[split] 0.999... equals 1

[Violet]

Take your time

[fr0d0]

Hey I just figured out that 3 is equal to 1 using this proof:

1*0 = 3*0

Divide both sides by 0 and you get:

1 = 3

Math is bollocks! Bin it all!

Rhizo

You just proved the trinity!

(Although my wife tells me that's incorrect but insists I don't quote her proof!)

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My eureka moment came when trying to double the 'gap' to see the larger gap... the difference between 0.9r and 2 - 0.9r ...in both cases there is not ever convergence with '1', but there is still no number you can add to 0.9r to make it 2 - 0.9r ...there is no representation of infinity minus an infinitesimal amount which would = convergence.

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How do you get a calculator to display 0.9r ?

1 / 3 = 0.3r
0.3r/ 3 = 0.1r
0.1r x 9 = 1

8.99999999999999 / 9 is the closest I've got

[Violet]

My eureka moment came when trying to double the 'gap' to see the larger gap... the difference between 0.9r and 2 - 0.9r ...in both cases there is not ever convergence with '1', but there is still no number you can add to 0.9r to make it 2 - 0.9r ...there is no representation of infinity minus an infinitesimal amount which would = convergence.

But you still do need that infinitesimal before it will be 1. That's my 'hang-up', if it can be called that. I also reject the 'mathematics' that could claim an infinite to be finite. That's why I cited the infinite limit: that the infinite looks like a graph that grows ever steeper, with a repulsion to the finite it is endlessly approaching.

It's like being caught in the event horizon of a black hole

[Tiberius]

But you still do need that infinitesimal before it will be 1. That's my 'hang-up', if it can be called that. I also reject the 'mathematics' that could claim an infinite to be finite. That's why I cited the infinite limit: that the infinite looks like a graph that grows ever steeper, with a repulsion to the finite it is endlessly approaching.

It's like being caught in the event horizon of a black hole

For the last time, nobody is claiming that infinites are finite. 0.9... is another way of writing 1 in decimal form. Again (I feel the need to knock this into your head repeatedly), there is a difference between something that is infinitely large and infinitely long.

[Violet]

I have great difficulty fathoming a difference between the infinite of 'infinitely large' and the infinite of 'infinitely long', or the infinite of 'infinitely stupid', or even the infinite of 'infinitely happy'.

I honestly don't see the difference in the infinite concept... all the more so as something infinitely long would also be infinitely large, and something infinitely large would also be infinitely long.

Anyway, it is because of our numbering system (base ten) that we cannot perfectly represent 1/3, 1/7, etc. I made that point in a post somewhere on page #12 I'd guess. The fractions are very simple, quite perfect to be honest. If I have a pie, and I cut it into 3 equal pieces: 1/3 x 3/1, 100% of the pie. I cannot represent this number as perfectly with a base ten system however, and the closest accuracy I will be able to attain is .3^ x 3, 99.9^% of the pie.

This isn't a problem with the faction, and we know intuitively that 100% of the pie was really eaten... but we need to recognize that there is an infinitesimally small gap before we attain 100%. This is not because you can't divide one thing into 3 pieces (you usually can)... it is because the base 10 system (while simple) has limitations. I find that our knowledge more often advances from questioning answers, then from answering questions. That we have established our numbering system neither makes it perfect nor makes oversights go away.

[Tiberius]

There isn't any difference in the infinity concept, but you seem to repeatedly confuse the context in which the infinity is used; hence me repeatedly telling you so.

The difference in context is very important. An infinitely large number cannot have any number larger than it. An infinitely long number cannot have any number longer than it (and for sake of argument, we call all infinitely long numbers the same length so it doesn't contradict the definition).

Nobody is trying to turn an infinite into a finite, but you can use induction (and common sense) to show that since:

0.9 - 0.9 = 0
0.99 - 0.99 = 0
0.999 - 0.999 = 0

that if you continually add 9's to each side, you will get the same result...0.

0.999... - 0.999... is no different. Now I'll admit, I made a mistake earlier in the topic (way back) when I said that "infinity - infinity = 0". I wasn't thinking right at the time and was trying to get across another concept all together. However, this context is entirely different, and it matters not that infinity - inifinity =/= 0 for the proofs provided. This isn't about the subtraction of infinite numbers, but the subtraction of infinitely long numbers.

We can perfectly represent 1/3 and 1/7 in decimal. 1/3 = 0.333... and 1/7 = 0.142857... (repeating 142857). I fail to see why the fact that these numbers are infinitely long means they are inaccurate.

[theVOID]

Sae - you raise some good points, but the clincher is the proof - Adrian is right when you look at the mathematics

1 = 9/9 = 9 x 1/9 = 9 x 0.111...=0.999

This is a non intuitive fact, the idea that two decimals can represent the same number, but it works in the proof. As non intuitive as this is when you think about it there is no getting away from the proof.

So Sae - are you going to stick to your predefined notions on the nature of infantecimals, or believe the proof when it is inconsistent with your existing understanding? You'd be doing something those other people always do.......

[Violet]

An infinitely large number cannot have any number larger than it. An infinitely long number cannot have any number longer than it

I said something wrong too I was trying to note that as a number gets longer (ie 9 < 99 < 999, just as .9 < .99 < .999) it also gets larger, and as a number gets larger it often gets longer (IE 1 < 10 < 100.) (Neither of these is always true, but I thought it might be an interesting correlation to draw between the two.)

I understand that .9^ - .9^ = 0... And as I am sure I showed in a recent post (the long one with hide tags I think, though it might have been the one before or after that one), in doing so we have cancelled out an unsolvable. As with (1 + x)(1 + x)/(1 + x), you can cancel out difficult (and in some cases impossible), and make a finite number of infinite or otherwise unsolvable numbers, as in: (1 + x)/1.

As i've said before, 1/3 is perfect as a fraction... but not as a decimal. Fractions really are much more perfect than decimals... and calculate otherwise impossible to calculate things (See pi) accurately. Just as we could not hope to call Pi's decimal representation perfectly accurate, we cannot call .3^ or .142857^^ perfectly accurate. Infinitely long numbers (representing fractions) cannot be represented perfectly in decimal form... as they will always require an infinitesimally small round up. This is negligible for all practical applications... but an inequality is an inequality, no matter how small

And actually, why wouldn't infinity - infinity would cancel itself out into 0... just as much any other infinite concept? Unless the two infinites are of different length of course, which would make one of them not infinite.

---

Sae - you raise some good points, but the clincher is the proof - Adrian is right when you look at the mathematics

1 = 9/9 = 9 x 1/9 = 9 x 0.111...=0.999

This is a non intuitive fact, the idea that two decimals can represent the same number, but it works in the proof. As non intuitive as this is when you think about it there is no getting away from the proof.

So Sae - are you going to stick to your predefined notions on the nature of infantecimals, or believe the proof when it is inconsistent with your existing understanding? You'd be doing something those other people always do.......

What's inconsistent about it, if you don't mind my asking? I don't doubt that the mathematicians probably have had the right idea... I simply don't see it yet Which other people specifically?

As far as I can see... the problem in understanding this comes not from the mathematics themselves... but from the decimal representation of them (base ten vs fractions). This is a lot like reading different languages... where in this language, I say A, B, C... and in the other language I say A, B, D&E. Translating between these two languages is not perfect, because there is no C in one, and the other lacks D&E. Some languages just lack C, and have no new letters.

From fraction to decimal is no different... and translation isn't perfect. Now, with both fraction and decimal cited (EG: 1/3x3=1 = .3^x3=1) [as many of the proofs are written]: It makes sense because of the conversion apparent between the two. However, decimal alone lacks the words we need to turn .9^ into 1... and leaves an infinitesimal gap between the two... just as .989^ leaves an infinitesimal gap between .989^ and .99. The gap is there with infinite decimals (note: or else there would be no use for them)... and that is just part of the 'grammar' of the language. That gap needs to be recognized, and condoned. It is negligible to the point that it doesn't matter in the slightest... but it remains in existence.

[Tiberius]

Why are you bringing infinity into it? This is not a discussion of infinity as a value (and I am well aware of the arguments that infinity is not a value...I agree wholeheartedly with them). Infinity is completely different from infinitely long numbers though. An infinitely long number doesn't equal infinity, as it could be 1.999999... (infinitely long string of 9's), making the number equal to 2 (or for sake of your argument, "just less" than 2).

Infinity is a concept tied very much to 1.9^. Infinity is an uncertain value... because of its endless nature. Until you remove the uncertainty from 1.9^... you cannot calculate with certainty. Infinity is not a number (value/whatever), it is a concept... and it is a concept present within 1.9^ that makes the number itself incalculable (because you have a value under the effects of an infinite concept, which makes the number undefined.).

Infinity is not an uncertain value. It isn't a value at all. You contradict yourself several times in this paragraph alone because you refer to infinity as a value and then say it isn't a number/value. Infinity as a value does not work, and the proof of this is easy enough. We aren't using it as a value however, so there isn't any panic.

If one has tied an infinite concept into a value, as one can see with .9^: then one has made the value of this number dependent upon infinity. The value of .9^ is based within something without calculable value. I thought I should bring this up before I disproved this 'equality'.

No it hasn't. The mere fact that we can write .9^ and know precisely what it means disproves your assertion that it isn't calculable. It might take an infinite amount of time to calculate it, but it is still calculable. Long division can show us that quite easily.

An infinitesimal is a number that cannot be measured because it is "so" small. It cannot be distinguished from 0. Your number 0.0^1 is not an infinitesimal number for two reasons:

1) You cannot logically or mathematically have an infinite string of 0's (as denoted by your 0^) which is then followed by a 1. We have been over this. An infinite string has no end, so you have no place to put the 1 on.
2) Even if this number were to exist, it is easily distinguishable from 0, since it is 0.0^1 away from 0. Ergo it does not have the attributes of an infinitesimal number.

An infinitesimal is 'the infinith' number of infinity. By the very definition of it, 0.0^1 is the definition of the infinitesimal. It is no easier to distinguish from 0 than .9^ is from 1. They are an equivalent distance away from 0 and 1. The reason infinitesimals exist is because the infinith number exists. Infinity is like an endless gap... between two pieces of land infinitely far away from each other (And that land exists, just so much as that gap exists). .9^12343 is no different in this concept. If you remove the infinite from the value: you can calculate that 12343. That is why such numbers are able to exist, assuming infinites can be calculated at all.

There is no such thing as the "infinith" number of infinity, mainly because with infinity you can always go one further. I'm going to take a guess at what happened here. You wanted to sound clever without doing much work, so you quickly skim read the Wikipedia article on infinitesimals. You saw the word "infinith" and decided that sounded pretty cool and smart, so you wrote that word down.

Shame you didn't read further, since you would have learned that:

The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.

This is what you get from reading Wikipedia without understanding basic concepts I guess. You contradict yourself again by saying Infinity is both "endless" and that it can have "12343" on the end.

If you don't like Wikipedia, I suggest you don't use it to try and argue about infinitesimals. If you want other sources, I suggest you use Google:

I didn't say that I don't like wikipedia... I said it is not always a reliable source. If mathematicians truly think this, then they are incorrect. So far as I can tell, and I have seen no reason to think otherwise: these 'proofs' are nothing but errors of mathematics.

The Wikipedia page was written by mathematicians...duh! I gave you links before to University mathematics departments on the proof that 0.999... = 1. Take it up with them if you like...but I wouldn't.

This is exactly your error in proving this inequality is equal. You have removed the infinite from the equation. An infinitely long string of numbers is infinite. It is undefinable, and it is endlessly long (thus endlessly gargantuan). Is a hot fire not fire? Then is something infinitely long not infinite? (The Identity's Equality). Not at all. The concept is exactly the same. In 'infinitely long' you are describing the length as being infinite. And here again you are tying an infinite into the definition (and thus value) of something. That's what is being done here .0^1 & .9^

No, an infinitely long string of numbers is not infinite. It is infinitely long (I thought I'd covered that with the original descriptor). Is something infinitely long not infinite? Yes, of course it is. Here is a proof:

0.999... is an infinitely long number
2 (or any number above 1) is greater than it.
Infinity cannot have any numbers greater than it.
Ergo 0.999... is not infinite.
Q . E . fuckin' D.

What you are doing is exactly what you accused me of doing before, taking an infinite and removing it, making it finite. .0^1 as a number cannot exist. You cannot have an infinite number with a finite end.

This is precisely why these 'proofs' fail to prove what they are trying to do: they are removing the infinite from the equation to prove it. That is fallacious reasoning to me. And some people do not believe that one even can remove an infinite.

There was no infinite to begin with. I've been over that.

And thus you are no longer proving that the infinitely long number is equal to a finite number... because you have cancelled out specifically that which you are trying to prove. I do not deny that this is basic math... that it is simple basic math that 1 ≠ .9^ is my point entirely in fact.

Ok, so by that logic if I wanted to prove a fraction was equal to another fraction, canceling out the fraction means the proof is wrong since I've canceled out specifically that which I was trying to prove? Ok, so maths is broken...

1/2 = 2/4

(multiple both sides by 4)

2 = 2

Oh noes! The fractions are gone! 2 =/= 2!

Yeah...erm...bullshit.

Umm... 2.2^x9 = 19.9^8. which is ≠ to 20 (even further from equality than .9^ is from 1). In one of the equations, the infinitely long number remains. In the other: it has been neutralized. This discrepancy between otherwise equivalent methods is because an infinite is an undefined, and until you remove the undefined: you cannot have a defined. In one method it was removed, and the finite answer of 20 is available. In the other: it was not removed, and 19.9^8 is the closest approximate we can get. It remains undefined.

Ok, please get out your calculator and type in 20 / 9. The answer? 2.2^

Now do some basic algebra:

20 / 9 = 2.2^
(multiple both sides by 9)
20 = 2.2^ x 9.

Again, your lovely example had yet more errors, with the number 19.9^8 which doesn't exist, because *takes a deep breath* YOU CAN'T HAVE AN INFINITELY LONG NUMBER WITH A FINITE END...

Citing my gapped land example above: yes, you can. You can have an infinitely long value separating other values. There is no impossibility about it. An infinitesimal is infinitely small number, which can easily be represented by 0.0^1... infinite zeros to show that it is infinitely small... and a 1 for the infinith digit to show that it is different than zero. Once again: nothing impossible about it.

You are indisputably a moron. Instead of reading misreading Wikipedia, please go and ask some Mathematics professor (preferably one who has a PhD) to teach you something correct for a change.

And actually, infinity - infinity would cancel itself out into 0... just as much any other infinite concept. Unless the two infinites are of different length of course, which would make one of them not infinite.

Ok, so apparently you didn't even pass Math at age 15, where I learned the proof that this doesn't work at all. Here it is for your enjoyment.

Imagine a positive integer number line (all possible integer values from 0 and above). The number line tends to infinity. For every integer you have, you can always add one to it (1 2 3 4 5 ...). In this set, which we shall call P, there are an infinite number of integers.

Now imagine a negative integer number line (all possible integer values below 0). The number line tends to - infinity. For every integer you have, you can always subtract 1. (-1 -2 -3 -4 -5...). In this set, which we shall call N, there are an infinite number of integers.

Good so far? Ok, so what about the entire set of integers, spanning from - infinity to + infinity. It's an infinitely large set. It isn't 2 x infinity, since such mathematics (if it even worked...which it doesn't) simply ends up with a value of infinity again.

To reduce it even more. There are an infinite amount of integers (both negative and positive) in the set of all integers. However between 0 and 1, there are an infinite amount of real numbers. So the set of all real numbers is the set of all integers (infinitely large) multiplied by infinity, which equals infinity?

No. Infinity is simply not a value. You could argue that in the equation "Infinity - Infinity", the first Infinity we are talking about is the set of all real numbers, and the second Infinity is the set of all real number between 0 and 1. So we are left with an infinitely large set made up of the sets of real numbers between every integer and it's neighbours, apart from 0 and 1.

It just doesn't work. Infinity is not a value.

[Ephrium]

Some here cannot accept the fact 0.99999999........ minus 0.9999999999....... =0

I tell you, to look at it from a philosophical viewpoint.


What is 1,2,3,4 etc. They are not some magical fairies. They simply represent a certain quantity.

Likewise, 0.99999999.... is a quantity. Whn this quantity minus the same quantity, itself, it becomes 0.

Looked at from my viewpoint it is easy.

Recall I have high IQ. When you try to answer a question, a very basic approach is to figure exactly what are the things involved in the question, what is the nature of it.