2=1.999...infinity! What are your thoughts? [Archive]

jxyama

Jan 24, 2004, 09:36 PM

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

Jan 24, 2004, 09:41 PM

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

Jan 24, 2004, 09:52 PM

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

Jan 24, 2004, 10:14 PM

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

Jan 24, 2004, 10:26 PM

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

Jan 25, 2004, 12:11 AM

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

Jan 25, 2004, 12:48 AM

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

Jan 25, 2004, 01:12 AM

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

isus

Jan 25, 2004, 01:35 AM

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

Jan 25, 2004, 02:53 AM

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

Jan 25, 2004, 04:14 AM

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

Veldek

Jan 25, 2004, 04:30 AM

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

Jan 25, 2004, 06:24 AM

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

Jan 25, 2004, 07:53 AM

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

Jan 25, 2004, 09:39 AM

But you can't divide by zero!

kevin49093

Jan 25, 2004, 09:48 AM

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

Jan 25, 2004, 10:33 AM

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

Jan 25, 2004, 10:42 AM

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

jxyama

Jan 25, 2004, 12:13 PM

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

Jan 25, 2004, 12:18 PM

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

Jan 25, 2004, 12:55 PM

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

Jan 25, 2004, 01:39 PM

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

Jan 25, 2004, 01:55 PM

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

Jan 25, 2004, 01:57 PM

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

Jan 25, 2004, 02:01 PM

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

Jan 25, 2004, 02:36 PM

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

Jan 25, 2004, 05:21 PM

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

Jan 26, 2004, 08:06 PM

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

Jan 26, 2004, 08:26 PM

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

Jan 26, 2004, 09:39 PM

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

Jan 26, 2004, 09:51 PM

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

Jan 26, 2004, 10:32 PM

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

Jan 26, 2004, 10:55 PM

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

Jan 26, 2004, 11:03 PM

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

Jan 26, 2004, 11:43 PM

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

Jan 26, 2004, 11:51 PM

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

Jan 26, 2004, 11:57 PM

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

Jan 26, 2004, 11:58 PM

cddonline

Jan 27, 2004, 12:09 AM

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

Jan 27, 2004, 12:12 AM

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

Jan 27, 2004, 12:55 AM

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

Oct 13, 2004, 01:52 PM

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

Oct 13, 2004, 02:19 PM

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

Oct 13, 2004, 02:20 PM

Don't panic

Oct 13, 2004, 02:53 PM

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

Oct 13, 2004, 03:00 PM

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

Oct 13, 2004, 03:17 PM

mpw

Oct 13, 2004, 03:34 PM

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

Oct 13, 2004, 03:39 PM

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

Oct 13, 2004, 06:16 PM

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

Oct 13, 2004, 06:23 PM

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

Oct 13, 2004, 06:42 PM

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

Oct 13, 2004, 08:30 PM

YOU'RE ALL NERDS!!!!!

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

Roger1

Oct 13, 2004, 08:49 PM

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

Oct 13, 2004, 08:50 PM

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

Oct 13, 2004, 10:34 PM

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

Counterfit

Oct 13, 2004, 10:45 PM

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

For very large values of 2.

Doctor Q

Oct 14, 2004, 02:57 PM

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

Oct 14, 2004, 04:14 PM

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

Oct 14, 2004, 04:36 PM

(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

Oct 14, 2004, 04:51 PM

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

Oct 14, 2004, 06:07 PM

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

Naught again... :p

pianojoe

Oct 14, 2004, 07:09 PM

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

Oct 14, 2004, 07:32 PM

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

Oct 23, 2004, 05:01 AM

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

AmigoMac

Oct 23, 2004, 07:35 AM

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

billtanderson

Oct 23, 2004, 09:49 AM

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

Oct 23, 2004, 11:49 AM

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

Oct 13, 2005, 11:37 PM

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

Oct 13, 2005, 11:58 PM

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

Oct 14, 2005, 12:20 AM

tobefirst

Oct 14, 2005, 12:26 AM

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

jsw

Oct 14, 2005, 12:34 AM

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

Oct 14, 2005, 12:40 AM

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

clayj

Oct 14, 2005, 12:41 AM

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

jsw

Oct 14, 2005, 12:42 AM

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

Oct 14, 2005, 12:45 AM

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

Oct 14, 2005, 12:47 AM

Doctor Q is correct.

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

clayj

Oct 14, 2005, 12:57 AM

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

Oct 14, 2005, 12:59 AM

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

clayj

Oct 14, 2005, 01:03 AM

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

Oct 14, 2005, 01:06 AM

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

Oct 14, 2005, 01:07 AM

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

Oct 14, 2005, 01:14 AM

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

Oct 14, 2005, 01:15 AM

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

Oct 14, 2005, 01:22 AM

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

Oct 14, 2005, 01:22 AM

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

Oct 14, 2005, 01:28 AM

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

Oct 14, 2005, 01:33 AM

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

Oct 14, 2005, 02:01 AM

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

Oct 14, 2005, 02:04 AM

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

Oct 14, 2005, 02:10 AM

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

Oct 14, 2005, 02:20 AM

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

Oct 14, 2005, 02:31 AM

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

Oct 14, 2005, 03:22 AM

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

Oct 14, 2005, 08:12 AM

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

Oct 14, 2005, 09:18 AM

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

Oct 14, 2005, 12:33 PM

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

Oct 14, 2005, 01:05 PM

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

Oct 14, 2005, 05:26 PM

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

Oct 14, 2005, 05:29 PM

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

Oct 15, 2005, 04:54 PM

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

Oct 15, 2005, 06:25 PM

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

Oct 17, 2005, 10:32 AM

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

Oct 18, 2005, 02:09 AM

$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

Oct 18, 2005, 02:52 AM

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

Oct 18, 2005, 03:13 AM

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

Oct 18, 2005, 03:28 AM

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

Oct 18, 2005, 11:05 AM

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

Oct 18, 2005, 11:37 AM

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

oops. :o

obeygiant

Oct 18, 2005, 11:39 AM

3 is a magic number!

RHutch

Oct 28, 2005, 11:08 AM

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

Oct 28, 2005, 11:31 AM

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

Oct 28, 2005, 12:09 PM

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

Nov 16, 2005, 01:19 AM

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

Nov 16, 2005, 02:18 AM

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

Oct 19, 2006, 01:53 PM

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

Oct 19, 2006, 02:26 PM

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

Oct 19, 2006, 02:27 PM

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

Oct 19, 2006, 02:50 PM

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

Oct 19, 2006, 02:55 PM

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

Oct 19, 2006, 02:57 PM

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

Oct 19, 2006, 02:58 PM

Nerds.

monke

Oct 19, 2006, 02:59 PM

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

Oct 19, 2006, 03:01 PM

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

Oct 19, 2006, 04:12 PM

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

Oct 19, 2006, 04:23 PM

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

Oct 19, 2006, 04:52 PM

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

Oct 20, 2006, 12:36 PM

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

Oct 20, 2006, 12:51 PM

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

Oct 20, 2006, 02:49 PM

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

Oct 20, 2006, 02:50 PM

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

bearbo

Oct 20, 2006, 03:20 PM

Maxx Power

Oct 24, 2006, 03:42 PM

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

Oct 25, 2006, 07:38 PM

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

Oct 26, 2006, 02:37 AM

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

Oct 26, 2006, 03:22 AM

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

Oct 26, 2006, 03:45 AM

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

Oct 26, 2006, 03:54 AM

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

Oct 26, 2006, 04:06 AM

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.

Josh1488

Oct 7, 2009, 12:49 AM

was googling something about infinity and this came up and i couldn't resist enlightening anyone interested, as a teacher with a math degree.

2 is in fact equal to 1.9999999...

i won't get into a rigorous proof, however, here's a quick sketch

1.9999999... is actually 1 + 9/10 + 9/100 + 9/1000 + ...
this is called a geometric series, that is, an infinite sum of the form ar^n

it can be shown that the geometric series converges if and only if |r|<1, and it converges exactly to ar/(1-r)

now we can think of the terms after the 1 as the terms to the right of the decimal place, and those terms form a geometric series with a=9 and r=(1/10), so if our series converges to x, we can essentially say 2=1.x whatever x is

now since i already have said that geometric series converge to a/(1-r) you can plug in our a and r to get:
((9)(1/10))/(1-(1/10)=(9/10)/(9/10)=1

adding that previous 1 term in we have 1+1=2
so there you have it, a rough sketch of a rigorous proof to show that 2 = 1.9999999999999

there are many other interesting facts that arise on the subject of infinite sums, this being one of the less noteworthy ones, i'd urge anyone with an interest in math to skim through a calculus textbook or something on real analysis for more information concerning these topics

sammich

Oct 7, 2009, 01:02 AM

My turn, if Im repeating someone I guess it's just they way they expressed it was different from my way of thinking (EDIT: as I now realise, this thread is actually 4 pages instead of just the first one i read)

x = 1 - 1/n lim n -> inf
10x = 10 - 10/n lim n -> inf

Now:
10/n lim n -> inf = 1/n lim n -> inf (*)

Hence:

10x = 10 - 1/n lim n -> inf

Therefore:

10x - x = (10 - 1/n) - (1 - 1/n) lim n -> inf
9x = 9
x = 1

EDIT: it made sense in my head when I started...but now I'm not so sure. At (*) because it only holds true when you actually take the limit?

sammich

Oct 7, 2009, 01:19 AM

Holy cow this is an old thread.

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.

That's like saying that π (pi) doesn't have any meaning. The use of numbers is purely theoretical until you apply it to something. If you have a cubic metre/foot of sand, and you took a single grain out, then you would have yourself 0.9999999...(arbitary) cubic units of sand. Using an apple is difficult because of what you define what a single unit of an apple is.

c-Row

Oct 7, 2009, 04:43 AM

And I always thought the saying "If zero is really huge, it's almost a bit one" was only a joke... :confused: :D

Ttownbeast

Oct 7, 2009, 01:52 PM

Look at it on a graph think about it visually zooming in infinitely .99999999999999999infinite does not equal 1 it's a vertical line on a graph that is .infinite01 away from 1 almost equaling 1 but not 1 the proper symbol for this relationship would be the exact use of the equal sign as a pair of wavy lines like this "≈" this making 1≈.9999999999999999999999999........... the truest statement. Though stating 1≠.999999999999999............ is a rougher guess just as is stating 1=.9999999999999999..........

the problem I've seen here is incorrect usage of the mathematical symbols.

Signal-11

Oct 7, 2009, 03:20 PM

gessoandgold

Oct 7, 2009, 03:37 PM

Are you an aspirin salesperson by any chance? :cool:

Demosthenes X

Oct 7, 2009, 08:47 PM

dukebound85

Oct 7, 2009, 08:52 PM

they are the same by the very definition of what consisitutes a number

Capt Crunch

Oct 7, 2009, 10:12 PM

I love all the people who were condescending in their belief that 1.9999 was different than 2.

Here's a tip kids, if you claim educational superiority, be sure you're right.

Ttownbeast

Oct 7, 2009, 10:37 PM

In my first university calculus class, the professor wrote on the board:

0.999... infinity
1

And asked us if they were different numbers. Per her argument, they are not, since there's no number in between 0.999... and 1. Ergo, they're the same. Or, more technically,

Then is .8888infinity the same as .99999infinity? therefore .8888888infinity equals .99999999infinity? Since there is no number in between? a remainder exists doesn't it? though It is a remainder that is as irrational as the two numbers it is between in this case, just as it is an irrational number between .99999 inf and 1. Using the logic that .999999infinite is the same as 1 somebody could argue that all decimals of this nature are equal to one by relation that they are equal to each other in sequence. in this case there is always a roughly 1/10th or 1/10th of infinity of an amount missing not making it actually 1 but almost 1. Stating that it absolutely equals 1 is just for convenience sake I remember one important lesson my calculus professor taught us about practicality that even the best of mathematicians are lazy so 1=.99999999etc for practical purposes, but those who were going on to study in certain fields like quantum physics the matter that such things are different. Numbers have plenty of importance when it comes to calculating on the smallest known scales. My calculus was horrible in practice (writing out the math)when I was taking these courses, but my understanding of the math (comprehension and analysis of others work) was good enough to grasp the concepts of the field. A physicist cannot be lazy as a mathematician can. The deeper they go the more they have to know and apply to the experiments to properly interpret the data correctly. Rounding things does not work well at the quantum level for accuracy even if things have to be rounded the information is more accurate the more decimal places are used in which it is able to be calculated.

dukebound85

Oct 7, 2009, 10:43 PM

i know its wiki but its true nonetheless

http://en.wikipedia.org/wiki/0.999...

read up on it

Ttownbeast

Oct 7, 2009, 10:46 PM

i know its wiki but its true nonetheless

http://en.wikipedia.org/wiki/0.999...

read up on it

If you know its a wiki....oh never mind.

dukebound85

Oct 7, 2009, 10:48 PM

If you know its a wiki....oh never mind.

Yea, never mind:rolleyes:

Burden is on YOU to proof they are not the same after proofs proving they are the same have been established

Also, feel free to look up all the references on the topic the wiki conviently provides proving they ARE The same by the way

Ttownbeast

Oct 7, 2009, 10:59 PM

Yea, never mind:rolleyes:

Burden is on YOU to proof they are not the same after proofs proving they are the same have been established

Also, feel free to look up all the references on the topic the wiki conviently provides proving they ARE The same by the way

Reference something juried for the proofs, that's all I'd like to see for a change Wikipedia is notoriously chaotic. But that doesn't necessarily make it true either are the references primary? secondary?....?

the vj

Oct 7, 2009, 11:34 PM

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 arrived here late but...

You both need to go back to school and at same time pay attention to the movie Contact.

You are dividing prime numbers who just can be divided between themselves and 1. Like the signals in the movie where prime numbers.

And 9 between 9 is 1, not 0,9999.

Pay attention to the numbers and do not guess because you will have loads of financial problems IF you ever manage to get enough money to get one.

Mr. lax

Oct 7, 2009, 11:50 PM

Think about this in the form of conics (using cones) When a cone is stretched to infinity it becomes a cylinder...

benlangdon

Oct 7, 2009, 11:56 PM

k don't know if anyone said this.
INFINITY IS NOT A NUMBER

my teacher has a shirt that says this.
o and repeating digits has a vinculum on top of it, not w/e infinity.

to indicate a properly repeating digit ... is sufficient.

OP, what math have you taken

omg, im not even going to try, im seeing all kinds of messed up things here. dam people. unsubscribe. i don't even want to know what happens with this.

Ttownbeast

Oct 8, 2009, 12:00 AM

Think about this in the form of conics (using cones) When a cone is stretched to infinity it becomes a cylinder... How so? maybe a section of the cone seems like a cylinder but there is still a slight angle--perceptibly yes it is for all practical purposes a cylinder, But as part of the whole not the section at some point as it is being infinitely stretched it is still a cone.

Signal-11

Oct 8, 2009, 12:03 AM

Reference something juried for the proofs, that's all I'd like to see for a change Wikipedia is notoriously chaotic. But that doesn't necessarily make it true either are the references primary? secondary?....?

Doesn't matter. That Wiki page references a number of simple, easy to follow proofs (we'll call 'em proofs for now) that demonstrate that 0.(9) = 1.

This is not a debatable subject. By mathematical definition, 0.(9) = 1.

And no, 0.9~ is not an irrational number. Because it = 1.

There is no last 9. Your gut is telling you that there is a last 9 in the sequence, because infinity is a difficult concept to understand on an intuitive level. What's happening here is your intuition is overriding your intellect.

You said you took calculus. Review your definition of a limit, then infinity. The maths are there, as plain as day, yelling at you that 0.(9) = 1. You're just not listening because you're looking for the last 9.

Ttownbeast

Oct 8, 2009, 12:11 AM

k don't know if anyone said this.
BUT INFINITY IS NOT A NUMBER

my teacher has a shirt that says this.
o and repeating digits has a vinculum on top of it, not to the infinity.

Only using that infinity thing to relate homie. I just couldn't find the little line over the number in key caps on my mac. Like I said I am bad at writing the math but I do interpret others data very well--you are correct infinity is not a number it is more an expression or concept if you will of something neverending--I was terrible at story problems in mathematics too I won't lie. Saying "to the infinity" is an easier way to convey the concept to stoners of something that goes on and on such as this .99999 debate thingy will--it will probably outlast humanity itself (if some other intelligence should bother with it).

Ttownbeast

Oct 8, 2009, 12:15 AM

Doesn't matter. That Wiki page references a number of simple, easy to follow proofs (we'll call 'em proofs for now) that demonstrate that 0.(9) = 1.

This is not a debatable subject. By mathematical definition, 0.(9) = 1.

And no, 0.9~ is not an irrational number. Because it = 1.

There is no last 9. Your gut is telling you that there is a last 9 in the sequence, because infinity is a difficult concept to understand on an intuitive level. What's happening here is your intuition is overriding your intellect.

You said you took calculus. Review your definition of a limit, then infinity. The maths are there, as plain as day, yelling at you that 0.(9) = 1. You're just not listening because you're looking for the last 9.

it's a chase I am not looking for a last 9 I just keep chasing knowing i'll never reach an end

Doctor Q

Oct 8, 2009, 04:32 PM

True or false? - "A regular polygon with an infinite number of sides is a circle."

Here's why it's true:Consider the original question in this thread. When we say that 1=0.9... or that 1 is 0.9..., the definitions of "=" and "is" are "there is no difference between the values".

One way we know that there is no difference is that if you tell me any number D>0, I can show that the absolute value of the difference is smaller than D. In other words, I can show you that you are wrong if you pick any positive value. Therefore, the difference D can't be more than zero, so it must be zero.

In the same way, if you tell me any number D, I can make an N-sided regular polygon inscribed in a unit circle so that the difference in area between the circle and the polygon is less than D. All I have to do is make N big enough. Therefore, by making the number of sides infinite, I can arrange that polygon=circle.

Here's why it's false:By definition, a regular polygon is a figure in a plane consisting of a finite number of end-to-end line segments that have equal lengths and form equal angles. (There are many other ways to say this.)

Therefore, an infinite polygon can't exist by definition.
So apparently the statement is false even though you can prove that it's true. ;)

fireshot91

Oct 8, 2009, 04:51 PM

Then is .8888infinity the same as .99999infinity? therefore .8888888infinity equals .99999999infinity?

I think 0.888 (repeating) is (apparently) equal to 0.9, not 0.99999 (repeating).

But, Doctor Q, care to explain how D=0, when you said that D>0? I mean, does D stand for a specific unit in your polygon problem, or just any variable?

Like, would it be the same if I said X>0, and you proved me wrong by saying that X=0?

Signal-11

Oct 8, 2009, 05:33 PM

I think 0.888 (repeating) is (apparently) equal to 0.9, not 0.99999 (repeating).

But, Doctor Q, care to explain how D=0, when you said that D>0? I mean, does D stand for a specific unit in your polygon problem, or just any variable?

Like, would it be the same if I said X>0, and you proved me wrong by saying that X=0?

Fireshot, think about it this way, which also addresses an error in your post.

What positive numerical value fits between 0.(9) and 1?

What positive, numerical value fits between 0.(8) and 9? It's greater than 0, therefore, the two are not equal. 0.8(9) = 0.(9), not 0.(8)

fireshot91

Oct 8, 2009, 05:38 PM

Fireshot, think about it this way, which also addresses an error in your post.

What positive numerical value fits between 0.(9) and 1?

What positive, numerical value fits between 0.(8) and 9? It's greater than 0, therefore, the two are not equal. 0.8(9) = 0.(9), not 0.(8)

Ah yes, I meant 0.8999..repeating, not 0.9.My fault.

siurpeeman

Oct 8, 2009, 05:48 PM

I think 0.888 (repeating) is (apparently) equal to 0.9, not 0.99999 (repeating).

0.888... isn't euqal to 0.9. 0.888... is 8/9; 0.9 is 9/10.

Signal-11

Oct 8, 2009, 05:50 PM

Doctor Q

Oct 8, 2009, 06:28 PM

That's horse puckey, man.

Looked at simply, how large is the set of all even numbers? How large is the set of all numbers divisible by 43564?

How large is the set of all regular polygons? Take the last one in the set. That's your circle ;)Sets have "last ones"? You learn something new every day! :)

Signal-11

Oct 8, 2009, 06:50 PM

Sets have "last ones"? You learn something new every day! :)

I was wondering if anybody'd pick up on it, considering I just wrote this (http://forums.macrumors.com/showpost.php?p=8614533&postcount=167) a couple posts up the page.:D

Capt Crunch

Oct 8, 2009, 07:18 PM

Looked at simply, how large is the set of all even numbers? How large is the set of all numbers divisible by 43564?

Those two sets have the same cardinality, that is, they have the same size. The reason is that there is a bijective (http://en.wikipedia.org/wiki/Bijection) map from one set to the other, namely 2x gets mapped to 42564x.

It doesn't matter that one set is a subset of the other. Quite weird.

Signal-11

Oct 8, 2009, 08:17 PM

Those two sets have the same cardinality, that is, they have the same size. The reason is that there is a bijective (http://en.wikipedia.org/wiki/Bijection) map from one set to the other, namely 2x gets mapped to 42564x.

It doesn't matter that one set is a subset of the other. Quite weird.

This goes back to the thing about there being no last 9. Most people don't have a good grasp on the concept of of infinity and this is one of those issues that trips them up.

I know plenty of engineers that can't grasp that the set of all even numbers is the same size as the set of all whole numbers. To me, this is a good mathematical shibboleth on whether a person actually fundamentally grasps calculus or if he is just a technician with a degree who does what the numbers tell him to do.

Ttownbeast

Oct 10, 2009, 02:06 AM

This goes back to the thing about there being no last 9. Most people don't have a good grasp on the concept of of infinity and this is one of those issues that trips them up.

I know plenty of engineers that can't grasp that the set of all even numbers is the same size as the set of all whole numbers. To me, this is a good mathematical shibboleth on whether a person actually fundamentally grasps calculus or if he is just a technician with a degree who does what the numbers tell him to do.
What if anything anyone has said leads you to an assumption that people are looking for a "last number" in a decimal that clearly is on going (eternal, forever, infinitely, whatever the term for endless in general you want to nitpick about concerning definitions) and still not so clearly equal to one _ pretty damn close by a constantly narrowing margin but still not 1 this .9 thing is a bit of mystery otherwise until someone figures out a way to express it accurately. You speak as if all the rules of numbers have been established and there is nothing that's going to change them. This is a case of an anomaly where no solid rule has been figured out so for sheer laziness on the part of mathematicians for the time being 1=.9(and so on) for practical purposes.

dukebound85

Oct 10, 2009, 02:12 AM

What if anything anyone has said leads you to an assumption that people are looking for a "last number" in a decimal that clearly is on going (eternal, forever, infinitely, whatever the term for endless in general you want to nitpick about concerning definitions) and still not so clearly equal to one _ pretty damn close by a constantly narrowing margin but still not 1 this .9 thing is a bit of mystery otherwise until someone figures out a way to express it accurately. You speak as if all the rules of numbers have been established and there is nothing that's going to change them. This is a case of an anomaly where no solid rule has been figured out so for sheer laziness on the part of mathematicians for the time being 1=.9(and so on) for practical purposes.

It has been proven

This is no anomaly.

This exercise is getting to the basics of what is the definition of what a rational number is plain and simple. It is not "laziness" by any means

In other words, if one cant understand that .9 repeating =1, than they don't understand the very definition of a rational number

Ttownbeast

Oct 10, 2009, 02:26 AM

It has been proven

I don't see how you are confused to be honest

This is no anomaly.

This exercise is getting to the basics of what is the definition of what a rational number is plain and simple

Not proven just supported, like a theory, all it has is data supporting it for now, and like any theory is not immune to becoming falsified by further research at some point should it become possible. Speaking of such things in a case closed fashion does no justice to progress pushing this topic toward that ugly little "D" word I prefer not to use.

dukebound85

Oct 10, 2009, 02:33 AM

Not proven just supported, like a theory, all it has is data supporting it for now, and like any theory is not immune to becoming falsified by further research at some point should it become possible. Speaking of such things in a case closed fashion does no justice to progress pushing this topic toward that ugly little "D" word I prefer not to use.

It is proven. It is not a "theory"

As I said, this is the very definition of what a rational number is.

Propeties of a rational number

The set \mathbb{Q}, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the integers \mathbb{Z}.

The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of \mathbb{Q}. The rational numbers are therefore the prime field for characteristic zero.

The algebraic closure of \mathbb{Q}, i.e. the field of roots of rational polynomials, is the algebraic numbers.

The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.

The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones. Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.

Emphasis mine

By using the very definition of a rational number, if .99999 and 1 are not identical, there must exist another rational number between them. There isn't, and by that fact, and that alone, they must be the same

It is not even questioned in the field of mathematics if .9999repeating=1 or not. It's more of a conceptual notion that tests if people understand the defintion of what exactly a rational number is

I am not making this up, I am giving you mathematical definitions that prove they are identical.

dukebound85

Oct 10, 2009, 02:49 AM

Here is a scenario for you to help you understand

Do you agree that 5 and 6 are separate rational numbers?

Yes, obviously but why?

Because there are an infinite amount of rational numbers seperating them, such as 5.5, 5.2, 5.1, 5.1111, etc.

Now take .99repeating and 1

Show me a number that is between .999repeating and 1.

You can not. As a result, they by defintion, are equal

The only way they cannot be the same is if you could find a number that is between them which can not be done

going back to an earlier post you had
Then is .8888infinity the same as .99999infinity? therefore .8888888infinity equals .99999999infinity? Since there is no number in between?

yes, clearly 0.9 is in between. As is .91, .92, .921345, , .9222222222, and an infinite amount of others

there is not however, anything between .9999999repeating and 1

Ttownbeast

Oct 10, 2009, 03:02 AM

It is proven. It is not a "theory"

As I said, this is the very definition of what a rational number is.

Propeties of a rational number

Emphasis mine

By using the very definition of a rational number, if .99999 and 1 are not identical, there must exist another rational number between them. There isn't, and by that fact, and that alone, they must be the same

It is not even questioned in the field of mathematics if .9999repeating=1 or not. It's more of a conceptual notion that tests if people understand the defintion of what exactly a rational number is

I am not making this up, I am giving you mathematical definitions that prove they are identical.

OK of course that is all this math based on a standard of ten symbols for single digit whole numbers we assume is a universal standard? (i think this is called base 10 but I may be incorrect here I'm rusty)perhaps there is a rule involving other standards that applies to what in this standard is freakish and causes these arguments we're having here with our system. I'm thinking about that dude a few posts back talking about "Contact" and recalling something I read once (not clear where) about how non-universal our standard may actually be, not once have I ever encountered mathematics outside our system in my daily life just more complex ways of manipulating it within the limits of those ten digits. Of course there is binary code and other standards with less digits. The problem here of course would be switching over to another standard (kind of like switching from standard measurement to metric in construction to give a rough analogy) What if under another base system this is not a problem and can clearly define the difference between the two in this system and given an equation to translate between the standards (kind of like calculating degrees F to and from Degrees C does) could show they are different?

Ttownbeast

Oct 10, 2009, 03:09 AM

there is not however, anything between .9999999repeating and 1
Not by our common system believing they are the same doesn't necessarily make them the same.

Signal-11

Oct 10, 2009, 08:26 AM

What if anything anyone has said leads you to an assumption that people are looking for a "last number" in a decimal that clearly is on going (eternal, forever, infinitely, whatever the term for endless in general you want to nitpick about concerning definitions) and still not so clearly equal to one _ pretty damn close by a constantly narrowing margin but still not 1 this .9 thing is a bit of mystery otherwise until someone figures out a way to express it accurately. You speak as if all the rules of numbers have been established and there is nothing that's going to change them. This is a case of an anomaly where no solid rule has been figured out so for sheer laziness on the part of mathematicians for the time being 1=.9(and so on) for practical purposes.

I'm trying to be polite here, but at this point, you're way off the rails and the things that you're writing make no sense. It's obvious at this point that you don't understand the underlying mathematics. I also won't argue mathematics with someone who doesn't understand what constitutes a mathematical proof.

Not by our common system believing they are the same doesn't necessarily make them the same.

This statement speaks very clearly to your level of understanding. 0.(9) is another way to write "1." This is a very useful concept for dealing with curved lines. Another way to write 1 would -cos(pi).

Or better yet, -(e^i(pi)).

Ttownbeast

Oct 10, 2009, 01:03 PM

Duke: Between the two there is something, it's undefined but there is something just because we have no "number" as a symbol for it doesn't make it any less probable that they are two separate numbers. Signal:I am not saying that anyone here is a bad mathematician because I mention laziness--of course by definition .9'whatever' =1 as my last instructor stated by definition and we went over all the exercises and proofs concerning this in our coursework but he also stated they are not equal in the sense that working the problem out is a waste of time better spent on math that can be applied to our reality (this from a gentleman who started learning calculus before the days of the pocket calculator, doing everything by a wooden slide rule) he went on to state that mathematicians are lazy (not my words)and who knows when or if a solution will be discovered resolving this issue, but you're taking such comments personally when it is merely criticism of a tiny flaw in a concept that has been taught and assumed to be perfect.

dukebound85

Oct 10, 2009, 01:49 PM

OK of course that is all this math based on a standard of ten symbols for single digit whole numbers we assume is a universal standard? (i think this is called base 10 but I may be incorrect here I'm rusty)perhaps there is a rule involving other standards that applies to what in this standard is freakish and causes these arguments we're having here with our system. I'm thinking about that dude a few posts back talking about "Contact" and recalling something I read once (not clear where) about how non-universal our standard may actually be, not once have I ever encountered mathematics outside our system in my daily life just more complex ways of manipulating it within the limits of those ten digits. Of course there is binary code and other standards with less digits. The problem here of course would be switching over to another standard (kind of like switching from standard measurement to metric in construction to give a rough analogy) What if under another base system this is not a problem and can clearly define the difference between the two in this system and given an equation to translate between the standards (kind of like calculating degrees F to and from Degrees C does) could show they are different?

sorry but if your argument is based off a movie i really dont know what else to say

If you actually take time to research it, you will see that the base has nothing to do with this.

Mathematics is completely independent of base
Not by our common system believing they are the same doesn't necessarily make them the same.

They ARE the same. Why are you refusing to understand the very definition of what constitutes a rational number?

Duke: Between the two there is something, it's undefined but there is something just because we have no "number" as a symbol for it doesn't make it any less probable that they are two separate numbers.

There is nothing between them. Thats the concept of .(9) going on for infinity

I have provided you mathematical proofs and definitions.

You have provided me your opinion and an argument based off a science fiction movie. You need to do better than that

Signal:I am not saying that anyone here is a bad mathematician because I mention laziness--of course by definition .9'whatever' =1 as my last instructor stated by definition and we went over all the exercises and proofs concerning this in our coursework but he also stated they are not equal in the sense that working the problem out is a waste of time better spent on math that can be applied to our reality (this from a gentleman who started learning calculus before the days of the pocket calculator, doing everything by a wooden slide rule) he went on to state that mathematicians are lazy (not my words)and who knows when or if a solution will be discovered resolving this issue, but you're taking such comments personally when it is merely criticism of a tiny flaw in a concept that has been taught and assumed to be perfect.

There is no flaw in the concept. It's a definition

Your instructor seems incompetent if he is unable to come to terms with this.

By the way, many people have learned calculus before the calculator. Hardly gives him any "superior" credibility just as a fyi

Ttownbeast

Oct 10, 2009, 11:12 PM

sorry but if your argument is based off a movie i really dont know what else to say

Not based on a movie I stated essentially that the mention of the movie reminded me of an issue with mathematics that ours may not be as universal a language as we think (the movie uses the assumption that it is universal for the plot--Sagan needed a device to help tie the story together in this instance), not that I got it from the flick you might be over-analyzing things just a bit here.

dukebound85

Oct 11, 2009, 05:24 PM

Not based on a movie I stated essentially that the mention of the movie reminded me of an issue with mathematics that ours may not be as universal a language as we think (the movie uses the assumption that it is universal for the plot--Sagan needed a device to help tie the story together in this instance), not that I got it from the flick you might be over-analyzing things just a bit here.

You earlier wanted proofs, which were supplied, along with mathematical definitions, showing how they are the same.

You use opinion and a vague notion from a scifi movie for your argument yet seem to accept that over proof..

You have yet to provide ANY evidence stating otherwise (but demand proof for the counter argument, which was given), yet still cling to the notion you are correct...why?

You can start with evidence showing this issue you speak of. I want proofs, not "opinions" with no supporting evidence:cool:
Not based on a movie I stated essentially that the mention of the movie reminded me of an issue with mathematics that ours may not be as universal a language as we think

stridemat

Oct 11, 2009, 05:33 PM

Wow I just wasted 30 mins of my life reading this, and I still do not understand it. Now Im not exactly stupid but this is so over my head.:D

Eraserhead

Oct 11, 2009, 05:50 PM

1.99999...9 = 2. /end thread.

They are mathematically equivalent, this is first year University maths, and has been rigorously proven.

The reason is because you cannot find a rational number (i.e. a fraction) between 1.99999...9 (to infinity) and 2, and between any two distinct real numbers you can always find a rational number (as is explained here (http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2002;task=show_msg;msg=0284.0002)).

You earlier wanted proofs, which were supplied, along with mathematical definitions, showing how they are the same.

Exactly. To be honest I don't see how you can argue with mathematical proofs. They make even evolution or gravity look like flaky theories in comparison.

c-Row

Oct 12, 2009, 03:57 PM

I thought of a math problem which probably has been chewed through before, but still...

So, a man sits in his car, 100 miles from his destination. His car goes exactly the distance to his destination per hour, so when he's 99 miles away from point B, he goes 99 mph only, and so forth. Keep in mind that he's constantly getting slower*, not breaking down from 100 to 99 mph, so when the distance is 99.3 miles for examples, the car goes 99.3 mph.

Q. Will he ever reach his destination? (* he would if he would drive the last mile with one mph, that's why I added the clarification)

This is somewhat similar to a short story by either Asimov or Dick I read some time ago, where a scientist in a tube gets shrunk more and more...

Eraserhead

Oct 12, 2009, 04:02 PM

Q. Will he ever reach his destination?

Yes, but it would take an infinite amount of time to do so.

nick9191

Oct 12, 2009, 05:07 PM

Here is a scenario for you to help you understand

Do you agree that 5 and 6 are separate rational numbers?

Yes, obviously but why?

Because there are an infinite amount of rational numbers seperating them, such as 5.5, 5.2, 5.1, 5.1111, etc.

Now take .99repeating and 1

Show me a number that is between .999repeating and 1.

You can not. As a result, they by defintion, are equal

The only way they cannot be the same is if you could find a number that is between them which can not be done
That would then imply that 0.99.. is not a number, but merely represents an unknown number. Therefore they cannot be equal.

Eraserhead

Oct 12, 2009, 05:30 PM

Therefore they cannot be equal.

This is simply false. Mathematicians have proven rigorously that they are equal.

dukebound85

Oct 12, 2009, 05:34 PM

That would then imply that 0.99.. is not a number, but merely represents an unknown number. Therefore they cannot be equal.

No, it is exactly 1

benbondu

Oct 12, 2009, 06:27 PM

Doctor Q

Oct 12, 2009, 06:30 PM

If an object moves from point A to point B, it must have been in motion for some length of time, starting at time S and ending at time T. But it can't actually move during that time period because at any given instant between S and T it is not moving. Why? Because an instant is an arbitrarily small lengthlof time during which we can take a snapshot and see that the rate of speed is zero.

Mathematically: We know that distance = speed x time. For any speed and positive number N that you specify, I can make sure the instant (time) is short enough so that distance will be less than N. Therefore, distance (which is a physical measurement so it must be non-negative) is smaller than any positive number. Hence distance is 0 at any instant.

No matter how many zeros you sum (even an infinite number of them) you still get zero. We know that the object is not moving at any given moment (instant). So across all moments from time S to time T, the object has not moved.

And with that I have proved why the cars on the 405 Freeway in Los Angeles never move!

anywhere08

Oct 12, 2009, 06:59 PM

2 x 17 = 34

1.999999999999 x 17 = 33.9999999999998

Those two aren't equal, thus 2 =/= 1.9999...infinity. Am I missing something here?

benbondu

Oct 12, 2009, 07:09 PM

Because an instant is an arbitrarily small length of time.

I take slight issue with your definition of instant.

instant (http://www.merriam-webster.com/dictionary/instant)

Arbitrarily small is not the same as infinitesimal, just like arbitrarily large is not the same as infinite.

Eric5h5

Oct 12, 2009, 07:18 PM

2 x 17 = 34

1.999999999999 x 17 = 33.9999999999998

Those two aren't equal, thus 2 =/= 1.9999...infinity. Am I missing something here?

Er, yes...you missed the fact that 1.999999999999 is not at all the same thing as 1.99999...9. You just rounded it off, and made a different number.

--Eric

savar

Oct 12, 2009, 07:20 PM

2 x 17 = 34

1.999999999999 x 17 = 33.9999999999998

Those two aren't equal, thus 2 =/= 1.9999...infinity. Am I missing something here?

Yes, but:

http://xkcd.com/386/

This thread really surprises me. I learned this stuff in 9th grade. It's stuck with me ever since. I went to a pretty good school, but surely you folks learned this kind of math at some point in high school? Or even college?

anywhere08

Oct 12, 2009, 09:05 PM

Yes, but:

http://xkcd.com/386/

This thread really surprises me. I learned this stuff in 9th grade. It's stuck with me ever since. I went to a pretty good school, but surely you folks learned this kind of math at some point in high school? Or even college?

http://drunkenachura.files.wordpress.com/2009/07/internet-serious-business.jpg

Eraserhead

Oct 13, 2009, 08:22 AM

Am I missing something here?

1.99999999999 with a finite number of 9's is not the same as 1.999...999 with an infinite number of 9's. (I've been beaten by ages by Eric5h5 though :o)

This thread really surprises me. I learned this stuff in 9th grade. It's stuck with me ever since. I went to a pretty good school, but surely you folks learned this kind of math at some point in high school? Or even college?

Its poor scientific literacy.

Doctor Q

Oct 13, 2009, 03:33 PM

Its poor scientific literacy.
I think many people learn this kind of thing in school and then forget it -- unless it's related to their further studies or they find math to be fun and interesting on its own.

Speaking of fun, tell your friends that the average person has 1.9999... parents and see if they believe you.

benbondu

Oct 13, 2009, 06:30 PM

0.(9) is another way to write "1." This is a very useful concept for dealing with curved lines.

You've got me interested. Please elaborate.

BTW, not to nitpick, but isn't "curved lines" sort of an oxymoron? Aren't they just called curves?

Signal-11

Oct 13, 2009, 08:06 PM

You've got me interested. Please elaborate.

BTW, not to nitpick, but isn't "curved lines" sort of an oxymoron? Aren't they just called curves?

Yes and no. In strict mathematical terminology, a 'curve' can include straight lines.

Anyway, if you are interested, you should consider taking some analytical geometry and calculus courses to learn the fundamentals. Of course, judging from some of the participants of this thread, this isn't a guarantee of understanding. Specifically, if you understand what a limit is and what it represents, then you will understand why 0.(9) = 1. (There is no last 9!)

As I mentioned, there are different ways of representing 1 depending on the type of mathematical problem with which you are concerned. If you're working with circles, your representation might include the term pi.

benbondu

Oct 13, 2009, 11:17 PM

Yes and no. In strict mathematical terminology, a 'curve' can include straight lines.

That was sort of my point. A line is a straight curve. To say straight line is redundant and to say curved line is oxymoronic; in strict mathematical terminology as you say. But at any rate, that's not important.

As I mentioned, there are different ways of representing 1 depending on the type of mathematical problem with which you are concerned. If you're working with circles, your representation might include the term pi.

It sounds like you're referring to using a symbol, like "pi" or "e", to represent a number that is not possible to represent exactly with simple decimal notation. Were you talking about "pi" and circles when you said "This is a very useful concept for dealing with curved lines."? If so, there's no need to expound anymore. I was just curious if it was something else that I might not have heard of before.

I would probably make a distinction between a number like 1, pi, or e, and an expression like -cos(pi) that evaluates to 1. I personally would call [-(e^i(pi))] an expression rather than a number. Only because it involves operations like multiplication and exponentiation.

I'm a little torn though about whether or not I would call 0.(9) a number. I can see a case for it, but really, when I look at it, I see a representation of an infinite series that evaluates to 1. Thus in my mind, 0.(9) = 1, but I wouldn't necessarily say 0.(9) is 1. Probably a silly and possibly an incorrect distinction.