Why does 1/3 = 0.3333. . . . 3333.
It is an infinite number, one out of 3 of something is a finite fraction.
Why do this and certain other fractions have infinite values?

Why does 1/3 = 0.3333. . . . 3333.
It is an infinite number, one out of 3 of something is a finite fraction.
Why do this and certain other fractions have infinite values?
Ummm 1 divided by 3 or 1 divided by 9Originally Posted by cool skill
Fractions are just division problems
It's just an artifact of the decimal system.
Oh. And. It's not 1.333333333...
It's 0.3333333....
1 1/3 is 1.333333333...
2/3 is 0.666666666....
As insanity said, it's just a division problem. But, as I said, the infinitely repeating decimal is an artifact of the number system. If we used a system based on the number 9 then it wouldn't be. But, then we'd start running into problems with other numbers. Every possible number system would have such artifacts.
It's not infinite. How can it be  it is <1?Originally Posted by cool skill
If we used a base 9 system, 1/3 = 0.3. But 1/2  0.4444. . . . 4444.Originally Posted by invert_nexus
It isn't infinity because it doesn't go all the way past all of the numbers.
But it does keep going on and on and on infinitely depending on which flawed number system we use.
Therefore, what kind of number system should we use that doesn't do this?
Please, quit writing repeating decimals like there's a "last place", 1/2 base 9 is 0.444..., not what you have.Originally Posted by cool skill
Can you explain why you think this is flawed? Some rational numbers have infinite repeating decimal expansions I don't see the big deal. If it makes you uncomfortable keep all your rational numbers as fractions and approximate your irrationals by fractions (which you're doing anyway anythime you truncate an irrationals decimal expansion).Originally Posted by cool skill
There is no such representation. It is easy to show for integers; just choose an integer (other than 1) that is relatively prime to your chosen base and look at its multiplicative inverse. Because representations are determined by integral powers of base, and all positive powers of that base are again relatively prime to the chosen integer, you'll never be able to get rid of the remainder, so to speak. Try it yourself and it'll become pretty obvious.Originally Posted by cool skill
As for other possibilities, you could choose a noninteger as your base, but then you end up with multiple representations for a single number, which is generally considered a bad trait. And I'm fairly sure it doesn't solve your problem.
You can use factorials or fibonacci numbers to make some rather unusual bases, but they have their drawbacks, so it'd be a hard sell.
One option you have is to count all the rationals in the manner Cantor used many years ago to show that the set of rationals and the set of positive integers were of the same size. Then you have a onetoone correspondence between all those "faulty" fractions (and all others) and integers. Then just represent each integer with the corresponding number of tally marks. For example, 1/3 is the 5th number (I think) so you would write it 11111.
Or, for that matter, just assign your own symbol to every rational. It'd be just a useful.
Basically, you just have to give up the idea that this trait is a fault. It's just a convenient representation system, nothing more.
How is it convenient if it keeps going on and on?
How is that not a flawed representation?
Using a base 10 system, 1/3 does not equal an exact number. In logical reality, a unit split into 3 parts is a unit split into 3 finite parts.
The base 10 system does not represent logical reality.
That is why these number systems seem to be flawed.
If you use it logically 'as is', it doesn't work. If you take it as approximations, it comes close infinitely close, but it never gets anywhere. But you can still use it practically.
It still makes no sense.
Suppose we stop using decimals, and depict fractions as simply fractions. What would be the implications?
0.333... is no less 'exact' than 0.5 is.Originally Posted by cool skill
The decimal system is convenient in the sense that it gives us a way of representing any real number we like. If you like another view, the decimals are the real numbers, or at least a model of them. No one has claimed that these infinite decimals (or the real numbers themselves) were reality. They are handy in modeling of realityyes, but realityno.
If you're doing "real world" computations, like finding the area of a circle, you are going to be approximating any irrational number by a truncated decimal (i.e. a fraction) there's no way around this unless you're willing to accept placeholders (like the symbol for pi) in your answer. You can think of truncated decimals as converting every fraction to the same base (actually power of the same base), this makes them much easier to work with.
What makes no sense? That we have to approximate things? Get used to it. You aren't going to have an exact fraction for pi, or any other irrational number (hence the name).Originally Posted by cool skill
Everything you would use for a computation is a truncated decimal anyway (fraction). If you were using 0.333 (note it's truncated) to represent 1/3 you were hopefully already aware that this is an approximation and you always had the option of leaving this division by 3 until the end or using more decimal places if you needed more accuracy. If you didn't need more accuracy, than nothing is lost by using 0.333 instead.Originally Posted by cool skill
If you went and abolished decimals and used only whole fractions, many things would be a giant pain in the ass. Can you express pi neatly to within 1/10000? Are you just going to write 31415/10000? Maybe 333/106? How is this better than 3.1415?
Approximations do not make sense. In order for something to make sense, it has to be exact. It's human logic. I doesn't matter if you are einstein or not, these approximations defy logic period, and therefore make no sense to the human mind. 0.333... is not an exact number. It represents an approximation accorsing tot he decimal system. It does not represent logical reality.
The object is to represent reality correctly and logically. 0.5 equals 1/2. 0.3333.... does not equal 1/3. Hence the term irrational. It cannot be comprehended with the human mental limitations of logic.
Really? In that case, what does 0.333...  (1/3) equal? I hope you're not going to propose that it equals an infinite number of zeros followed by a 1...Originally Posted by cool skillNo offense, but you're the only one who seems to be having trouble comprehending it.Hence the term irrational. It cannot be comprehended with the human mental limitations of logic.
If I was the only one having trouble comprehending it, it would not be deemed irrational.
It is deemed irrational because it cannot be comprehended by normal rational thinking. You have no idea what you are talking about. If you think you can comprehend irrational numbers, you suffer from serious mental delusion. No rational mind can comprehend irrational numbers.
Every reputable mathematition knows that 0.3333.... infinitelty approaches 1/3, but does not equal 1/3.
Approaching a number is not rationally the same thing as equal to a number.
I could be wrong. But, I don't think that it is an irrational number. An irrational number would be something like pi that never actually converges to a definite number. It keeps going and going but it does not do so in a repeating fashiong.
Let's look at a different number.
0.825825825825....
This too is a rational number, I think, because it is a pattern of repeating numbers.
Actually, I just looked up the defintion of an irrational number. I'm not sure if the latest number would be called one or not, but 0.33333... is definitely rational.
"An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q. Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational."
From: Wolfram's Math World.
And. I think that does cover the example I just gave as well as it is a periodic repeating decimal.
So. Relax. It's all rational.
I'm afraid you are the one who doesn't know what they are talking about. A rational number is socalled because it is the ratio of two numbers. An irrational therefore is not such a ratio.Originally Posted by cool skill
Look: 1/3 x 3 = 1
0.333... x 3 = 0.999...
1/9 = 0.111...
0.999.../9 = 0.111...
so 0.999... = 1
It's all to do with the continuity of the number line.
Wrong. Irrelevant.
0.3333.... infinitelty approaches 1/3, but does not equal 1/3.
You have yet to prove otherwise.
OK, boss.Originally Posted by cool skill
Coolskill. It's you that is wrong. And, as usual, you just refuse to accept the fact.
It's part of the definition. 1/3=0.333333.....
Deal with it.
Edit:
Hey, Guitarist. Still ignoring me? Didn't I call you an anus at one point or another? Guess who was the original anus? Mr. CoolSkill here. And. Now seeing you two side by side... I take back the anus on your behalf. There's no comparison...
Heh.
If 0.333... infinitely approaches 1/3, then the difference between the two is infinitely small. That means that the difference is zero. “Infinitely small” and “zero” are the same number. If there was any actual difference, then the difference wouldn’t be infinitely small. Since there is no difference, they are the same number.Originally Posted by cool skill
I don’t know why this is so hard for you to understand. I’m still anxiously awaiting to hear from you as to what (0.333…) – (1/3) equals.
At which point, I remove you from my (overlong) ignore list.Originally Posted by invert_nexus
Thanks for the sorry.
Moron Troll. Retard.Originally Posted by invert_nexus
A number that infinitely approaches 0 does not equal zero. There is a huge difference between a number and infinitely approaching a number.
Who said that there was no difference. The fact that there is a difference is the whole issue. Oh wait you must be making up your own false mathematics.Originally Posted by Scifor Refugee
Hey. I was nice in here for quite some time. It wasn't until I grew tired of you just ignoring what everyone said and just ignoring the mistakes you'd made in definitions and whatever that I finally decided to call an anus an anus.Moron Troll. Retard.
Again.
Well. There's a lot of mathematicians who would differe with you on that subject.A number that infinitely approaches 0 does not equal zero. There is a huge difference between a number and infinitely approaching a number.
1/3=0.3333......
Simple as that.
An infinitesimal difference. But it's not really a difference in fact rather than an idiosyncracy of the decimal system. If there was a problem with the number then it would show up in every number system. Not just one.Who said that there was no difference. The fact that there is a difference is the whole issue. Oh wait you must be making up your own false mathematics.
And what are you saying about false mathematics? Something about your whacko definition of irrational and rational numbers?
Heh.
Know what you're doing?
It was your favorite word a few weeks ago.
Will you say it?
Or should I?
Heh. This is so wonderfully ironic.
No. I will now simply cut and past my earlier post for you, since you apparently didn't read it:Originally Posted by cool skill
If 0.333... infinitely approaches 1/3, then the difference between the two is infinitely small. That means that the difference is zero. “Infinitely small” and “zero” are the same number. If there was any actual difference, then the difference wouldn’t be infinitely small. Since there is no difference, they are the same number.
Now before you say anything else in this thread about how 0.333... doesn't exactly equal 1/3, please tell us what 0.333...  (1/3) equals. Since you propose that they aren't equal, there must be some nonzero value that results when you subtract one from the other.
Moron Troll. Retard.Originally Posted by invert_nexus
Fine. I'll say it.Moron Troll. Retard.
You.
Are.
Caviling.
Muahhahahahahahhahahahaha!!
HA!!
Seriously. That's so sweetly ironic one would almost think you did it on purpose.
Anyway.
Why don't you speak to the point rather than respond with your usual witty response?
What's 1/30.333333....?
Hmm?
I already saw this and responded to this. It is WRONG.Originally Posted by Scifor Refugee
Infinitely approaching a number is not the same as being equal to a number.
Approaching someting is not the same as equaling something. You attempt to ask what it equals. It doesn't equal anything. It is simply approaching something.
First you mention that the difference is infinitely small.
Therefore, that difference is zero because infintiely small and zero are the same number. WRONG.
Infinitely small and zero are not the same number. If they are, you have yet to explain how they can logically be the same. Otherwise, I would be better off assuming that you have no idea what you are talking about. Show some proof that infinitely small and zero are logically the same thing.
That is the whole point of this thread, how is it that infinitely approaching can be equal to a number logically without being a paradox? Stop reiterating the point, and prove it.
Moron Troll. Retard. Get a clue. Oh I forgot. You're way too damn dumb!Originally Posted by invert_nexus
Cool skill,
You're being irrational. And as we all know from your outpouring of wisdom: "No rational mind can comprehend irrational numbers." That goes for you as well. No rational mind can comprehend an irrational retard, such as yourself.
Maybe you are being confused by the use of the term “approaches”. If a number infinitely approaches something, then it’s already there. You can consider it to have finished with it’s “approach” and to have moved on to “arrival”. You can check out to infinity, and it will always have the same value. If two numbers have the same value, then they are equal by definition.
I guess I'll ask for the third time what you think 0.333...  1/3 equals.
This would be the same as saying what does 1/3  1/3 equal. It's zero.Originally Posted by Scifor Refugee
Yeah, that's my point. But he doesn't think so.
Of course not. Reiterating that infinitely approaching a number is the same thing as equaling a number is not proof.
I have already stated this. Yet you keep reiterating the same thing. You are using circular reasoning to prove your point. Circular reasoning doesn't work. Insulting people doesn't work. Either provide an explaination or not.
Infinitely approaching a number means that you continue to get closer and closer to it without ever touching it. Note: WITHOUT EVER TOUCHING IT.
In other words, you get infinitely closer and closer without ever equaling it.
You are saying that getting infinitely closer and closer to a number without equaling it is the same thing as being equal to it.
Think of it this way:Originally Posted by cool skill
When you write 0.333... it doesn't mean that the repeating 3s are racing out in length forever, always increasing in number but never quite equaling 1/3. That is not what is meant by an infinite repeating decimal like 0.333...
I could sit down and write 0.3 and then keep adding 3s after the decimal point for years and I would, as you put it, continue to get closer and closer to the value of 1/3 without ever touching it. However, that's not what 0.333... means. When you have a repeating decimal like 0.333..., it means that the sequence of 3s is already at infinite length. It's not "continuing to get closer and closer," because it's already infinite. It would be meaningless to "get closer" by adding another 3 to the end of the sequence, just like it would be meaningless to add 1 to infinity.
It's already been pointed out that irrational means it's not a ratio of two whole numbers. The term "irrational" may have initially been chosen also to imply a connotation of "unreasonable" or "unfathomable", but you'd do well to ignore this usage of the word. Mathematicians today have no trouble at all with irrational numbers, the same way they're just as comfortable with "imaginary" numbers as they are with "real" numbers (another unfortunate naming scheme).Originally Posted by cool skill
This is false. You'd be hard pressed to find a mathematician who thinks 0.3333.... is anything but 1/3. I suggest you do some research on what the real numbers actually are and what the decimals are supposed to represent. Take a look at the construction of the reals from the rationals (in most intro analysis texts or something like "Calculus" by Spivak). In particular you should try to find precisely what 0.333... is supposed to represent. In short it's the real number that is the limit of the sequence of rational numbers 0.3, 0.33, 0.333, ... That this limit exists and is in fact 1/3 will live in the details of the construction of the reals.Originally Posted by cool skill
Even though 0.333...=1/3 will make sense with no mention of the real number system (i.e. you don't have to leave the rationals) I suggest you examine the reals more carefully to get a better understanding of what a decimal actually is, since this more general setting will be required for decimals in general.
I'd also suggest you attempt to figure out if your idea of an "infinitely small number" fits in with the modern definitions of the real number system, possibly compare with the so called nonstandard analysis approach (or hyperreals).
hehe  you're a pretty funny guy, coolspill.
Did you know the Earth is round regardless of what you see out your window?
It's because the real numbers are really only a set of equivalence classes of Cauchy sequences. Any two sequences (A_n) and (B_n) are in the same equivalence class if and only if ( (A_n)  (B_n) ) > 0. That's why we can write 3 as both 3.000... and 2.999...
The exact same logic applies to 1/3. There is no arguing it.
Let's see what we can find out
1)x = 0.444..
2)10x = 4.444..
Subtract equation two from 1
9x = 4
Therefore: x = 4/9
Simple, really!

When we take your value x=0.9999.. , we find that
x>!1
That being said:
x=1  x<1
If x<1, then there exists an intermediary value y (between x and 1)
In plain English, we may say that if 0.99999 is less than 1, then there exists a larger value (y) which is even still greater than 0.9999.. and yet less than 1.
0.999.. < y < 1
Now, coolskill, just in case the earlier proof is not sufficient, what value y can you name which is both greater than 0.999.. and less than 1?
I await your answer.
 Sincerely, Hogwarts Department of Mathematics
Did you know your mother is rounder than the entire earth?Originally Posted by (Q)
If I am filling a glass with water, and wish to fill it to 4/9 cups, I could not do so using the decimal system. Because when you do the long division, you get 9 into 4.0 wich is 0 with remainder 4. When you continue to do the long division, you continue the process for the rest of your life.Originally Posted by Albus Dumbledore
First by adding 0.4. Then by adding 0.04. Then by adding 0.004. When you continue to fill the cup, you keep adding and adding a smaller amount of water. The point is, you will never ever ever no matter what hit the actual 4/9 point if you continue to add water for the rest of your life and through infinity if you could live forever.
x<1Originally Posted by Albus Dumbledore
This is correct. You have been closer to proving the situation than anybody else in this thread, but the situation remains unclear.Originally Posted by Albus Dumbledore
What I am having trouble with is that 0.999... is not a solid number. It keeps on going. That is why it is difficult to place a number y in the term 0.999... < y < 1. This is why I titled the thread paradox.
0.999... < 1. Therefore, it is not equal to 1. Normally, when 2 numbers are not equal. There is an infinity of fractions between them. But it does not apply in this case. It seems like a paradox. However, people here claim that it is not.
Tell me, what is a "solid number"? I'll repeat myself, the decimal 0.999.... is the real number (that also happens to be an integer in this case) that is the limit of the sequence begining 0.9, 0.99, 0.999,... It's no more vague or ambiguous than anything else you'd like to call a number. I guess I'm assuming that you understand limits if you learned some "intuitive" version of limits in an intro calculus class forget it and do it properly from an analysis text (Spivak's "Calculus" would work as well)..Originally Posted by cool skill
Have you bothered to look at a construction of the real numbers from the rationals? Either the Dedekind cut way or the equivalence class of Cauchy sequence ways should prove enlightening (though with your hellbent view of decimals, I'd suggest the latter).
ps. your cup filling example is nothing more than zeno's paradox. If you are adding water at a contstant rate the time to add each stage 0.4, 0.04, 0.004, ... etc, decreases exponentially and hey, geometric series converge so you'll get your 4/9 of a cup in a finite amount of time.
Any mathematician will tell you that 0.999… exactly equals 1. You are arguing that it doesn’t equal 1 by presupposing that it’s smaller, which isn’t valid reasoning. Can you prove that 0.999… is less than 1? I doubt it. In fact, it’s pretty easy to prove that 0.999 exactly equals 1 using the same method that Dumbledor just demonstrated.Originally Posted by cool skill
Umm, this sounds a lot illogical to me. Almost like a "gimme" in golf. Sure it's very very close but it's not 1. They can say what they want, close enough is not exact science.Originally Posted by Scifor Refugee
Here is a proof that 0.999… exactly equals 1:Originally Posted by (In)Sanity
Let x = 0.999…
10x = 9.999…
10x – x = 9
9x = 9
x = 9/9 = 1
So, if x = 0.999..., then it must also equal 1. They are the same number, but expressed with different representations.
This gedanken reminds me of Zeno's paradox. It is impossible to get 4/9 cup exactly in real life  just as it is impossible to get 1/2 cup exactly. Get a piece of paper and draw 10 vertical marks at regular intervals. It seems obvious at first that the 5th mark represents 4/9 of the length (assuming the first mark represents zero). But this is not true in terms of accuracy  that is, if we want 4/9 and no more, no less.Originally Posted by cool skill
Every mark on that paper has a certain thicknesss to it. Now as you and I know, the left and right boundaries of that 5th mark do not represent 4/9 of the distance because they do not lie on the same point. What then? If we say the middle of the mark represents 4/9 (that would be arbitrary) then we would have to draw another mark to indicate that midway point. Doing so however creates the problem anew. So now you see that it is not even possible to measure out 1/2 cup. We might get 0.499993 but we can rarely (in the physical world) be certain we have reached 0.5 exactly for this reason. This relates to the uncertainty principle in physics. That's why we estimate in every day life (not to mention such an experiment would ignore the problems given by water's property of surface tension). Points in space are an abstraction; not all meter intervals are exactly the same. If we were to be supremely accurate, you would find that one man's kilometer of dirt rode would not be equal to another man's kilometer of road in the mathematical sense. They would only be approximately equal. But this has little to do with the topic..
But here, where we don't want to deal with approximately equal..
Logically speaking, if 0.999.. is not greater than 1 and cannot be shown to be less than 1, then it can only be equal to one  that is the only remaining option. The only way to show that 0.999... < 1 is to provide a y for 0.999.. < y < 1. Since this cannot be done:This is correct. You have been closer to proving the situation than anybody else in this thread, but the situation remains unclear.
What I am having trouble with is that 0.999... is not a solid number. It keeps on going. That is why it is difficult to place a number y in the term 0.999... < y < 1. This is why I titled the thread paradox.
0.999... < 1. Therefore, it is not equal to 1. Normally, when 2 numbers are not equal. There is an infinity of fractions between them. But it does not apply in this case. It seems like a paradox. However, people here claim that it is not.
0.999.. !< 1
and
0.999 !> 1
Therefore:
0.999.. = 1
If 0.999.. is not a solid number, then neither is the square root of two, or pi, or e.
The problem you are having is that:
0.99999... + 0.00000... 1 = 1
But if
0.9999.. = 1
then
0.99999... + 0.00000... 1 = 1 + 0.00000.. 1
but it seems intuitively like the left equation equals one while the right equation is greater than one so that there is a paradox. But this is not so because the apparent problem can be resolved using the earlier method shown.
We have already seen that 0.999.. is neither greater than or less than 1.
Using the same method, we can show that for
0.0000.. 1 > y > 0
There is no real number y.
0.0000.. 1 must therefore be equal to 0, blasphemous as it may seem. We know that it certainly is not negative, but if we say it is positive then we are left with the impossible problem of finding a value y to satisfy the expression.
I suspect the problem lies in the understanding of numbers.
What is your definition of a number?
For the nth time, 0.999.. is defined as the real number that is the limit of the sequence begining 0.9, 0.99, 0.999,.... This is EXACTLY 1. There are no ifs ands or buts or anything inexact about it. Note this is COMPLETELY different then claiming that there is some number in the sequence 0.9, 0.99, ..etc that equals 1, which is distinctly not what any mathematician is doing when they say 0.999...=1.Originally Posted by (In)Sanity
Seriously, any people who have with this find yourself an intro analysis text and go brush up on what the real numbers and decimals actual mean to mathematicians.
I beg you, this thread is silly enough, please don't add nonsense by writing things like 0.000...1 that have no meaning whatsoever. There is no "last place" in an infinite decimal expansion like this notation is trying to suggest.Originally Posted by Albus Dumbledore
Nought point nine nine nine recurring equals one for the same reason that 1/3 = 0.333.....
0.999.... only exists because of the operations of arithmetic within the decimal system. And the decimal system is a convenience for certain kinds of mathematical operations. If we didn't have it, certainly we could use the rational form of numbers to perform our operations, it's slightly less conventient, but still we could.
In the realm of the irrationals, however, is where decimal expansion comes into its own. Just because you can't ever get to the last digit of pi, e or the root of any prime number, doesn't mean that the fact that decimal notation (or rather powerbase notation, which isn't dependent on the number Ten) helps us to understand the concept of irrationals (and transcendentals) better than any other system could. There is a sum formula for pi which is a large operation of mulitplies, adds, roots, divides, etc., all multiplied by 1/16<sup>n</sup>. This means that by plugging in any value for n into the part after 1/16<sup>n</sup> will get you the n<sup>th</sup> hexadecimal digit of pi, whether it's the fifth, the one hundredth or the four hundred billion trillionth. I think this is a massive tribute to the usefulness of powerbase expansions, even if we cannot ever get to the end of them.
I beg you, this thread is silly enough, please don't add nonsense by writing things like 0.000...1 that have no meaning whatsoever. There is no "last place" in an infinite decimal expansion like this notation is trying to suggest.
Are you an idiot or do you not understand that I was using it to show him where his reason was incorrect?
As I said in conclusion to him:
I suspect the problem lies in the understanding of numbers.
You were treating 0.000...1 as if it were some kind of meaningful number, it's not. If you know this then why in the hell would you try to do this?Originally Posted by Albus Dumbledore
If the problem is his understanding of numbers (and I agree it is, I've been suggesting all along that he take a look at some foundations) then it's not going to be solved with incorrect and meaningless notions like 0.000...1, this is only going to encourage crappy habits and be misleading.
Just my opinion though, I don't think you should use bad maths to correct bad maths.
I see the reasoning behind 0.999... = 1. I still find that logic to be imperfect (while accepted). The nines could go on forever thus making it never actually reach 1. So from a logical standpoint 0.999... can never equal 1, from a flawed mathematical standpoint I see how it can. Even if it is common practice you really can't argue with the logic. Almost is never exactly there yet. But still, the more you push the digits out the closer it gets to 1. So I'll concede that it's close enough and would not produce a measurable amount of error.
The logic is not imperfect or flawed at all. 0.99... is DEFINED as the point that is the limit of the sequence 0.9, 0.99,.... This is the definition of this decimal expansion. It's what the symbol means. To assign any other meaning to 0.99..., specifically that it's some kind of continually evolving thing, amounts to changing it's definition and you are no longer talking about the same 0.99... that everyone else is.
Note the distinction between the limit of a sequence and the actual sequence itself.
I understand, the definition makes it what it is.Originally Posted by shmoe
If you take the set of 0  .999... and 1 then it would be hard to see how 0.999... equals 1. I won't argue common practice or definitions as it's a loosing battle. Logic is absolute, not open to debate by opinions, surveys, definitions or practice. 0.999 with an infinite number of 9's is never going to be 1. 0.999... however being defined as 1 should just be called 1 and not 0.999... Someone at some point in time had of said that it's impossible for us to calculate out that far (infinity) so we may as well just call it 1. I get it, and from a practical standpoint it makes sense.
Here you are assigning some other meaning to 0.999... I believe this is where your problem is, before you can even begin to talk about whether 0.999... is or isn't 1 you have to have a precise notion of what you mean by 0.999.... This definition must precede any attempts you have of applying 'logic' to 0.999... If you don't know what the thing you are talking about even is then it's impossible to draw any meaningful conclusions.Originally Posted by (In)Sanity
0.999 with an infinite number of 9's repeating will become infinitely closer to 1 as it becomes longer yet never really reach 1. That is what I'm referring to. 0.999... may in fact have it's own definition outside of infinite repetition.Originally Posted by shmoe
You're viewing 0.999... as a changing thing, why? If 0.999... is representing a number (maybe you think it isn't?) then it's not moving. It's either 1 or it isn't.Originally Posted by (In)Sanity
This is really the distinction I was talking about between the sequence and it's limit. The limit does not change at all, it's fixed (for a given sequence). The sequence 0.9, 0.99, ... does have these properties you're talking about, the terms themselves are getting closer and closer to 1 but no term actually equals 1, but the limit of the sequence is 1, plain and simple. Do you see the difference? This may just be a terminology thing, I'm not sure.
It has one widely accepted definition. If you want 0.999... to mean something other than the accepted mathematical one, that's fine, but then you're no longer talking about the same mathematics everyone else is.Originally Posted by (In)Sanity
I see where your coming from on the limit, the next step after 0.999... has to be 1. Because of it's infinite repetition it is in a sense 1. I'm just being highly critical. I would argue that because it doesn't start with 1.000... that is can't really be a true 1.000... The math and the accepted view works 0.999... to be 1. Again, I'm just being highly critical.Originally Posted by shmoe
Again, there is no next step, because 0.999... !< 1. 0.999... is 1. There is no "next step" because with 0.999... you are dealing with every digit of 0.999... in one go. It is the sum of 0.333... + 0.333... + 0.333..., and 0.333... = 1/3 + 1/3 + 1/3 = 1.
The problem here is one of perceptions.
The human mind is boggled by this notion of an infinite number of 9's stretching off in the distance. The human mind is one of motion and of expanding awareness.
Insanity is visualizing the 9's a 9 at a time and therefore sees it in a fluid manner. The more he looks, the more it grows. But, the number is a number and is not changing. It's not expanding. It's not doing anything other than being exactly what it is.
It's been shown through various methods (not least of which is the literal definition from Shmoe) that 0.999... is 1. But, the human mind insists on only accepting that portion of the number which it's mind has encompassed. And visualizes the number moving towards 1 but never actually reaching it.
I must admit that I've taken only a few introductory tutorials on calculus and keep getting distracted before getting to the real deal, and I believe that my understanding of limits is flawed in the way Shmoe has demonstrated it to be. One day, I will have to delve more seriously into the topic and see if I can't understand it from the ground up rather than the top down.
Shmoe, what do you think of the infinitessimal approach? I've got a textbook using infinitessimals and hyperreal numbers to begin the course rather than limits which is usually the case... I forget the address... Let me find it.
Here it is: http://www.math.wisc.edu/~keisler/calc.html
Elementary Calculus: An Approach Using Infinitesimals
I have to admit to not making it very far into this book either... I'm afraid that math is hard pressed to capture my attention for very long. I'm going to have to cattle prod myself into it eventually though. I was just wondering if you've heard of this book and/or if you think it would be better to learn through standard techniques.
Edit: And. What of the irrational numbers such as the square root of 2 and pi and such? They are numbers that do not fall into a predictable pattern. So. Is it a number like 0.9999... is? Is it a particular number that can be said to be stationary and not changing? It's possible to accept the definition of 0.999... as stated. One needn't keep calculating more and more 9's to see the number. But what of the irrationals?
A tricky concept for the layman, I think.
Something has been lostthere's no "next step" here. There's no last term in an infinite sequence.Originally Posted by (In)Sanity
Do you agree that the limit of the sequence 0.9, 0.99, ... is equal to 1? I don't mean about 1, or nearly 1, I mean actually exactly 1. If you don't believe this, can you give me the precise definition of the limit of a sequence and use it to show it's not 1? You have to not believe this if you believe 0.999... is somehow not a "true 1.000...". That 0.999... is defined as the limit of this sequence isn't really up for debate, maybe there's some other kink in the process that you're seeing?
You can be as highly critical as you like. The mathematics is bullet proof. The foundations of analysis were beaten to death and made rigorous in the days of Weierstrauss, et al. This sort of thing is very solid. I would make the claim that what you usually see in the typical high school/first year uni math classess does a poor job of preparing you to cope with the finer points of the real numbers. If you want a more rigorous view of how mathematics actually works these days, pick up an intro analysis text (eg. Rudin, Pfaffenberger, or others) or a rigorous Calculus text (eg Spivak's) and get reading.
Well as I said before it becomes infinity closer and closer to 1 each and every 9 that is added. At some point the limitations of our own ability compute this number causes it to equal 1. It still never actually truly hits the number 1 even if it becomes closer and closer for infinity. I understand the concept and can envision why one would conclude that it does in fact equal out to 1, no matter how hard I try however I can never agree that it holds up to pure logic. It's more a victim of our own inability to calculate this infinite number. No matter how far we let it go there is always going to be that smaller and smaller fraction of a fragment away from true 1. Despite all those who have concluded otherwise with whatever tests they have run I still say it's not so.It's been shown through various methods (not least of which is the literal definition from Shmoe) that 0.999... is 1. But, the human mind insists on only accepting that portion of the number which it's mind has encompassed. And visualizes the number moving towards 1 but never actually reaching it.
It would be like trying to reach the bottom of a bottomless pit, in theory you get closer and closer with each passing second, yet you never actually get to the bottom.
For all practical purposes and calculations that man could ever dream of computing I will say 0.999... does in fact equal 1. For pure logic I would have to say it doesn't.
I know I just frustrate the crap out of people when I go off on stuff like this. My mind works much different then many people.
Ahh, thanks for the link. I knew there was a first calculus course textbook following the NonStandard Analysis approach (or hypperreals or infinitessimals), but didn't know of a free electronic version.Originally Posted by invert_nexus
I've peeked at Robinson's book before and been to a number of intro lectures on this approach and in some ways it's very nice, others kinda icky. It takes more work to build up basic machinery, but once you get that out of the way, many things are simpler. I'm very curious to see how they arrange it for an elementary approach.
If you've already got some "standard" calculus behind you, it might be easier to go back and learn it properly before leaping into the infintessimal approach, I'm not really sure though. Whichever can hold your attention to the end might be the best choice!
Actually, you're just following the intuitive approach. The one that most people fall into thinking. I suspect it's misconceptions such as the one you're experiencing that professors spend a good deal of time digging out of people's heads.I know I just frustrate the crap out of people when I go off on stuff like this. My mind works much different then many people.
You're stuck on the idea of motion. Getting closer and closer and closer.
The number is. It's not moving at all. It's not getting closer to anything.
Then I will say without qualifiers that your pure logic is wrong (or is applied to something that isn't mathematics).Originally Posted by (In)Sanity
This has nothing to do with anyone's ability to write out billions of 9's after a decimal point. This has nothing to do with a human's limitations to finite calculations or computations. There is a unique real number that is the limit of 0.9, 0.99, .... Always remember that even though we call these numbers "real" they aren't actually something you can hold on to. We use them to model reality, but they aren't reality. You limitations as a human have no impact on their behavior.
editbefore someone points it out, I lied in my first sentence and added a qualifier. I sometimes forget what I've typed at the start of a sentence by the time I reach the end.
I guess it all goes back to definitions.Originally Posted by invert_nexus
If we started with 0.111 and added another 1 it becomes closer to 0.1112 if we add another 1 it becomes closer still.
0.999... Would not move or count, it is a static number. This number of course could never be printed as it would consume the universe and more. This part I understand. It's that infinite size that makes it equal out to be 1. Because it could never be worked with in it's true form. Still using pure logic and unlimited time and resources it would never actually equal 1, even if it consumed the entire universe and beyond with decimal places.
Do you see what I'm getting at?
Sure, but do you also believe that there are only a finite number of integers since we'll never be able to explicitly write down an infinite number of them?Originally Posted by (In)Sanity
Well, I guess I sort of have to. My beliefs however don't impact logic. Logic would say there is no limit.Originally Posted by shmoe
I can accept that 0.999... equals 1 for all our purposes. In an unlimited world it would not. Just as I don't believe the speed of light is an absolute perfect constant (another topic).
Does this not trouble you at all? If there is only a finite number of integers, there must be a largest, say M. What's M+1 then?Originally Posted by (In)Sanity
What's an "unlimited world"? Remember mathematics is not reality.
You haven't answered my question about the limit of 0.9, 0.99,... Have you seen the precise definition of the limit of a sequence? There's really nothing dodgy at all about the statement 0.99..=1 once you understanding all the definitions involved.
My problem is I'm going with a perfect absolute world. In my perfect absolute world math has no limits.Originally Posted by shmoe
So 0.9 does not equal 1 nor does 0.999 or 0.9999999999. So how many decimal places out does 0.999... equal 1 ? I'm not trying to be difficult or change the way people have defined things, I'm simply trying to use that perfect world model to try to understand how 0.999... could ever equal 1. With the not so perfect world model I could see it happening.
I'm not sure what you mean by math having no limits in your perfect absolute world.Originally Posted by (In)Sanity
Remember that 0.999... isn't just a number like 0.9, or 0.999999, but with an infinite number of nines. I keep saying this over and over, but it's because it's important, 0.999... is a limit. This is the way we deal with a number with an "infinite" number of places, by defining it to be a limit. This is a common theme when dealing with the infinite. We define an 'infinite' thing to be the thing that our sequence of finite things is getting arbitrarily close to (sounds vague, but we do make this precise, both what we mean by "thing" and "arbitrarily close").
You're probably happy with decimals with a finite number of nonzero digits. The notation 0.99... is suggestive in that it looks like it's one of these, except with an infinite number of digits. I can't stress this enough that this is merely a suggestive analogy, and you should think of 0.999... as the limit point of the relevant sequence because that's what it is, nothing else.
Well I suppose the concept is that because it's 0.9999999 forever that anything more would equal 1. Because it always goes on forever it's easy to conclude that it must be 1 as it's impossible to have anything between 0.999... and 1. You can't add another 9 to an infinite line of 9's.Originally Posted by shmoe
You're continually avoiding talking about 0.99... as a limit. Are you comfortable with limits? Have you ever studied them rigorously (the intuitive approach of most basic calculus classes doesn't count as rigorous)?
There's really no shame in answering no to these. If the answer is no and it would make you more comfortable, I can make a very large list of things I don't understand (it might be a long post though).
Well in my above post I stated you could not get any other number between 0.999... and 1. So yes I understand the limits.Originally Posted by shmoe
Ok. So here's something that's been occurring to me to ask.
The limit of 0.999... being one is really easy to see.
What about 0.333...? What is it then? It's not 0.4. Nor is is 0.34. Or 0.334. Or any of those.
So. What would you say it is? Does it equal anything other than itself? 0.333...?
And how does this relate to irrational numbers such as pi?
Edit: Or would you just say 1/3 as it's limit? Is saying that 1 is the limit of 0.999... the same as saying 1/3 is the limit of 0.333...?
0.333...=1/3.Originally Posted by invert_nexus
0.3, 0.33, ... are the partial sums of a geometric sequence. The nth term is 3/10*(11/10^n)/(11/10)=(11/10^n)/3. As n>infinity, this >1/3.
A number is rational if and only if it has a periodic decimal expansion. That is to say it eventually repeats. e.g 1/7=.142857 142857 142857 ..., spaces to show the period, but the decimal expansion of pi will never repeat like this.
That statement to me isn't proof of understanding of limits, but I'll let it go. I'll just say that after my first calculus class I would have also believed that I understood limits and real numbers, etc. It was a real enlightening experience when I took my first analysis class and actually did things "properly".Originally Posted by (In)Sanity
Ok. That's the conclusion I came to (and which, ironically enough, is what started this whole thread...).0.333...=1/3.
See. Here's a thing that I think might be confusing people. When I asked the question, I was thinking along the lines of 1 being part of the decimal system. I.e. 0.8, 0.9. 1.0, 1.1, etc... But. I think this is where some of the confusion lies.
0.999... is to 1 as 0.333... is to 1/3. It's like a translation between two different systems.
Am I on the right track now?
It's almost unfortunate that 1 occurs in both system and causes this particular confusion... We could say that it's not really 1 so much as 1/1 or some other ratio. This would clear up the confusion that I had.
Or. Am I on the completely wrong track here?
And. What I was asking about the irrationals is simply... Well. Nothing simply about it.
It's like I said earlier, it's relatively easy to encompass an understanding of a number like 0.33... or 0.99.... because calculation isn't needed to know that the same number (or series of numbers in a case such as 0.142587...) is repeated in an infinite sequence. But, with an irrational (or transcendant?) number like pi the number must be calculated and therefore it leads one to consider if numbers such as this actually exist as a... single number or if they exist more in a state of flux.
Of course, I imagine the answer to this is that of course it's a single number, but it's far more difficult to comprehend than a repeating decimal.
Tricky.
You can think of it like this if you really like. I mentioned earlier that 0.333... makes sense inside the rationals, but the full picture of decimals requires an understanding of the reals. Decimals give a way of representing any real number. The rationals sit inside the reals (or a "copy" of them does), so each will have a decimal expansion (or two).Originally Posted by invert_nexus
The thing that seems to cause confusion is that a decimal expansion is not necessarily unique, like the 0.999...=1.000... case. This shouldn't be troubling unless you have a preconceived notion that it should be unique. Another way to construct the real numbers is that they are the decimals, that is the set of all infinite sequences composed of 0's to 9's. You then go on to define addition, multiplication, etc. and show this object behaves like we'd hope. One thing you do here is seemingly arbitrarily identify the different looking things 0.999... and 1.000.... This is necessary if we want this construction to have all the usual properties. This unfortunately causes much confusion, but it really shouldn't. Everyone has no problems identifying the symbols 10/6 and 5/3 in the rationals, and this is a similar thing we're doing (maybe everyone should have to go through a construction of the rationals as equivalence classes of pairs of integers, etc.).
pi is transcendental, all transcendentals are irrational. In general showing something is transcendental is much harder than showing it's irrational. As to whether it exists or not, well does 1 exist? 2? 0? Anything we call a "number"?
One difference I see between 1/3 or 0.333... and 0.999... is that with 0.999... is that there is not a number at all between 0.999... and the next whole number like there is with 0.333...
Not sure why I'm still on this subject, guess I'm just curious about my misunderstandings.
By whole number I assume you mean naturals like 1, 2, 3, etc. then the next whole number after 0.333... is 1, so there's lot's in between. This isn't the difference between 1/2=0.49999... and 1/3=0.333... though. (I'm not sure this is what you mean?)Originally Posted by (In)Sanity
Keep in mind that in the reals, there is no "next number". That is there is no number that comes immediately after 0, say. In other words there is no smallest positive real number (same is true for rationals).
Here we go again, why represent 1/2 as 0.4999... ? 1 divided by 2 has always been 0.50000. If this suppose to be more precise?Originally Posted by shmoe
But yes I see there is no space between 0.4999... and 0.5
I was referring to a natural number like 1 on the first part.
It's no more or less precise, but 0.4999... is a decimal representation of 1/2.
I wasn't sure what you were getting at by saying there are numbers between 0.333... and the next whole number 1. I mean this is obvious isn't it? The usual problem with 0.999... is that it is equal to another decimal expansion, 1.000... in the same way that 0.4999...=0.5000... or any other example like this. You also have plenty of numbers between 0.4999... and the next whole number 1, so I don't understand what point you were trying to make in the part I had quoted in my last post.
Pretty much my point was that whenever you have x.x999... that the next number in the sequence will be radically shifted left in decimal places. 0.999... to 1 or 0.4999... to 0.5 unlike 0.333... that could just have a 4 tossed on the last digit somewhere out at the end of infinity. Pointless point I guess, just an observation.Originally Posted by shmoe
What do you mean by "next number in the sequence"? Please try to be precise if you answer this.Originally Posted by (In)Sanity
ps. I realize that it may seem like I'm being a pedantic bastard and if so, I do apologize. You did seem willing to probe your understanding and all of my questions are aimed at this goal.
Well it's not as if I could write out what I'm referring to. Then again it may also be impossible to add to the last digit on an infinite stream as their is no clear last digit.Originally Posted by shmoe
For demonstration sake I'll try..not assuming it's correct of course.
0.333...33333 if you add 1 to the last digit (that doesn't exist). You might get 0.333...33334.
With 0.4999...99999 if you add 1 to the last digit it would cause it to roll over to 0.5
Same with 0.999...9999999 adding 1 to the last digit pushes it to 1.
Now I understand it's impossible to really add 1 to that last digit, more of a hypothetical then anything.
It's good that you realize there is no "last digit" to add 1 to, and I'd try to avoid thinking about a last digit if possible. Let me attempt another way to explain what you're driving at, attempting to avoid any allusions to a "last decimal place".
The nth number in the sequence 0.9, 0.99, 0.999, ... is 10^(n) away from a number that has a terminating decimal expansion (in this case 1=1.000...), that is the nth term+10^(n)=1. As n grows this number we have to add to "roll over" the digits goes to zero, so the limit of 0.9, 0.99,... is 1. We aren't adding anything to 0.999.. to get 1, but the thing we need to add to the terms in the sequence to get 1, goes to zero.
For 0.3, 0.33,.. at each number in this sequence, if you want to "roll over" all the digits, you have to add 0.66..667 (n1 6's followed by a 7). There's nothing "small" we can add like 10^(n) in the .99.. case, so 0.333... is not going to equal any terminating decimal. You could aim at a smaller terminating decimal by having less 6's, like 0.0067+0.3333=0.334, but you'll hit the same problem, 0.0067, 0.00667, ... does not tend to 0. You could try to add more 0's to this as you moved along the sequence, say 0.333+0.067=0.34, 0.3333+0.0067=0.334. At each step you'd be "rolling over" to a number with more and more digits, so you won't get a terminating thing this way either. The end result is we've failed to show 0.333... is also equal to a terminating decimal expansion (this isn't meant to be a rigorous proof that 1/3 does not equal any terminating decimal, though you can prove this).
The problem only comes with terminating decimals (ones that are eventually zeros, like 0.5000...). They will always have an alternate decimal representation ending with 9's. This is not true for repeating decimals (unless they are already repeating 9's at the end of course).
So. What we have is something like dueling limits.We aren't adding anything to 0.999.. to get 1, but the thing we need to add to the terms in the sequence to get 1, goes to zero.
Well. Not exactly dueling... Complementary.
0.99999... limits to 1.
Because the number that is needed to be added to that number limits to 0.
Which goes back to what was being said long ago about 1/30.333...=0.
If there is no seperation between 0.999... and 1 then they are the same number.
Ah, but there is a separation. In fact, for all real numbers the 'separation' is infinite.If there is no seperation between 0.999... and 1 then they are the same number.
Infinitesimal, you mean. Which means, that strange mathematical construction which is simulatneously nothing and yet not nothing, and the foundation of all calculus.
A number that is not a paradox is a number that does not go on forever.
It's impossible to divide 1/3 in the decimal system without the paradox occurring. It's a flawed representation of 1/3. 1/3 is an exact portion that cannot be represented as an exact portion with the decimal system because the decimal system paradoxes when it attempts to do so. If it did not do so, no paradox would occur.
Ok. I wasn’t sure what that whole situation you were talking about meant.Originally Posted by Albus Dumbledore
What are you talking about? This is completely illogical.Originally Posted by Albus Dumbledore
True: If a number is not less than X, and not greater than X, it must equal X.
This 0.9999… is not greater than 1.
1. You are saying that this number cannot “be shown” to be less than 1. Therefore, it is equal to 1.
2. Anybody can say it either way: It cannot be shown to equal 1. Therefore, it must be less than 1.
How can you just randomly choose between #1 and #2, and assume, #1 is correct?
The fact is, 0.9999…. is less than 1.
This has nothing to do with Xeno’s wordplay paradox.
Your proof makes no sense so far. Because the fact is this:
0.9999 < 1.
0.9999… + .0000…1 makes no sense. How can you have a number such as 0.0000….1?
I have not been using any other definition than what we all know as a number. Irrelevant.Originally Posted by Albus Dumbledore
If I knew the reason why 0.9999…. = 1, I wouldn’t be posting here. From what I know about it thus far, 0.9999…. < 1. Telling a person to read any book that says 0.9999…. = 1 is not proof. It’s fallacy. Nothing but hearsay. Restating an assertion, and pointing out text that restates the assertion is not proving. Wow ok. You say it is equal to 1. The book says it is equal to 1. Yet you cannot properly show how.Originally Posted by shmoe
The number does not equal 1 in any logical sense. It approaches 1 without ever equaling it.
Right.Originally Posted by (In)Sanity
What he seems to be arguing is that it is defined to equal 1. Restating that it is defined to do so is pure hearsay. It is not proof. how can you prove this so called definition?
Sigh.
You're incapable of ever admitting failure. Aren't you?
How many times does Shmoe have to tell you the definition of 0.99... before you accept it?
Why even bother asking if you're not going to listen?
You make no sense whatsoever.
I guess the point of this thread is supposed to be something along the lines of, "Oooh. Cool Skill. You're so smart. You've found a fatal flaw in our mathematical system. And look. He's not even educated in mathematics. He did it all on his own! What a genius. Refined reinvention, indeed. Make love to me, Cool Skill!"
Uh. Did you know that mathematics is a system of agreed upon definitions?how can you prove this so called definition?
You... did... didn't you?
You do realize that the number system is an artifical creation of Man's mind, right?
Everything you need to know to understand this is in standard textbooks. You will have to do some searching to find a textbook that flat out declares 0.999...=1, even more searching for one that doesn't also contain the tools needed to understand this yourself. I've suggested you actually take some time to understand the definitions involved, once you do it's quite trivial. You are apparently too friggin lazy to go and do some research on your own, so why on earth do you expect me to spend my time producing a rigorous argument?Originally Posted by cool skill
You aren't going to be satisfied until someone holds your hand and does it for you, so read the following:
http://planetmath.org/encyclopedia/Sequence.html
http://planetmath.org/encyclopedia/Series.html
http://planetmath.org/encyclopedia/C...tSequence.html
http://planetmath.org/encyclopedia/D...Expansion.html
(In the third link, our metric d(x,y) is the usual one on the reals, xy. In the last one, note how a decimal is defined as a series, i.e. the limit of the sequence of partial sums. It also only talks about rational numbers, but equation (1) is how an infinite decimal is defined in general)
Having read the above, you now know that 0.999.... is defined as the limit of the sequence a(1)=0.9, a(2)=0.99, a(3)=0.999, ....
Now 1a(n)=10^(n). Let epsilon>0, then take N such that epsilon<10^N. Then if n>N, 1a(n)=10^(n)<10^(N)<epsilon. Therefore the limit of a(n) as n>infinity is 1, i.e. 0.999...=1
That's it. There is absolutely no arguing with the definitions mathematicians use. You might claim that they don't make sense, but this is totally irrelevant it's what they have been using to communicate with (and you, though you apparently haven't really bothered to understand what you even mean by 0.999...). The people who taught you what a decimal was knew it's meaning to be what's implied by the above, you just recieved a filtered distilled version. Or you were taught by people who didn't understand fully because they had also recieved a filtered, distilled version, and the chain of ignorance added another link.
ps. I much prefer textbooks to internet links, but since you are unwilling to go look them up on your own and I can't actually hold your hand to walk you across the street to the library and get some books for you, this will have to do. If you are genuinely interested in learning the mathematics involved in more detail and can convince me that you're actually willing to go look up some books, I'd be happy to provide a detailed list of references and would also be happy to discuss their contents.
2/7 has a nice decimal representation:
0.285714285714285714285714285714285714285714285714 28571428571
42857142857142857142857142857142857142857142857142 85714285714
28571428571428571428571428571428571428571428571428 57142857142
85714285714285714285714285714285714285714285714285 71428571428
57142857142857142857142857142857142857142857142857 14285714285
71428571428571428571428571428571428571428571428571 42857142857
14285714285714285714285714285714285714285714285714 28571428571
42857142857142857142857142857142857142857142857142 85714285714
285714285714285714. It goes on forever, of course. It's an attractive example of a RATIONAL number.
On the other hand Pi:
3.141592653589793238462643383279502884197169399375 10582097494
45923078164062862089986280348253421170679821480865 13282306647
09384460955058223172535940812848111745028410270193 85211055596
44622948954930381964428810975665933446128475648233 78678316527
12019091456485669234603486104543266482133936072602 49141273724
58700660631558817488152092096282925409171536436789 25903600113
30530548820466521384146951941511609433057270365759 59195309218
61173819326117931051185480744623799627495673518857 52724891227938183011. That goes on and on too. I don't know how many places it been calculated to, but it doesn't repeat itself. An example of a transcendental IRRATIONAL number
:wink:
Hey! Who suggested studying the text books! Whatever next? Have you no consideration for us sadists who delight in witnessing all this emotion about logic?
What failure?Originally Posted by invert_nexus
Admitting that "1 = 0.9999...." makes no sense?
Are you on crack?
You're a moron.Originally Posted by invert_nexus
Where did I ever state a flaw?
OF course it is relevant.Originally Posted by shmoe
Only an idiot would think that it is ok for definitions to not make sense.
Only a retard would think that whether a definition makes sense or not is irrelevant.
We already know what the definition is. The question from the beginning is how can we prove that this definition is correct and not a paradox?
Your failure to convince anybody that 0.999999... < 1.
But it can be shown to equal 1.Originally Posted by cool skill
1/3 = 0.3333333333....
3 x 0.33333333...... = 0.999999....
3 x 1/3 = 1
Therefore 0.999999..... = 1
QED.
Cool Skill, lets keep it cool.
Invert, if you turn back up to answer, please do likewise.
To the topic in hand. Would you agree that there is a difference between
9.99999 and 9.99999. ? If so, what is that difference?
Bear with my treatment of the exhausted topic from above. At the end I will throw in something new and interesting.
It is a fact that there are many different ways of representing the same number.
2, 53, 6/3, 1.9999..., sqrt(4), etc...
One way we have of saying these representations are the same number is that the difference between them is zero.
So what is difference between 2 and 1.9999...
that is what does 2  1.9999... equal
We use limits to handle questions like this.
We can say that 2  1.9999... is the limit of the following sequence
.1, .01, .001, .0001, .00001, ...
Anyone with an knowledge of calculus can tell you that the limit of this sequences is 0. Which means that 2 and 1.9999... are the same.
Well enough of that. While we are talking about representations of numbers with an infinitely repeating pattern (in this case a repeating pattern of digits). Are any of you aware of the fact that you can represent square roots which are irrational with a different kind of repeating pattern? I am talking about continued fractions, the passion of the great mathematician Ramanujan.
sqrt(2) = 1 + 1/(2+ 1/(2 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + .........))))))
sqrt(3) = 1 + 1/(1+ 1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + .........))))))
sqrt(5) = 2 + 1/(4+ 1/(4 + 1/(4 + 1/(4 + 1/(4 + 1/(4 + .........))))))
These are not easy to compute even when you truncate at a finite number of steps, but for example if we replace the ......... with 0 in the above we get,
sqrt(2) ~ 1.4142011834319526 whereas sqrt(2) = 1.4142135623730951...
sqrt(3) ~ 1.7317073170731707 whereas sqrt(3) = 1.7320508075688772...
sqrt(5) ~ 2.236067970034716 whereas sqrt(5) = 2.23606797749979...
Yet each of the above continued fractions have an obvious repeating pattern from which you can get an increasingly accurate estimate (quite rapidly) of the square roots by extending the pattern farther. By the way, I did not compute these truncated continued fractions by hand (I used emacs lisp)!
You may not think this is so great because after all you can get an increasingly accurate estimate of these square roots with the binomial theorem. But in these cases it is not such a simple repeating pattern although once you have the binomial series computed the calculation is considerably easier. For example,
sqrt(2) = 1 + 1/2  1/8 + 1/16  5/128 + 7/256  21/1024 = 1.405273438
As you can see, in this case it does not converge very fast at all.
sqrt(5) = 2 + 1/4  1/64 + 1/512  5/16384 + 7/131072  21/2097152
= 2.236066341
This one converges faster than sqrt(2) but not as fast as the continued fraction for sqrt(5).
It is totally irrelevant that YOU claim they don't make sense. They make perfect sense to mathematicians, and anyone who understands what a limt is and what the real numbers are.Originally Posted by cool skill
You've already demonstrated that YOU don't know what the definition of a decimal expansion is, and what we mean by an infinite number of repeating digits. Read the links I gave. Bettter yet go find an intro analysis textbook. Do some work to actually understand basic analysis, or just accept the fact that you will never have a clue.Originally Posted by cool skill
I have no idea where you think there's a paradox (is it in the definition of a limit? a series? the real numbers themselves? what?) or how you hope to prove that a definition is "correct". Which is the "correct" definition of the natural numbers, {0, 1, 2, 3, ...} or {1, 2, 3, ...}? Answer: neither. Both will be used by different authors.
Honestly, do you expect me to build up the machinery from the ground level here for you? Do some work.
Hard to beleive this is still being discussed! Anyway, let's try this......
Those who think that 0.999... < 1 have two choices. Either they tell us what number lies between 0.999... and 1 or they admit that the real line is not continuous.
As to the latter, let me simply say that the "real line" is socalled because it reflects our everyday experience, that railwaylines, roads etc are continuous, i.e. to every point on the line I can assign a real number, no gaps. This would conflict with the insistence that 0.999... < 1
Mathematicians have always accepted that 1/3=0.33...........333 just as they accept 3/3=0.999..............999 and not 1.
Why did you dig up a thread that had been dead four months, just to post a manifestly incorrect statement?Originally Posted by chamilton333
1) You put in a "terminating digit" when there isn't one (see pg 1 of the thread).
2) You stated that mathematicians accept that 3/3 <> "100%" by which I assume you mean 1.0, when in fact they believe that 3/3 = 1.0, and also 0.9999.... = 1.0.
proving 0,999.....=1 is quite simple
x=0,999...
10x=9,9999...
9x=9,9999...0,9999...=9
therefor x=1
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