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786 Comments
- dirtyfratboy, on 10/12/2007, -23/+164"if you actually believe that then your a ***** moron and should not be teaching anyone
that's whats wrong with the public education system in the United States dumb ass teachers like you"
"i don't have to disprove anything dirtyfratboy's logic makes no sense what so ever"
First off, learn about punctuation before hating on the U.S. public education system.
Secondly, you're a stupid idiot. This isn't my blog. I just submitted a link to this theory. - ComputerWiz, on 10/12/2007, -6/+49Just look at the infinite series equation. If anyone knows calculus, this will probably make sense.
.99999... = 9/10 9/100 9/1000 ...
just use a/(1 - r)
Also, isn't any rational number able to be written as a fraction with a finite number as the numerator and denominator? If so, how would you write 0.9999...? Think about that.
Question: If any number is infinitly close to another does that make it equal to that number? Now I'm not sure because isn't that the main idea behind calculus that it isn't? - twollamalove, on 10/12/2007, -18/+61You know what, ***** it. Today I've lost my faith in digg completely.
- D4V1S, on 10/12/2007, -15/+56I wouldn't say that unless I could disprove his theory. Can you?
update: without reading KineticFlow's comment....:-) - joeshlub, on 10/12/2007, -32/+73.99999... Isn't a number. Thats like saying infinity isn't a number. Its an innacurate representation. .99999.... is a decimal approximation. Case closed. .99999 doesn't equal one because .99999 is an approximation. So all of this stuff with using fractions doesn't apply, because 8/9 DOES NOT ACTUALLY EQUAL .8888..... because it is an approximation.
So simply put, infinity isn't an actual number, its a concept. .9 repeating isn't a number either, its just a concept, specifically an APPROXIMATION. if it could really exist, it perhaps would be equal to one. But try writing it. just like infinity, its impossible. Concepts don't equal numbers. Case Closed. Try refuting that. Seriously, don't dig me down unless you can refute it. And if you can, At least reply, I'll be happy to hear about it.
At best, it is, yes, APPROXIMATELY one. - twollamalove, on 10/12/2007, -18/+50To be fair have not yet read the article, but I believe the headline is referring to 0.9 (repeating), which is indeed mathematically equatable to 1.
- KineticFlow, on 10/12/2007, -37/+61This article tries to explain that 0.99999... = 1, but all his arguments are conversely supporting that those two rational numbers are not equal at the same time. Non-mathematical arguments made, such as "looking for patterns" and inaccurate "definitions" of words.. and he didn't learn this until he went to university?
He also has an ignorant attitude against those who criticize his opinions, saying that ..
"...sum of an infinite series (go talk to some math professors, and see how far you get) or to deny the very existence of the number .9999.... "
What? SUM of an infinite SERIES? Series, by definition, is the sum of a sequence of terms.
He's suffering in the world of real numbers (not even irrational, but rational numbers).
I wonder what he would say to his students when he has to teach complex numbers. - Mohonri, on 10/12/2007, -5/+27I'll take a different tack from everyone else here. Instead of proving that .999... = 1 (which it does), I challenge you to prove that it does not.
If .9999.... != 1, then they must be two distinct numbers.
If they are two distinct numbers, then there *must* be a third number X such that .99999.... < X < 1.
If there is no such number X, then .99999.... = 1.
Let me know when you find it. - vileS, on 10/12/2007, -2/+22THERE ARE FOUR LIGHTS!!
- Momoru, on 10/12/2007, -5/+25The bottom line is that there is nothing you can subtract from 1 to get .9999999...
- twollamalove, on 10/12/2007, -6/+25This is no different that a converging integral in calculus. No one would ever try to tell their calculus teacher that just because two lines never actually touch (they get infinitely close) the area between them cannot be determined.
The point is that
lim n-> infinity PI(m=0->n) (0.9 * 10^(-m)) = 1
0.9 (repeating) is infinitely close to 1. Better put 0.9 (repeating) = 1. - johndi, on 10/12/2007, -31/+47Many mathematicians like to play number games. Unfortunately some people buy into it and preach it like a religion, and have great faith in their beliefs. It's really sad when teachers do this, and then try to browbeat students with their dogma.
I've seen ''proofs'' that all horses are white, 1+1=3, and that women are the root of all evil (sure it uses words, but the concept is the same). The numbers work out, but you can lie with numbers as well as words.
Here is an interesting concept I've seen math geeks use to bend people's minds. There are an infinite number of infinities. After all if you start counting at one and go up you will never run out of numbers, the same is true if you start counting at 6, 7 or 42. If you're really adventurous you'll notice you can do this counting forwards and backwards. The truly talented will be able to do both at the same time, and it's likely a good way to throw oneself at the ground and miss ; ) Anyways, here are some math jokes to amuse.
http://www2.cs.uregina.ca/~cowles/MathJoke.html - mystagogue, on 10/12/2007, -4/+20i'm surprised that so many on digg have a problem accepting this. maybe just because i have a math degree, but i remember encountering this before college. between every two rational numbers is another rational number. if .9999... didn't equal one then tell me what number is between it and one?
- Xeworlebi, on 07/25/2008, -1/+17And thats why mathematics don't like endless decimals, but use fractions, its not presice enough. Thats also why we use letters and signs for non-repeating decimals like pi, i, ... otherwise you decide where it stops, but it doesn't stop.
- Jakelshark, on 10/12/2007, -7/+22there is such thing as infinite geometric series, its algebra 2 stuff
- swalter, on 10/12/2007, -6/+21@thebigmagu
"dumb ass teachers" aren't the problem with the public education system in the United States, it's ignorant parents and people like yourself. Why don't you try to sue him or something.
If you read the article, the author gives numerous examples proving his theory. - Alegis, on 10/12/2007, -16/+30Agreed. The problem with such theories is that one calculates with "infinity" as if it's a normal number.
The definition of 0.9 (- above the 9) is that it its a series of infinite 9. By definition, it isn't even 1.
If you can prove otherwise, it means those calculation methods are invalid. Like dividing by 0 for example, or tangens of 90 degrees. - exipolar, on 10/12/2007, -2/+15What I really find funny is the fact that, since our numbering system is a base 10 exponential system, it can allow for such definitions of different numbers.
what people don't realize is that .99999 does not equal 1, duh! but when we say .99999 repeating, that DOES equal 1. it's the fact that we express it as an infinite term that allows it to be equal to 1. that said, if we had a base 4 numbering system, we would argue whether or not .3 repeating was equal to one, which in a base 4 system it would be.
whoever, the real fact of the matter is that we have to live with the fact that there are more than one way of expressing the same thing. numbers the way we represent them are simply compressed base 10 poly nomials
like 4829= 4*10^3+8*10^2+2*10^1+9*10^0
when you realize that, it's just obvious that .9 repeating is just another way of saying
sum from n=1 to infinity 9/10^n
which is what the article goes over
when you think about it, it's just an odd way of saying one
just like e^(i*pi/2+2pi*n) is a very odd way of saying 1 - LucasOman, on 10/12/2007, -5/+18It's not a religion. It's fact. What are your qualifications to argue with these mathematical proofs? The "proof" that 1+1=3 contains an arithmetic flaw that is simply hard to identify. Those false proofs rely on such flaws to trick their readers. Can you find any flaws in this teacher's proofs? I challenge you to do so. There are none. And there are not infinite infinities, there are infinite sets that have infinite size. There are, however, only two sizes of infinity, rationals and reals.
Honestly, I'm amazed that this article has caused such a stir on digg. I would think that only 8th graders would feel the need--and have such a lack of knowledge--to argue with such a solid fact in mathematics. - pfunked, on 10/12/2007, -23/+35> .9999... does not equal 1, there a basic fundamental of math that says each and every number (including .999... and 1) have their own equal and seperate placing on the mathmatical numberline
Nonsense. .999... and 1 are just different ways of writing the precisely same number, with the same place on the numberline. The same way that 3/3 is another way to write 1. - whoutz, on 10/12/2007, -1/+13I quote the recurring_decimal page on wikipedia which puts it best, "The method of calculating fractions from repeated decimals, especially the case of 1 = 0.99999..., is sometimes contested by the mathematically naive:"
Hasn't anyone had freshman calculus? Here's the proof on wikipedia article:
http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1
I also suggest you read about power series and reimann sums. - umrgregg, on 10/12/2007, -3/+150.99... = 1.
Answer without gee wiz tricks or arrogance here: http://mathforum.org/dr.math/faq/faq.0.9999.html - MMNManiac, on 10/12/2007, -6/+18wow. forget the proof. i think this thread proves more that technical people doesnt necessarily mean smart people.
it's very, very simple folks:
1/3 + 1/3 + 1/3 = 1, correct?
translate into base-10 decimal:
.3333... + .3333... + .3333... = .9999... = 1
I don't see why there's an "argument" over this. You either get it or you don't. It's not up for debate, it's just is.
It's not a math "trick" - those who don't understand simply have a misunderstanding of math. It's just is. Not subject to discussion.
~MMN Maniac - LucasOman, on 10/12/2007, -2/+14LISTEN PEOPLE. We're not talking about limits here. This number doesn't APPROACH 1. It's a number, not a series. It IS 0.999... followed by infinite 9s. It doesn't "grow". It doesn't "get longer". It IS long. Infinitely long.
If .999... is not equal to 1, then there HAS to be a number between it and 1.0. Find it. You can't. - ae3145, on 10/12/2007, -1/+13Hi.
I have a math degree. These are my conclusions of the day: Digg for the quick news, Slashdot for the commentary.
This is high school stuff guys, don't embarrass yourselves. If you want something to think about, consider what e^(pi i) = -1 actually means. Google it, do some research, learn, enjoy.
Take care,
g - aristotle1990, on 10/12/2007, -9/+21There's an extremely simple proof to show that 0.999.... = 1.
1/9 = 0.111...
2/9 = 0.222...
8/9 = 0.888...
9/9 = 0.999...
9/9 = 1.
0.999... is the furthest you can get to one. Precisely because it doesn't end, it equals one. - inactive, on 10/12/2007, -4/+15http://mathforum.org/dr.math/faq/faq.0.9999.html
http://www.cut-the-knot.org/arithmetic/999999.shtml
it is not like he invented this or pulled it out his butt.
This is famous, well proven and well accepted
find me one proof against the arguement... one not made up by a commenter.
Do you know what it means to be infinately close to one? means there is no finite difference between the two. There is a meta phyisical one but not a finite one. - pfunked, on 10/12/2007, -3/+14The trick here is that it's not "next to" 1 at all. It actually is "1".
Imagine conversely trying to represent an "infinitely small" number. 0.000... (zeroes going on forever). Is this equal to 0 or "next to" 0? We know we can just as easily write 0.0 as 0.000..., and there's no value at the end of those infinite zeroes, so 0.000... *is* zero. The same concept with 0.9999...., just a bit harder to swallow. - LucasOman, on 10/12/2007, -0/+11It's NOT a limit. It's NOT a series. It's a NUMBER. It doesn't "grow". It doesn't "get longer". It is what it is. It is a number that has nothing on the numberline between itself and 1, meaning that it is equal to 1.
- benhocking, on 10/12/2007, -5/+16@psiit: Actually, you're just reinforcing its consistency. You can't stick a 1 after an _infinite_ series of 9's. And, 8.9999... is, in fact, 9.
- Norweed, on 10/12/2007, -4/+15LOLOLOL....
This thread is soooo funny. I can see all these people that think they know math, but in fact are just guessing and spouting off stupid crap. Take a few college level math classes and it'll become PAINFULLY obvious that infact 3/3 = .99999... = 1
I don't really know how to dumb it down any more than this, but see if you understand this. The fact that there are an infinite number of 9's makes it equal one. If there were ANY finite number of 9's then it would NOT equal 1, however once this list is declared as going on forever it equals one.
Hell, this is pre-calc limits stuff. Look up any number of series that approach a number. They will never equal that number untill the limit is taken to infinity. When this happens it actually GETS to the number, not just really really close. Hence why 3/3 = .9999999..... = 1
If you don't get it then at least stop making yourself look stupid. - Virak, on 10/12/2007, -5/+16"Each 9 you add on to the end does bring the number closer to 1."
How, exactly, would you add something on to the end of something that HAS NO ***** END? IT'S INFINITE! - aluhain, on 10/12/2007, -3/+13As an actual mathematican I find these sort of debates pretty amusing. If anyone actually wants to know why .999...repeating is actually equal to 1, what you need to do is take a Real Analysis class. Even an undergraduate level course should cover this. Not calculus, not algebra, etc. The arguments that the author of the article used are amusing, but most of them are more sleight of hand than anything else. The fact of the matter is though that the two numbers are in fact equal, the same, etc. However you want to say it. Some people have been taking the approach of .9 .99 .999 ..... etc. This is sort of the calculus level thinking, and it is not quite the correct way to approach this. The limit of that sequence (not series) is .999... repeating. Why? Because there are an infinite number of 9s. However, numerically this is also approaching 1 from below, and 1 is clearly an upper bound. In fact, 1 is the least upper bound for this set. Anyway, there are several ways to approach it at this point, the fact that limits are unique in the Real Numbers is easy to prove, etc. and even you calculus level people should remember that. So that's one way that you can prove this fact rigorously, from the Zermelo-Frenkel (With or Without Axiom of Choice) axioms (basic logical building blocks that the number systems which don't make your head explode are built from).
There is a bit of play in the axiomatic system though, and many times mathematicians will simply define these two numbers to be equal to begin with. So no proof necessary, they are simply equal. In fact, you NEED these two numbers to be equal for your number system to work out right. As above, it isn't necessary to make this definition and sidestep things, but it is one way around things. - simpleid, on 10/12/2007, -1/+11I'm sorry, it's a fact. It equals 1. You'll be ok. I didn't say it was up for debate, so don't try. It's been debated. It's been decided.
This isn't new. It's old. It's 1. Waste all the time any of you want.
Why don't you just go varify by e-mailing any mathematicians you can find instead of trying to argue so pointlessly. Why? Because you obviously don't 'get' it. - zentraedi, on 10/12/2007, -4/+14His proofs are 100% right. I had no idea math educuation has fallen so far.
I knew that before I even went to university for math. I remember losing a percent on my grade 10 biology mark because my teacher didn't understand it. - twollamalove, on 10/12/2007, -7/+17@romhom: you're saying that 0.9... - 0.9... != 0?
@TheRonald: Nice tactic. Why back anything up when you can just blurt claims like that out? - drwtsn32, on 10/12/2007, -4/+13psiit:
8.999999..... does indeed = 9
The number isn't "8.9999.........1". If the 9's are infinitely repeating, you can't all of the sudden stop and put a 1 at the end. - zentraedi, on 10/12/2007, -2/+11Wow, I can't believe there are morons who are so confident in their stupidity they would mark this article as innacurate.
- hessamz, on 10/12/2007, -10/+19ahhh the rounding error of life
- interiot, on 10/12/2007, -2/+11Proof... that beating a dead horse sometimes IS fun. :) Critics pwned.
- interiot, on 10/12/2007, -1/+10the Crowd versus the Expert. And this is the age of Web 2.0, so of course the crowd is right.....
- drwtsn32, on 10/12/2007, -2/+11Wrong. 0.99999...... + 0.0000000.... = 1. You can't have a number like 0.00000000....1. How can an infinitely repeating zeros suddenly change and end with a 1?
- drwtsn32, on 10/12/2007, -4/+13"but when multiplying by the 10 you remove 1 less 9 from the infinite number of 9s"
Um, how do you have one less than infinite? If one less than infinite is not infinite, then what number is it exactly?
I thought this guy saying 0.999999...=1 must obviously be wrong when I first saw this on digg, but after reading his reasons they certainly do make sense. - BenjaminJoffe, on 10/12/2007, -11/+20I read digg all the time but this is the first time that I have felt compelled to comment.
First of all please do not comment on this article unless your opinion is well formed. I see many comments by people who have obviously not studied maths to a high degree. What seems right to you is irrelevant unless you can understand the underlying mathematics.
For those who disagree with the article and continue to believe that 0.999... and 1 and different I propose another way of thinking about it.
For two numbers to be distinct we must be able to observe that they are distinct. If these two numbers where "plotted on the number line" as many of you state, then they would clearly be very close together. So close in fact that they would appear the same, zoom in a little, a little further, keep doing that until the world ends and the two points would still appear at the same place.
So if there is no way of ever knowing these two numbers to be different then they are not different. - mystagogue, on 10/12/2007, -3/+12no. where does the five go? the nines are repeating forever.
- Dimensio, on 10/12/2007, -0/+9"It's entertaining to read the comments and flawed logic of those here who think it is not true: it reflects the general lack of rigor in this country."
The problem is likely that it does not "feel" true to those who ojbect. The reality and truth of the situation is irrelevant; how they feel about the presentation is all that matters when evaluating the validity of the claim. Unfortunately for these individuals, neither mathematics nor reality are determined by how an individual feels about a particular point. - Norweed, on 10/12/2007, -1/+9You're missing the fact that .999... is EXACTLY THE SAME as 1. Not close...not really really close....EXACTLY THE SAME.
- drwtsn32, on 10/12/2007, -2/+10That's exactly his point! He is showing that there can't possibly be something "after" an infinite number of 9's.
- jejones, on 10/12/2007, -0/+8Oy vey. The sum of an infinite series is defined to be the limit of the sequence of sums of the first n terms as n goes to infinity. (Please don't go on about "infinity is not a number"; if you take calculus, you'll find that all the places where the infinity symbol is used are defined in ways that don't use it. It's just a convenient shorthand.)
By .9999... we mean the sum of 1/10^n from n = 1 to infinity. We can all agree that .9999.... = -log10(x), the sum of 1/10^n from n = 1 to M is between 1 - x and 1.
Sorry, folks, .999... is equal to 1. - Canthros, on 10/12/2007, -12/+20Very, very simple proof:
1/9 = 0.111...
9*(1/9) = 9*(0.111...) = 0.999... = 9/9 = 1
QED.
(Wish I could claim to have made that realization myself, but I picked it up in the last argument of this sort I saw.)
picaman: The naysayers are basically arguing Zeno. The important thing to remember about Zeno's Paradox is that an empirical test would quite quickly prove it wrong, i.e. Achilles would win the race. -
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