that it can be debated again???

[jsw]

The one thing I'm sure of is that the quantity (2 - 1.999...) is a precise measure of my concern for the correctness of any post in this thread.

[igucl]

How does a thread that hasn't been touched in a year suddenly find new life?

[clayj]

How does a thread that hasn't been touched in a year suddenly find new life?Ask the person who resurrected it ("twenties1234")...

[jsw]

How does a thread that hasn't been touched in a year suddenly find new life?
Well, actually it was something like 0.99 years, so...

oh, never mind. ;)

[Santiago]

I think it's pretty obvious that 2 != 1.99999999....9

If you assume the number of 9's in 1.99999999....9 is "x", then the difference between 2 and 1.99999999....9 is 0.["x-1" number of 0's]...1.

The difference maybe infinitely small, but it is a difference nonetheless.

There is no last nine. The difference is zero. They are two ways of writing the same quantity. If you want to claim otherwise, you need to produce a real number that lies between the two, and no such number exists.

[gekko513]

Doctor Q is correct.

The limit of the series equals 2, but the series itself doesn't equal 2.

[clayj]

There is no last nine. The difference is zero. They are two ways of writing the same quantity. If you want to claim otherwise, you need to produce a real number that lies between the two, and no such number exists.Well, I only included a terminal 9 in order to make the number look nicer... :) I realize that in an infinite number of 9's, there is no last 9.

But surely it's obvious that for ANY number "x", x != x-1?:

1 != 0
2 != 1
3 != 2
...
infinity != infinity - 1

If you multiply both sides of the equation

1.99999999... = 2

by 10000000... (1 followed by infinity zeros), you'd get

199999999... = 200000000...

and that's wrong, too.

[jsw]

I am an engineer. In my world, they are equal.

[clayj]

I am an engineer. In my world, they are equal.Well, of course... you're dealing with matter and energy, neither of which exist in or deal with infinities.

From a purely mathematical perspective, however, I would have to say that they are NOT equal. It doesn't matter if you can't find a real number between "x" and "y"... they're not the same. The difference may be infinitely small, but that's OK because we're talking about decimals that stretch to infinity. Besides, infinity's not a real number (in that it cannot be expressed using digits) anyway... it's an abstract concept that has no logical basis in this universe.

[jsw]

Well, of course... you're dealing with matter and energy, neither of which exist in or deal with infinities.

From a purely mathematical perspective, however, I would have to say that they are NOT equal. It doesn't matter if you can't find a real number between "x" and "y"... they're not the same. The difference may be infinitely small, but that's OK because we're talking about decimals that stretch to infinity. Besides, infinity's not a real number (in that it cannot be expressed using digits) anyway... it's an abstract concept that has no logical basis in this universe.
Actually, infinities do exist in the world of physics, for example when dealing with black holes, which are real... pending, of course, certain m-brane theories to the contrary.

But, agreed, for the most part, it's an abstract concept.

[tdhurst]

Besides, infinity's not a real number (in that it cannot be expressed using digits) anyway... it's an abstract concept that has no logical basis in this universe.

I disagree with that...it's just that WE lack the knowledge to express it.

[clayj]

I disagree with that...it's just that WE lack the knowledge to express it.OK, leaving God out of it, there's not sufficient computational power available in the entire Universe to express infinity.

The Bekenstein Bound sets an absolute limit on the amount of information that can be stored within a given region of space... so, even if we had godlike power and could harness all of the matter and energy within the Universe to computational purposes, the fact that the Universe has a finite size dictates that we would NOT have infinite computational power.

How do we know the Universe is not infinite in size? Easy. The sky is black at night. An infinite Universe would have stars in EVERY possible direction, so the sky would be white at night. A black sky means the Universe is finite.

So, if we had godlike power, we could talk about virtual infinitude (defined as "infinite for all PRACTICAL purposes"), but true infinity will ALWAYS be beyond our reach.

[dukebound85]

1.999infinity=2

Theres a basic reason to this

If you can not seperate 2 numbers by another number then the numbers by definition have to be equal since there is never a dividing point between the numbers. Sorry for the laymans tallk but im tired. I think the biggest problem is that people cant handle the concept of infinity. I mean take the concept of Fourier Series. This pretty much lets you rewrite any perodic function as a combination of sines and cosines...... and when taken to infinity, it WILL become that function. Same idea applies here. People just need to become more knowledgeable about infinity. The concept of infinite series is pretty much standard curriculum in any calculus 2 course.

Night,
Jonathan

[jsw]

How do we know the Universe is not infinite in size? Easy. The sky is black at night. An infinite Universe would have stars in EVERY possible direction, so the sky would be white at night. A black sky means the Universe is finite.
Not true.

Expansion theories postulate that the universe expanded faster than the speed of light for a period, meaning light from the most distant stars has not and may never reach us. Given sufficient time, current theory is that the universe will be bathed in light.

Also, there is no basis for the argument that the universe is finite, especially given current 11-dimensional m-brane/string theory, which provides lots of places to store infinite space.

[tdhurst]

OK, leaving God out of it, there's not sufficient computational power available in the entire Universe to express infinity.

How do we know the Universe is not infinite in size? Easy. The sky is black at night. An infinite Universe would have stars in EVERY possible direction, so the sky would be white at night. A black sky means the Universe is finite.


But, if the universe WAS infinite, many stars would be infinitely far away and we know that our sight is NOT infinite.

So we can never truly know because we lack the ability to compute it.

[clayj]

Not true.

Expansion theories postulate that the universe expanded faster than the speed of light for a period, meaning light from the most distant stars has not and may never reach us. Given sufficient time, current theory is that the universe will be bathed in light.Where did you read THAT? I've read Brian Greene's books, and I've never heard of any supraluminal expansion period.

And the Universe is already bathed in the 3° K background radiation from the Big Bang...

Also, there is no basis for the argument that the universe is finite, especially given current 11-dimensional m-brane/string theory, which provides lots of places to store infinite space.OK, theory. I'm more interested in what's been scientifically proven... the farthest distant known object at 13.5 billion light-years (forgive me if the figure is slightly off), the known rate at which it's receding (based on red shift), etc. Theory may postulate an infinite space packed into extra dimensions, but what we know empirically is that the Universe is finite...

Anyway, I still think that 1.99999999.... != 2. ;)

[clayj]

But, if the universe WAS infinite, many stars would be infinitely far away and we know that our sight is NOT infinite.

So we can never truly know because we lack the ability to compute it.The problem, of course, is that our understanding of the Universe is not what we'd like for it to be.

But the thing is, if we assume that all of spacetime is behaving similarly, then the fact that objects 13 billion light-years away are receding at 97% of the speed of light implies that the Universe CAN'T be much larger than that, because no object could be so far away that its speed of recession is greater than c (unless, of course, Einstein was wrong... and I ain't going there). Ergo, the Universe CAN'T be infinite, UNLESS it's not all expanding. But everywhere we look, we see signs of expansion.

Occam's Razor, then, says that the Universe is finite. Ergo, we cannot actually express infinity, except as an abstract (albeit useful) mathematical concept.

[plinkoman]

ok, time to think about this logically

the difference between 2 and 1.99... is a 1 preceded by an infinate number of zeros, infinite means never ending, so the zeros never end, thus the 1 never comes, so if the difference is an infinate number of zeros, and nothing but zeros(hence infinite), then the difference is zero.

there is no difference ;)

[greatdevourer]

They are different, but only in theory. The moment you start trying to do summat with it, it becomes 2, because the difference is that small, and it's easier to work with

[dukebound85]

Haha they are not different in theory thats the whole pt of this arguement which isn't much of an arguement to begin with.

Jonathan

[greatdevourer]

Haha they are not different in theory thats the whole pt of this arguement which isn't much of an arguement to begin with.

Jonathan They are different. It may be 0.00r1, but they're still different, and that *could* make a difference to certain parts of theoretical physics

[plinkoman]

They are different. It may be 0.00r1, but they're still different, and that *could* make a difference to certain parts of theoretical physics

ok, lets take a different example, .333... is a rational number, it is equal to 1/3, and as far as i know, no one has ever disputed that, so, .333...x3 equals .999... whereas 1/3x3 equals 1

if you take two equal numbers and do the same things to them, the answers will be the same, meaning .999... is equal to 1

like i said before, the difference would be a 1 after infinite zeros, and since the zeros go on to infinity before stopping, thus they don't ever stop, thus the 1 never comes; the difference is just alot of zeros, so there is no difference, not logically, not mathmatically, not theroetically, not physically, and i'd like to see someone try to find some theoretical equation where that wasn't so

[Doctor Q]

But surely it's obvious that for ANY number "x", x != x-1?:

1 != 0
2 != 1
3 != 2
...
infinity != infinity - 1

If you multiply both sides of the equation

1.99999999... = 2

by 10000000... (1 followed by infinity zeros), you'd get

199999999... = 200000000...

and that's wrong, too.The mistake in this argument is that 1 followed by infinity zeros is an infinitely large number, in other words infinity (more precisely, Aleph-0 (http://mathworld.wolfram.com/Aleph-0.html)), and Aleph-0 does not share arithmetic properties with real numbers such as 1.99... and 2. For example, Aleph-0 is equal to one less than itself and knowing that Aleph-0 times A is equal to Aleph-0 times B does not imply that A=B.

Your values 199999999... and 200000000... are also infinite, so you can't call them unequal based on integer properties.

The reason that 1.99... and 2 are equal is that there is no value between them. They are unequal if and only if there is a value between them (which you might find by averaging). But that's not the case, for the following reason:

Suppose N exists with 1.99... < N < 2. Being less than 2 means it must begin with 1 and some decimal digits. If any one of those digits is other than 9, then N=1.999...D... for some digit D, but then N=1.999...D... < 1.999...9... because D<9, so N is not greater than 1.99... (contradiction). Therefore, every decimal digit of N must be 9, so N=1.99... and again N is not greater than 1.99... (contradiction). Since our supposition leads only to contradictions, the claim that N exists is false.

I love the song they quote:"Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take one down, and pass it around, Aleph-null bottles of beer on the wall".

[MongoTheGeek]

I remember the first time I ran into this problem. 8th grade. We were pounding through conversion of decimal to rational numbers.

I got the answer. I didn't like it. I checked the teachers book and still didn't like it. It was like someone killed Santa and stoned the Easter Bunny.

Then I sat down and did the 1/3 +2/3 and I saw the hole. That the numbers were overly complete and there were some times when more than one number was the same thing.

Santa was still dead but the world made sense.

Since then I've been much more comfortable with numbers and yes, as an engineer I have rounded pi off to 10.

[tobefirst]

Since then I've been much more comfortable with numbers and yes, as an engineer I have rounded pi off to 10.

That seems like some awfully generous rounding! 3.1415...all the way up to 10??? (: Hahaha.

(I'm assuming...hoping...for our safety and wellbeing, you mean decimal places!) (:

[MongoTheGeek]

That seems like some awfully generous rounding! 3.1415...all the way up to 10??? (: Hahaha.

(I'm assuming...hoping...for our safety and wellbeing, you mean decimal places!) (:

Nope. I rounded it up to 10. I was just interested in the order of magnitude. It was a quick and dirty calculation to do wind load and see if something was sturdy enough. If the answer was no or maybe I would actually do the math. As was, the answer came back 100m/s and my roundings were in the safe direction so I said screw it.

[Josh]

Want to blow a child's mind?

Hold out your hands, fingers spread, and count the fingers on your left hand backwards from 10.

You will have: 10, 9, 8, 7 , 6. Emphasize the 6.

Then count the fingers on your right hand from 1:

1, 2, 3, 4 , 5.

Ask them what 5 + 6 is (11), and tell them you have 11 fingers.

Kids will think on that for an entire afternoon.

[Doctor Q]

Want to blow a child's mind?...Warning! This trick will not work on kids who actually have 11 fingers. If only you had given me this warning in time! Now the little kid I tried it on at school is in the corner crying that I'm making fun of his abnormality and his mother is outside my house pounding on the door while his very large dad is trying to get in the window. And there is a live TV news crew waiting for me to come out and they stepped all over my flower garden. I'm going to try tunneling out of the house to escape.

I hope the rest of you won't share my fate. Count the kid's fingers before trying this trick!

[eva01]

Warning! This trick will not work on kids who actually have 11 fingers. If only you had given me this warning in time! Now the little kid I tried it on at school is in the corner crying that I'm making fun of his abnormality and his mother is outside my house pounding on the door while his very large dad is trying to get in the window. And there is a live TV news crew waiting for me to come out and they stepped all over my flower garden. I'm going to try tunneling out of the house to escape.

I hope the rest of you won't share my fate. Count the kid's fingers before trying this trick!

ok that was the best thing i have read all day, it has made me much happier still have a horrible headache but i am happier

[Flying Llama]

Warning! This trick will not work on kids who actually have 11 fingers. If only you had given me this warning in time! Now the little kid I tried it on at school is in the corner crying that I'm making fun of his abnormality and his mother is outside my house pounding on the door while his very large dad is trying to get in the window. And there is a live TV news crew waiting for me to come out and they stepped all over my flower garden. I'm going to try tunneling out of the house to escape.

I hope the rest of you won't share my fate. Count the kid's fingers before trying this trick!

HAHA! Man that was funny, that made my day! :)

llama

[diddy]

Lorenz added an infinite series of fractions in a similar method as the first poster. So from the point of view of classical mathematics, 2 is a correct value. ;) But what do I know.

[jdechko]

That seems like some awfully generous rounding! 3.1415...all the way up to 10??? (: Hahaha.

That reminds me of the Simpsons episode where Lisa was explaining her nerd smell in front of the science committee and they all went crazy. Then the professor yelled "Pi is exactly 3!" and everyone freaked out. :D

In any case, I'll add my 1.999...infinity cents.... If someone gave me $2 billion (that is 2,000,000,000,000; my approximation of 2.0... infinity) and I only got $1,999,999,999,999 I would accept the "$2 billion" and not say a word about the missing fraction of a percent. :D

[bemylover]

$2 billion (that is 2,000,000,000,000; that's 2,000,000,000 in fact. other than that I totally agree to your argument :D

[ehurtley]

In college Advanced Calculus (MTH 353, if I remember correctly,) our professor (a PhD in Mathematics from MIT, and a PhD in Theoretical Mathematics from CalTech,) showed us a proof that 0.999... is, indeed, exactly equal to 1. Yes, an actual proof. Sorry, it's been WAAAY too long for me to remember the proof. Not quite 1.999...=2, but mathematically similar. (I figure it has to hold true for all numbers that end in .999...)

Aha! My Google-fu is strong:
Wikipedia (http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1)
Math Forum - Dr. Math (http://mathforum.org/library/drmath/view/55746.html)
Descmath (http://descmath.com/diag/nines.html)
Ask a Scientist at the Division of Educational Programs at Argonne National Laboratory (http://www.newton.dep.anl.gov/askasci/math99/math99167.htm)

[gekko513]

that's 2,000,000,000 in fact. other than that I totally agree to your argument :D
On that note ... in English the number system goes like this:
thousand 1,000,
million 1,000,000,
billion 1,000,000,000,
trillion 1,000,000,000 ... right?

In norwegian it is ...
tusen 1 000,
million 1 000 000,
milliard 1 000 000 000,
billion 1 000 000 000 000,
billiard 1 000 000 000 000 000,
trillion 1 000 000 000 000 000 000,
trilliard 1 000 000 000 000 000 000 000 ... and so on

Why the difference? What do other languages use?

[ehurtley]

On that note ... in English the number system goes like this:
thousand 1,000,
million 1,000,000,
billion 1,000,000,000,
trillion 1,000,000,000 ... right?

In norwegian it is ...
tusen 1 000,
million 1 000 000,
milliard 1 000 000 000,
billion 1 000 000 000 000,
billiard 1 000 000 000 000 000,
trillion 1 000 000 000 000 000 000,
trilliard 1 000 000 000 000 000 000 000 ... and so on

Why the difference? What do other languages use?

Even screwier, your 'English' units are only correct for American English. British English used to use slightly different units. (And some Britons still assume the old units, which are the same as your Norwegian above for milliard and billion.) The difference is properly called 'short scale' for the American versions, and 'long scale' for the former British system.

[bemylover]

all those terms come from one wod mille, which means thousand.

When Marco Pollo returned from China in 13th century, he was very impressed an tried to impress his fellow citizens by the wealths of China. The highest numbre expressed by one word was thousand(mille) at that time, which was not enough for him and that's when he invented a new one, million.

All these billions and trillions come from that million, so one should not look for a lot of meaning in those words.

[jdechko]

that's 2,000,000,000 in fact. other than that I totally agree to your argument :D

oops. :o

[obeygiant]

3 is a magic number!

[RHutch]

I really can't believe how much discussion this topic got.

I am also surprised at some of the outrageous claims made (that 1.9999.... is an irrational number).

I will just add that all of the people who have said that 1.99999.... = 2 are correct. There are many ways to show this, and many have been stated in this thread.

Not that this will be any more likely to make the non-believers believe, but I do have a BS and MS in pure mathematics, and my thesis was in the area of infinite series.

[applebum]

Ok - after seeing this thread, all I really want to know is

when I purchase a song from iTunes for .99, am I really paying $1??? If so, where did my penny go??

I hate math!

[RHutch]

Ok - after seeing this thread, all I really want to know is

when I purchase a song from iTunes for .99, am I really paying $1??? If so, where did my penny go??

I hate math!

No. But if you purchase a song from iTunes for $0.999999999....., then you will be paying a dollar.:p

[Doctor Q]

Ok - after seeing this thread, all I really want to know is

when I purchase a song from iTunes for .99, am I really paying $1??? If so, where did my penny go??The other penny goes to all the math geeks. We split it evenly amongst ourselves, using a very sharp chisel.

And now I'm sure this issue is settled.

[feakbeak]

People just need to become more knowledgeable about infinity.Infinity is a difficult concept to wrap your head around. What do you think drove Cantor insane? ;)

Another vote for 1.999... = 2 from another person with a math degree (BS in mathematics for secondary education).

As for those who have objections to infinity because it doesn't always have a practical purpose in engineering or in day-to-day tasks, you are missing the point. That is one of the beauties of mathematics, it not bounded by such practicalities.

For example, the importance of group theory was emphasized when some physicists using group theory predicted the existence of a particle that had never been observed before, and described the properties it should have. Later experiments proved that this particle really exists and has those properties.

Humanity learned a great deal about mathematics by making observations about the physical universe that we live in. Now, the tables have turned and we are understanding the nature of our physical universe by letting our minds run loose in the abstract world of mathematics. It's rather amazing if you think about it - reminds me of a quote by Einstein.

How can it be that mathematics, being after all a product of human thought, independent of experience, is so admirably adapted to the objects of reality?

[Iavhé]

It is true that 0.9 + 0.09 + 0.009 etc does equal 1. However the method in the original post for proving this is nonsense.

I do apologize though for assuming you were talking crazy talk. I'm a math major as well, and have often gotten into some interesting discussions when infinity came up. :-)
Johnnowak,
I am currently in college, and I am going to MAJOR IN MATHEMATICS. I am really sorry to repeat myself, but BY DEFINITION: 1 is equal to .9999bar
Another vote for 1.999... = 2 from another person with a math degree (BS in mathematics for secondary education).
And now I'm sure this issue is settled.
I think not ;)

I may not have a majore in mathematics but i still know 0.999...!=1

Lets say 0.999...=x

0.999...=(9/10)+(9/10^2)+...+(9/10^k)+...+(9/10^n)

0.999...*10=10x

0.999...*10=9+(9/10)+...+(9/10^k-1)+...+(9/10^n-1)

9x=(9+(9/10)+(9/10^2)+...+(9/10^k)+...+(9/10^n-1))-((9/10)+(9/10^2)+...+(9/10^k)+...+(9/10^n)) -->

9x=9-(9/10^n)

if 0.999...=1 shouldnt 9x=9 in this case? please proof me wrong

[atszyman]

if 0.999...=1 shouldnt 9x=9 in this case? please proof me wrong

You made the assumption of a finite sequence.

0.999...=(9/10)+(9/10^2)+...+(9/10^k)+...+(9/10^n)

should be

0.999...=(9/10)+(9/10^2)+...+(9/10^k)+...+(9/10^inf)

and

0.999...*10=9+(9/10)+...+(9/10^k-1)+...+(9/10^inf)

therefore

9x=(9+(9/10)+(9/10^2)+...+(9/10^k)+...+(9/10^inf))-((9/10)+(9/10^2)+...+(9/10^k)+...+(9/10^inf)) -->

9x=9

as described by Dr. Q's explanation of Aleph-0 (http://mathworld.wolfram.com/Aleph-0.html) earlier in this thread.

[eva01]

Surprised this popped up again. All you have to know is this

.1bar = 1/9
.2bar = 2/9
etc.
.8bar = 8/9

so

.9bar = 9/9 = 1

which will also work for 1 to 2

[monke]

Basically, this is the same as 1=.99999999

Two reasons:

.3333333333333333333333333333333333333333333333
times
3
equals
.9999999999999999999999999999999999999999999999

.3333333333333333333333333333333333333333333333
equals
(1/3)
times
3
equals
1

and this one:

1/3=0.33333.....
+2/3=0.66666.....
==============
3/3=0.99999.....=1

[Doctor Q]

At least we didn't spend 7 years finding a proof, announcing it, and then learning it was wrong. That's what happened to Andrew Wiles (http://en.wikipedia.org/wiki/Andrew_Wiles), who spent 7 years on a proof of Fermat's Last Theorem (which was not, in fact, Pierre de Fermat's last theorem). Wiles worked on his proof from 1986 to 1993 and then announced it to great fanfare. Soon after, a flaw it in was found. Luckily, he was able to work around the flaw and issue a corrected proof the next year.

Then there was the story of the mathematician from long ago who spent his entire career computing digits of pi, but made a mistake at one position, invalidating all of his remaining work. I couldn't find a reference to this story to find out any details. Perhaps somebody else knows.

[WildCowboy]

Then there was the story of the mathematician from long ago who spent his entire career computing digits of pi, but made a mistake at one position, invalidating all of his remaining work. I couldn't find a reference to this story to find out any details. Perhaps somebody else knows.

Ah yes...William Shanks (http://en.wikipedia.org/wiki/William_Shanks).

[vniow]

Nerds.

[monke]

At least we didn't spend 7 years finding a proof, announcing it, and then learning it was wrong.

That would have been horrible, but getting it in 3.5 years is still quite a while, though we didn't work on it day after day. 7 years trying to solve anything would be brutal, I would quit after a week. :p

Nerds.

We are all one vniow. :D

[WildCowboy]

7 years trying to solve anything would be brutal, I would quit after a week. :p

Welcome to my life in grad school...

[Rodimus Prime]

only problem with the


1/9 =.11bar and then going up to 9/9=.9bar is that it is not a mathatical proof. (those can use only letters. x/x=1 and that is always true.
Now do the math math by hand on 8/9 and you do get the .8 bar because of how it works out. You always have that remainder of 8 for every spot. But when you go up to 9/9 it a one. Now some where in that enteter .9bar one of the numbers was able to go up one and that cause the enter thing to go to 1.

Now for pratical reasons I could care less 1.9999....= 2 to me. heck I really for what I do I really only use 4-5 sig figures any way and just round the rest. so for examppel 1.9996=2 for all care and in the end my answer is still going to be close enough and will work because I well the knowns in the problems are really only with in maybe 5% of what they should be any way.

[WildCowboy]

There's an excellent series of five blog posts on this here (http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html).

[Doctor Q]

It's worth pointing out that Iavhé asked for evidence that a proof was wrong. Several of us posted other ways to prove that 1=.9bar. While that technically proved that Iavhé's proof is invalid, it didn't show why, which is more enlightening. Luckily, atszyman already gave the specific answer.

Somewhere in my collection of math stuff on one of my old Macs I have a trigonometry "proof" of a false claim. It's hard to spot the error, which is why I like it. I think it is in a ClarisWorks document. If I run across it, I'll post it for forum members to ponder.

[atszyman]

I think the concept of infinity is a bit of a hard concept to grasp in this thread. when you have .999999|9 there is always another 9 after the previous one. One way to illustrate this is to read up on Hilbert's Hotel (http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel). It's a fun logical exercise that illustrates some of the issues with an infinite sequence of 9s.

Like the fact that shifting one of the 9's over to be to the left of the decimal point does not result in a zero at the end of the sequence, but instead puts another 9 there.

It would seem to be possible to make place for an infinite (countable) number of new clients: just move the person occupying room 1 to room 2, occupying room 2 to room 4, occupying room 3 to room 6, etc., and all the odd-numbered new rooms will be free for the new guests. However, this is where the paradox lies. Even in the previous statement, if an infinite number of people fill the odd numbered rooms, then what amount is added to the infinite that was already there? Can one double an infinite? Also, for example, say the infinite number of new guests do come and fill all of the odd numbered rooms, and then the infinite number of guests in the even rooms leave. An infinite has just been subtracted from a still existing infinite, yet an infinite still exists. This is where Hilbert's Hotel is paradoxical.

By this method the set of all real integers has the same number of elements as the set of only the odd valued real integers...

Of course I would know almost nothing about this if we hadn't covered it only a month ago in my last grad school class.

[bearbo]

Aah, but 1 = 2!
How?

Well using the rules for algebra do the same thing to both sides (divide by zero).

1 = 2

: 1/0 = 2/0

so therefore infintity = infinity
this is (obviously) a faulty logic... simply because because infinity*0 does not equal to any fixed number, therefore

: 1/0 = 2/0

so therefore infintity = infinity

does not justify

1 = 2

now regarding to the hotel problem...

infinity is not a fixed number, it's a limit

[Doctor Q]

I've stayed at Hilbert's Hotel. It's a nice place, but rather expensive: 1 cent for the first minute, 2 cents for the next half minute, 4 cents for the next quarter minute, etc.

This reviewer (http://www.pagina12.com.ar/diario/contratapa/index-2006-07-07.html) actually got a photo of it!

[monke]

Q just completely confuzled me :confused: :) :confused:

[bearbo]

I've stayed at Hilbert's Hotel. It's a nice place, but rather expensive: 1 cent for the first minute, 2 cents for the next half minute, 4 cents for the next quarter minute, etc.

This reviewer (http://www.pagina12.com.ar/diario/contratapa/index-2006-07-07.html) actually got a photo of it!
cool... did you stay there past 2 minutes?

[Maxx Power]

definition of equal? smaller than any possible number? math is convoluted?

1, equal means equal.
2, impossible, there is always something smaller.
3, math is the language of pure and universal logic.

3. math is NOT the language of pure and universal logic.

It's A language of a particular class of universal logic at most.

There have been numerous attempts to reconstruct math by formally creating a logical system in which some or all of math would be a part of, therefore, abstractify math to the level of simple logic and algorithms. This is logicism, which created "Principia Mathematica" by Bertrand Russell and Alfred Whitehead, which recreated three volumes of math into simpler logic including set theory, number theory, and arithmetic, at which point a fundamental contradiction was realized (see also Godel's incompleteness theorem), and then logicism was abandoned. Which was too bad because this would unite mathematics with logic and analytical philosophy and give philosophy its much needed backing and solid proofs.

So now the general understanding of math is that it is an abstraction layer above simple logic and algorithmic language like cellular automata and your base-4 genetics. By the fact that the two systems can not interconvert perfectly, you can never arrive to the fact that Math itself can become the single, monolithic description of universal logic and being perfect.

[exabytes18]

Ha! I was looking at wikipedia's homepage today and believe it or not, the feature article is 0.999... (http://en.wikipedia.org/wiki/0.999...). :)

[Synapple]

This thread made my day yesterday... I was really bored at work and mad at my boss and decided enough was enough for the day: therefore, I had the chance to spend most of my day reading through this post and the other links found in this thread.

I obviously don't have a definitive answer on the issue but tend to agree with the math reasoning, assuming we take math as a number of rules and assumptions which lead to apparently contradictory results in particular situations.
I believe, under a mathematical reasoning, 0.9| definitely equals one, even if it seems to defeat the logic we have grown to associate to the representation of numbers in a decimal form.

[Dale Sorel]

Ha! I was looking at wikipedia's homepage today and believe it or not, the feature article is 0.999... (http://en.wikipedia.org/wiki/0.999...). :)
Thank you, as I knew from my experience with Calculus that was the correct answer.

[gnasher729]

I'm having a great discussion with a friend about the following...

Write down a clear definition what an "infinite decimal fraction" is and what it means. It is pointless to go any further before you have done that.

[mufflon]

this thread is rather impressive really - considering the fact that it sprung from errors in a calculator - where I believe I feel more at home than most people on this board - on account of building one, so kudos all around, this was very entertaining :)

[MrSmith]

I'll go a non-geeky way.

I have 2 apples on my desk. I can't have 1.999... apples on my desk. It isn't physically possible to cut some away leaving 1.999.... exactly. But it would be physically possible to cut my apple to any size if I had a thin enough knife, etc. Therefore 1.999... doesn't exist and must just be poncey talk for 2. But it's a theoretical number, you say. Well, numbers are only symbols representing matter on God's earth. What about infinity, you ask. It does exist. It's the size of the Universe. Or maybe it has no meaning. Everything is finite.

;)

I'm already looking forward to the "Does 2.99...=3" thread.