Does 0.9 repeated = 1?

Imaginary numbers are not less real then real numbers, actually both are subsets of complex numbers. Complex numbers are the most usefull things in mathematics, they are used in; Quantum mechanics, Engineering(they call it j for some reason), Cosmology(Stephen Hawking has put forth the idear that time was complex before the universe), Computer Science, Cryptology e.t.c.

Its debatable whether numbers are real or just part of a language. For example, there are two apples can be critised as there is no such thing as a standardized unit of apple. Also, 2cm can be critised too as there is something called the Heisenberg uncertainty principle, this disproves the idear that something sits in spacetime at a certain position.

Calculus is a trick subject, most of the stuff in it is not justified or formalised. This paradox you speak off is the fact that its hard to understand why, however, everything is formalised and made exact when you study Analysis.

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Originally Posted by Xei

My view also happens to agree with every piece of literature about the nature of mathematics I've ever read. I suggest you read some books about formal systems such as Godel Escher Bach or The Emperor's New Mind.

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If I was you I would read Chaitin's book Meta-Maths. You need to actually read a book that says the complete opposite to your Philosophy, as alot of Mathematician have said that Maths is not real and just a language. Rodger Penrose has been heavily critised for trying to say that Quantum Mechanics has a role in consciousness, his views are really on the frindge of Physics.

Formal systems I think will get replaced by Category theory.

Don't worry, I disagree with Penrose about quantum consciousness.

And he is actually a platonic realist, he says that the Mandelbrot set is evidence as it does not exist in physical reality or mental reality but is still real.

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Originally Posted by Xei

I know exactly what the answer is, thank you. You have just completely failed to comprehend what my post is about.

I got an A in A Level maths a year early and I managed to pass interview and get an offer to read maths at Cambridge university, which has the best and hardest maths course in the world.

So yes, I know what 2^-1 is.

._.

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sorry, im really bad with doing that, not understanding things. so were you saying that you dont see how it fits in with the pattern of 2*2 etc. ?

p.s. congrats on the uni thing, i hear cambridge is very good, but expensive am i right? of course, i live in scotland, so universities are free, yaay snp wooooo!!!!

I do see how it fits in with the pattern. It's just that you can't represent it in physical terms; we define it as the reciprocal because it gives a consistent and extremely useful system, but it's hard to give physical meaning to a negative number of numbers.

And argh, bloody Scots... it's really a complete travesty that the Labour government can constantly yell about promoting social mobility whilst also introducing ridiculous fees for university... the education system's completely fucked up, it's blatant that A Level grades are being eroded to cover up the rapidly falling standards, to the point that my qualifications are becoming worthless. Cambridge will be relatively cheap though, compared to the living costs of the Lodon unis I've applied to.

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Originally Posted by Xei

I do see how it fits in with the pattern. It's just that you can't represent it in physical terms; we define it as the reciprocal because it gives a consistent and extremely useful system, but it's hard to give physical meaning to a negative number of numbers.

And argh, bloody Scots... it's really a complete travesty that the Labour government can constantly yell about promoting social mobility whilst also introducing ridiculous fees for university... the education system's completely fucked up, it's blatant that A Level grades are being eroded to cover up the rapidly falling standards, to the point that my qualifications are becoming worthless. Cambridge will be relatively cheap though, compared to the living costs of the Lodon unis I've applied to.

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oh right i see what your getting at now.

thank god im scottish:mwahaha:

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Originally Posted by Xei

Admit what?

I said if you can't intuitively accept it then you'll just have to accept the proof.

I personally have no problem with it and see no paradox.

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If you can't see the seeming contradiction intuitively, then you will just have to accept that there is one.

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Originally Posted by Xei

I'm trying have a reasoned discussion here. There was nothing remotely inflammatory about my response. Why are you responding like that?

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Uh, because you made the shit-head comment that I don't know much about math? If you aren't aware of the fact that your ignorant comment was a blatant personal insult, you have a world of social skills training to undergo.

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Originally Posted by Xei

Yes sorry I meant 2^3 means 2 multiplied by itself 3-1 times, etecetera.

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Very good.

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Originally Posted by Xei

How can you 'represent' a number a negative amount of times? There is very little intuitive meaning you can give to this.

2^5 = 2 * 2 * 2 * 2 * 2
2^4 = 2 * 2 * 2 * 2
2^3 = 2 * 2 * 2
2^2 = 2 * 2
2^1 = 2

2^-1 = ?

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Its reciprocal. See my earlier post for the geometric change explanation.

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Originally Posted by Xei

The fact is negative numbers are just as 'unreal' as imaginary numbers. You can't physically represent a negative number of objects.

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There is a reason they are called real numbers. Do you think it is just some random word that was chosen? There are negative temperature degrees, you can owe negative amounts of money, and you can have negative amounts of money in an account. Negative numbers are real. But the square root of -1 flat out does not exist. It is only a hypothetical concept.

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Originally Posted by Xei

What on Earth is wrong with you?

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I don't respond very nicely to unnecessary assholishness. Please forgive yourself.

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Originally Posted by Xei

I wholeheartedly mean everything I say, otherwise I would not say it. You think I have some sort of agenda here?? My view also happens to agree with every piece of literature about the nature of mathematics I've ever read. I suggest you read some books about formal systems such as Godel Escher Bach or The Emperor's New Mind.

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I suggest you counter my arguments instead of appealing to Hofstadter and other eccentric writers.

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Originally Posted by Xei

Maths does not have monopoly on logic actually.

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No, but logic has a monopoly on math, actually.

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Originally Posted by Xei

There are true statements about numbers which cannot be proved with maths.

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Yes.

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Originally Posted by Xei

The first thing you should have been taught when learning about IMAGINARY numbers is that IMAGINARY is a technical term, not the adjective imaginary. The nomenclature comes from an old prejudice against French mathematics. It is now completely redundant.

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There is no square root of -1. Can you tell me what it is, aside from the letter i that is used to merely symbolize it? Can you tell me? You can't. That is why the word "imaginary" was used to describe the unit. There is no square root of a negative number.

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Originally Posted by Xei

pi is something which results from Euclid's postulates. Euclid's postulates are a model, and are actually physically wrong to high degrees of precision. But using Euclid's postulates and calculus, we can calculate that the number pi is equal to

4/1 - 4/3 + 4/5 - 4/7...

This is the only value that pi can have. It is not arbitrarily defined.

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Pi is something that results from the ratio of a circle's circumference to its diameter. Period. It is not some crazy number Euclid made up out of nowhere. It is REALITY. That is why we have to deal with the fact that the number is such a pain in the ass. We have no choice. Reality is what it is. Math is what it is. We only create the symbols.

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Originally Posted by Xei

Yes, but I'm not looking at the number '1' right now, am I?

You are using '1' to describe objects in physical reality.

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There is still 1 computer. That does not mean 1 is an object. Again, matter is not the only form of reality. 1 is a logical concept. The square root of -1 is an illogical concept.

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Originally Posted by Xei

Equally I could describe various physical phenomena using i (and could not without).

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Fill in the blank. One time, somebody had i ________.

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Originally Posted by Xei

I could also use 'quite long' to describe your post. This doesn't mean that 'quite long' exists.

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One is a vague description, and the other is an exact figure. However, if my post really is "quite long", then "quite long" exists.

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Originally Posted by Xei

1 and
10
01
are exactly the same thing. How can you claim that one way of representing it is real yet another is not? I can perform exactly the same operations upon it and get exactly the same answers.

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An operation you can do with the matrix gives the same figure. The matrix is a man made system of representing numbers, which existed long before we ever did.

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Originally Posted by Xei

Okay, real numbers do not have physical existence. Neither do imaginary numbers. This is obvious. But this 'other reality' which you hint at; why does i not belong there whilst 1 does?

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I will give you the answer again in i days.


The end of infinity is a paradox. I have yet to see a viable counterargument to that.

This is beginning to look vaguely like zeno's paradox.

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If you can't see the seeming contradiction intuitively, then you will just have to accept that there is one.

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No, that is the difference between proof and ituition, isn't it? You can't prove that there is a seeming contradiction, you can't even state it in words properly, and nobody else understands what you're talking about.

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Uh, because you made the shit-head comment that I don't know much about math? If you aren't aware of the fact that your ignorant comment was a blatant personal insult, you have a world of social skills training to undergo.

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I said that your understanding of the terms IMAGINARY and REAL are typical of somebody who hasn't ever used them, or at least had them explained properly. Have you ever actually done any maths with them?

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There is a reason they are called real numbers. Do you think it is just some random word that was chosen? There are negative temperature degrees, you can owe negative amounts of money, and you can have negative amounts of money in an account. Negative numbers are real. But the square root of -1 flat out does not exist. It is only a hypothetical concept.

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Yes, the term was coined by Descartes and was intended to be derogatory, because he didn't like them. That was 400 years ago and our knowledge of mathematics and physics has become extremely more advanced.

You are saying that real numbers are real because they exist in physical reality, essentially. You can use positive numbers to describe a set of similar objects.

But you must also use complex numbers to describe various aspects of reality. The mathematical formulation of quantum mechanics relies heavily on complex numbers; unless you don't think quantum physics is real, complex numbers are as real as real numbers. They're just a bit harder to find.

Fractals, too; do you think that the Mandelbrot set doesn't exist?

i can also be said to be a 90 degree anticlockwise rotation in 2D space. Are rotations not real?

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There is no square root of -1. Can you tell me what it is, aside from the letter i that is used to merely symbolize it? Can you tell me? You can't. That is why the word "imaginary" was used to describe the unit. There is no square root of a negative number.

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Can you tell me what 1 - 2 is without using negative numbers?

PS

0-1
1 0
*
0-1
1 0
=
-1 0
0 -1

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Pi is something that results from the ratio of a circle's circumference to its diameter. Period. It is not some crazy number Euclid made up out of nowhere. It is REALITY. That is why we have to deal with the fact that the number is such a pain in the ass. We have no choice. Reality is what it is. Math is what it is. We only create the symbols.

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It is not reality. In physical reality, the ratio of the circumference of a circle's diameter to its circumference is not exactly pi.

However, if you assume Euclid's (physically incorrect) postulates, then the only value you can possibly get for this value is 3.141...

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There is still 1 computer. That does not mean 1 is an object. Again, matter is not the only form of reality. 1 is a logical concept. The square root of -1 is an illogical concept.

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Okay, I'll try to explain it by looking at the history of maths:

We used to only have positive numbers. You could add them up, multiply them, and take away small ones from big ones.

Taking away big ones from small ones was not allowed.

However, with the development of algebra, it was realised that if you create a new kind of number, -1, it creates an extremely useful consistent system which can often be used to describe physical reality.

500 years ago, we only had real numbers. You could take roots of positive numbers, but taking roots of negative numbers was not allowed.

However, with the development of complex algebra, it was realised that creating a new number, i, creates an extremely useful and consistent system which can often be used to describe physical reality.

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Fill in the blank. One time, somebody had i ________.

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You can't have an imaginary number of things.

You can't have a negative number of things either.

Fill in the blank. One time, somebody had -1 ________.

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One is a vague description, and the other is an exact figure. However, if my post really is "quite long", then "quite long" exists.

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Well, I disagree. The object which is quite long exists, but not quitelongness.

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I will give you the answer again in i days.

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Considering we don't travel along the hypothesised imaginary time line I'll take this as meaning you can't answer.

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Originally Posted by Xei

No, that is the difference between proof and ituition, isn't it? You can't prove that there is a seeming contradiction, you can't even state it in words properly, and nobody else understands what you're talking about.

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I have proven that there is a seeming contradiction, and I have boldfaced my two clause statement of it several times. You still don't get what I am saying, but that does not mean nobody else does. Even if nobody at all gets it, what I am saying is logical. You act like this topic and Zeno's Paradox are complete non-issues, but they are both legendary because of their paradoxical natures. Even if you don't agree with what I am saying, you should still have enough class to not be such a schmuck about it. What went wrong in your life? This is an interesting topic. Stop trying to turn it into a sour grapes session.

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Originally Posted by Xei

I said that your understanding of the terms IMAGINARY and REAL are typical of somebody who hasn't ever used them, or at least had them explained properly. Have you ever actually done any maths with them?

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Oh, I was thinking you said this...

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Originally Posted by Xei

The stuff you say about imaginary numbers shows that you have very little understanding of what maths is

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Was your roommate using your account at the time?

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Originally Posted by Xei

Yes, the term was coined by Descartes and was intended to be derogatory, because he didn't like them. That was 400 years ago and our knowledge of mathematics and physics has become extremely more advanced.

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Well, negative numbers still don't exist. Maybe they will tomorrow or something. Try them out tomorrow and see if they are logical yet.

What is the square root of -1? Don't give me just a symbol that represents the imaginary principle. What is the square root of -1????????

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Originally Posted by Xei

You are saying that real numbers are real because they exist in physical reality, essentially. You can use positive numbers to describe a set of similar objects.

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No, I am not saying that. I am saying they can be applied to reality, logical reality and physical reality and probably other forms of reality. Imaginary numbers, on the other hand, are just hypotheticals that deal with impossibilities. They are not logical. Real numbers are logical.

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Originally Posted by Xei

But you must also use complex numbers to describe various aspects of reality. The mathematical formulation of quantum mechanics relies heavily on complex numbers; unless you don't think quantum physics is real, complex numbers are as real as real numbers. They're just a bit harder to find.

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I am not a physicist, but I have a lot of issues with quantum mechanics. I am a strict determinist, and I think that subjectivity is something that has very little place in science. You can see by my sig line what I think about subjectivity, so go figure. My prediction on quantum physics is that most of it will turn out to be a big crock of shit. The Heisenberg Principle should not even be famous. The idea that two particles in very different parts of the universe randomly become attracted to each other and move across light years to collide throws up a big question mark. There are no uncaused events, so Hawking's dice point is ridiculous. The idea that matter is not infinitely divisible really raises one of my eyebrows. Quantum physics so far strikes me as the left wing version of science, full of rationalizations and rebelliousness against the obvious for the pseudo-intellectual rush that comes with it. I don't quite buy into it, but I plan to study it a lot more. I am sure there is merit to some of it, but I am not sure what bits yet.

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Originally Posted by Xei

Can you tell me what 1 - 2 is without using negative numbers?

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No. It is -1. There is no way around it. If you owe me $5, I owe you $-5. That can be a matter of accounting and not dollar bills or checks. If I do something in -1 day from now, I did it yesterday.

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Originally Posted by Xei

It is not reality. In physical reality, the ratio of the circumference of a circle's diameter to its circumference is not exactly pi.

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If there are some differences in digits somewhere, pi is at least a very close approximation to the ratio. That is the origin of pi, and it is why we use it. We use it to calculate circumferences, diameters, and radii of circles, so go figure. There is a reason we can't suddenly decide, "Hey, let's make pi = 1. That will make things so much easier." We deal with mathematical reality as it is, not as we create it. We do not create mathematical reality. You have not explained your way around that point yet. Can you?

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Originally Posted by Xei

Okay, I'll try to explain it by looking at the history of maths:

We used to only have positive numbers. You could add them up, multiply them, and take away small ones from big ones.

Taking away big ones from small ones was not allowed.

However, with the development of algebra, it was realised that if you create a new kind of number, -1, it creates an extremely useful consistent system which can often be used to describe physical reality.

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That is when we discovered them.

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Originally Posted by Xei

500 years ago, we only had real numbers. You could take roots of positive numbers, but taking roots of negative numbers was not allowed.

However, with the development of complex algebra, it was realised that creating a new number, i, creates an extremely useful and consistent system which can often be used to describe physical reality.

You can't have an imaginary number of things.

You can't have a negative number of things either.

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Yes, you can have a negative number of things. I have explained that.

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Originally Posted by Xei

Fill in the blank. One time, somebody had -1 ________.

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Dollars. I have also had negative numbers of much greater absolute value than that for a bank account status. Negative numbers were sometimes a very real pain in the ass in my earlier adult life. I had to deal with them. They were extremely real.

If your rent was due yesterday, you have -1 day to pay your rent on time. I promise you.

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Originally Posted by Xei

Well, I disagree. The object which is quite long exists, but not quitelongness.

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The quality of the object does not exist?

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Originally Posted by Xei

Considering we don't travel along the hypothesised imaginary time line I'll take this as meaning you can't answer.

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My response spelled the answer right out. Imaginary numbers are not part of reality beyond the realities of fiction and imagination. Negative numbers are part of reality, as I illustrated. They are real numbers.


Now let's get back on topic. The 9's can never get to a point where they form 1, so it seems, yet they form 1. Infinity is reached, and that is a paradox.

Even if you disagree with that, at least admit that there is something at least a little bit interesting about the situation. Maybe say something like, "Yeah, that is maybe a bit strange." The, "Fuck you. You don't know shit," stuff is very unnecessary and childish. This is a really fascinating topic.

Infinity is reached. It says so right here:
http://img211.imageshack.us/img211/4874/999lu3.png

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Originally Posted by Xei

Infinity is reached. It says so right here:
http://img211.imageshack.us/img211/4874/999lu3.png

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How can a nonexistent limit be reached? Again, obviously it is. But HOW?????????

How? The question 'how' does not apply to numbers, does it?? There's not some kind of number building machine hidden behind the pixels, which we are trying to find out how it works. It is, because we can prove it to be so. There is no 'how'.

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Originally Posted by Xei

How? The question 'how' does not apply to numbers, does it?? There's not some kind of number building machine hidden behind the pixels, which we are trying to find out how it works. It is, because we can prove it to be so. There is no 'how'.

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:lol: That is what I have been asking this whole time. There we have your answer... There is no "how". Thanks.

How could there be no "how"?

Infinity is "reached" because you have to look at the number outside of the context of time.

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That is what I have been asking this whole time. There we have your answer... There is no "how". Thanks.

How could there be no "how"?

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Because there is no process going on here! There is no mobile adding of 9s, a number is a static thing.

You need a process to ask 'how', don't you?? That's what 'how' means; by what process?

'Bob eats a pie'.

HOW???

Fair enough.

'Bob is a pie'.

HOW???

What the hell?

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Originally Posted by Kushna Mufeed

Infinity is "reached" because you have to look at the number outside of the context of time.

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Even with time completely absent from the picture, there still seems to be a paradox.

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Originally Posted by Xei

Because there is no process going on here! There is no mobile adding of 9s, a number is a static thing.

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Does a limit that does not exist also exist, statically?

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Originally Posted by Xei

You need a process to ask 'how', don't you?? That's what 'how' means; by what process?

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Not at all. By what logic is what I am asking.

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Originally Posted by Xei

'Bob is a pie'.

HOW???

What the hell?

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If you found out your friend "Bob" is a human and also a pie, would you not ask, "How?" I would. I would say, "What the Hell? How is that possible?"

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If you found out your friend "Bob" is a human and also a pie, would you not ask, "How?" I would. I would say, "What the Hell? How is that possible?"

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Sarcasm I hope. I'll clarify for you:

Bob ate a pie.

HOW???

He puts it in his mouth, chews, and swallows.

Frederick is a rock.

HOW???

Uh...

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Not at all. By what logic is what I am asking.

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Very good, we've got there:

x = 0.999~
10x = 9.99~
10x - x = 9x = 9.99~ - 0.999~ = 9
x = 1

By that logic.

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Originally Posted by Xei

Frederick is a rock.

HOW???

Uh...

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So the answer would be, "Uh..."? I can assure you I would keep trying to get a better answer than that. I would say, "How does a non-thinking thing think?" A response like, "This picture and medical document prove that Frederick is a rock," would not answer the question.

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Originally Posted by Xei

Very good, we've got there:

x = 0.999~
10x = 9.99~
10x - x = 9x = 9.99~ - 0.999~ = 9
x = 1

By that logic.

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That does not address the specific paradox I keep talking about.

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So the answer would be, "Uh..."? I can assure you I would keep trying to get a better answer than that.

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Yes, precisely. You have shown us all that you will keep trying to get a better answer to a question which is completely meaningless. 'How is Frederick a rock' is completely and utterly meaningless.

You're asking 'how' of something which is not a process. How is this rock called Frederick? How is this 1 called 0.999~? Nonsense.

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Originally Posted by Xei

'How is Frederick a rock' is completely and utterly meaningless.

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The question of how a rock can think and be a human is completely legitimate. I don't think you would be satisfied with, "Uh... It's not a process."

Frederick isn't a human. Frederick is a rock.

:|

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Originally Posted by Xei

Frederick isn't a human. Frederick is a rock.

:|

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I am talking about a situation in which Frederick is apparently both a human and a rock. The question of, "How?" would be competely legitimate even though Frederick's status as both rock and human is not a process.

How means 'by what process', or the 'manner in which something is accomplished'. The question you should ask is 'why', which means 'for what reason'.

And of course, your analogy makes no sense, because rocks and humans are mutually exclusive objects.

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Originally Posted by Xei

How means 'by what process', or the 'manner in which something is accomplished'. The question you should ask is 'why', which means 'for what reason'.

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"How?" as in, "How is it possible?" is not an illegitimate question.

Person 1: "Frederick is a man AND a rock."
Person 2: "WTF????" How? How is such a thing possible?"
Person 1: "This proves that he is. That's how."
Person 2: "Huh? Is that all you have to say about it?"

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Originally Posted by Xei

And of course, your analogy makes no sense, because rocks and humans are mutually exclusive objects.

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Exactly, hence the relevance of the hypothetical.

Hypotheticals often involve concepts of the impossible, i.e. imaginary numbers. The fact that they are not possible realities does not make them illegitimate hypotheticals.

I do not get how you guys blow these things out of proportion. Simple answer:
0.9999~ does not equal one.
it is 0.~1 away from one.
This sounds impossible, but the fact that your here evens it out.

If you think about it, 0.9 is close, .09 makes it a little closer, .009 is very small and makes a miniscule difference, .0000000000000000000000000000000009 makes very little difference.