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Geek Culture / Is 10 = 9.9999999... ??

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ionstream
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Posted: 26th Aug 2008 00:24
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Mathematical Induction is not flawed because of the key phrase "true of all natural numbers" so that complex, imaginary, or even rational numbers do not apply (and of course infinity). If it said "any value" then yeah I'd say it was broken (of course who am I to say anything about such a widely used concept).

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Mahoney
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Posted: 26th Aug 2008 00:25
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Quote: "of course who am I to say anything about such a widely used concept"


Widely accepted != true.

But yes, Mathematical Induction is not flawed as far as I know. It simply doesn't include infinity.

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mamaji4
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Posted: 26th Aug 2008 00:29
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The new set of natural numbers has been defined by two new mathematicians Mahoney and ionstream and they do not include infinity. All ye rise and heed the words of the wise men.

I rest my case.
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Mahoney
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Posted: 26th Aug 2008 00:30
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Quote: "The new set of natural numbers has been defined by two new mathematicians Mahoney and ionstream and they do not include infinity."


I do hope you're being sarcastic.

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ionstream
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Posted: 26th Aug 2008 00:31
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Quote: "Widely accepted != true. "


Yeah but unless I spend a whole lot of time trying to disprove it I really wouldn't have a leg to stand on! I couldn't make any judgment unless I heavily study the thing.

Now I really need to get to programming, as Iron Man once said, "I'm working on something big!"

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mamaji4
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Posted: 26th Aug 2008 00:33
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Quote: "I do hope you're being sarcastic.
"


It was a good discussion, anyways, so no bad blood. Peace be to all.
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Mahoney
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Posted: 26th Aug 2008 00:35
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Quote: "Now I really need to get to programming, as Iron Man once said, "I'm working on something big!""


Lol. I know what you mean. Good luck on "something big."

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Seppuku Arts
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Posted: 26th Aug 2008 01:24 Edited at: 26th Aug 2008 01:25
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I suppose some of us can feel like bigger geeks now - we argued maths for 8 pages (well I gave up earlier on) Though nothing wrong with an 'intelligent' discussion as long as no one gets hurt or goes psychotic on us.

However, I have to say - I have proof that I was right, but it's such an amazing proof that your feeble minds won't understand, if you tried to interpret it your mind would explode! So I'll keep it safe from you by not showing you this proof, but trust me, it's right - plus it's proves the Japanese were the first to wear spandex - so it's an incredible proof.

You sir have the moral ambivalence of a mutated shrimp!
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Mahoney
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Posted: 26th Aug 2008 01:35
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Quote: "However, I have to say - I have proof that I was right, but it's such an amazing proof that your feeble minds won't understand, if you tried to interpret it your mind would explode! So I'll keep it safe from you by not showing you this proof, but trust me, it's right - plus it's proves the Japanese were the first to wear spandex - so it's an incredible proof."




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Seppuku Arts
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Posted: 26th Aug 2008 01:45
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Damn, now I wish I used my 'toff' impression for this one.

You sir have the moral ambivalence of a mutated shrimp!
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Chris K
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Posted: 26th Aug 2008 02:00 Edited at: 26th Aug 2008 02:08
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Hi, everyone.

I will try and clarify this debate, but let me first make clear that this is not an area of Mathematics over which there is any debate.

This is very simple Analysis. Analysis is a very important branch of pure Mathematics, it is very vigorous and confusing and so is normally only taught properly at university.

What most people here don't seem to know is that there is a VERY strict and COMPLETELY SOUND definition for an infinite limit. It is nothing to do with anything human, it does not rely at all on what seems natural or whatever...

-- Real Analysis Lesson --

A sequence, is a load of ordered real numbers a_1, a_2, ..., a_3, ..., a_n, ...

This sequence can be infinite, ie. for any n you give me, I can give you a_n.

We say a sequence tends to a limit - a - if.... (drum roll...)

***********
For any tiny number you give me....
I can find a point in my sequence...
so that....
Every number in the sequence AFTER this point, is less than this tiny number away from a
***********

------------

Here is an example.
Take the sequence, 1, 0.1, 0.01, 0.001, ....
where the nth term is equal to 1/10^n

I claim that this sequence tends to zero. I am therefore claiming the above definition.

So someone says, oh yeah, does it eventually stay within 0.0000123 of zero?

I say, yes, clearly every term after the billionth is at least that close to zero.

----------

For ANY bound they claim, I can find a point after which EVERY TERM is within this bound. That means that the sequence 1/10^n tends to zero, and we say the LIMIT of the sequence (as n tends to infinity), IS zero.

----------

Similarly, the number 0.999999999<infinite zeros> is really just the limit of a sequence. Therefore, people say that 0.999... = 1

What they mean is the limit of the sequence:

9/10, 9/10 + 9/100, 9/10 + 9/100 + 9/1000, ... is one.

There is no debate about this, there is no 'I don't really believe that', there is no 'humans can't comprehend it', it is undeniably true.

-------------

Bless mamaji4 and his crazy ways.
I hope you read this thread when you have learnt some Analysis.

-------------

For the record, you can't do normal arithmetic with recurring decimals, so the intuitive 'proofs' which are normally produced to settle this debate are in fact, not valid.
They do however often convince people of the right answer, even if in the wrong way.

The important thing to take away is the starred definition.

-= Out here in the fields, I fight for my meals =-
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Seppuku Arts
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Posted: 26th Aug 2008 02:11
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I'll accept that because I don't have a large amount of mathematical knowledge, but compared to what I was taught and what I could see, the argument appear completely illogical - however I'd probably have to learn more before understanding how 1 could equal 0.999~ or seeing why - the 'proofs' I saw people post didn't seem to surpass what seemed illogical. Considering it's mathematics that don't concern me, I'll take the experts' word for it. 'Being infinitely close to 1' seemed like the right answer, but appearently, it's not.

You sir have the moral ambivalence of a mutated shrimp!
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Chris K
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Posted: 26th Aug 2008 02:20
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The right answer is the rather inelegant, "for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |f(x) − L| < ε."

(this is the starred definition from above).

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Aaron Miller
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Posted: 26th Aug 2008 03:23
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@Chris K
I like your answer.

I'm not a dictator to those that do stuff for me by will. Only those who don't.
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Agent Dink
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Posted: 26th Aug 2008 04:07
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I'm with Sep on this one. I'll accept that mathematically all problems lead to the same answer that 9.9~ = 10, but it's still not logical to me. I look at it as a flaw with our number system more than anything.

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Mahoney
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Posted: 26th Aug 2008 04:12
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Quote: "I look at it as a flaw with our number system more than anything."


Mathematics isn't something that man "created". It's how the universe is designed. It's something concrete and immutable. You can't claim that it is flawed.

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DarkBasic Pro Guy
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Posted: 26th Aug 2008 04:24
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0.999... is different from just 0.999 I'm sure, but I have no idea what the dots mean. from wikipedia it seems like the dots just mean it's rounded. 0.9999999 (infinite 9s) rounds to 1.0
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Mahoney
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Posted: 26th Aug 2008 04:25
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Quote: "from wikipedia it seems like the dots just mean it's rounded. 0.9999999 (infinite 9s) rounds to 1.0 "


The ellipses mean that it equals 0.<infinite 9's>. 0.999... == 1, it is not rounded to 1.

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RedneckRambo
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Posted: 26th Aug 2008 04:28 Edited at: 26th Aug 2008 04:28
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Quote: "Mathematics isn't something that man "created". It's how the universe is designed. It's something concrete and immutable. You can't claim that it is flawed."

Yes we can because we claim that the universe is designed by math. Whether you, I, or anyone can believe it, there's a chance we're incorrect. I'm not saying I believe that, I'm just saying there's a chance. Whether you will accept that statement or not, I don't care, because there is that chance.

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Seppuku Arts
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Posted: 26th Aug 2008 13:12 Edited at: 26th Aug 2008 13:15
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Quote: "Mathematics isn't something that man "created". It's how the universe is designed. It's something concrete and immutable. You can't claim that it is flawed."


Mathematics is a part of the human perception; we just apply it as a logical principle, and it is a way we deal with explaning and predicting various things.

Mathematics, like logic is a product of reason, not the universe.

'2' doesn't exist, but it is a logic I apply to the universe - for example, I have 2 speakers, I'm just applying a number from my head and we'd agree that I have 2 speakers because it is logical. But it is all from reason.

You sir have the moral ambivalence of a mutated shrimp!
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tha_rami
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Posted: 26th Aug 2008 15:27
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Thank you, Seppuku, you prevented me from giving a huge logic/reason preaching.


A mod has been erased by your signature because it was larger than 600x120
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mamaji4
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Posted: 26th Aug 2008 19:07 Edited at: 26th Aug 2008 19:33
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Hey Chirs,

Glad to see you entering the discussion. We need a rock solid
understanding of pure math to carry this furhter.

Quote: "Bless mamaji4 and his crazy ways.
I hope you read this thread when you have learnt some Analysis.
"

I'm going to take that as a compliment even if you didn't mean it to be one.

I may have grown a bit rough round the edges since its been some time since I did my major in Pure Math, but I still manage to hold up my own.

Just for the record, wrt the last discussion we had, where I had non-rigorously proved that "an infinitely large number is greater than itself" , I just clobbered the proof myself in one of the posts on this thread.


Quote: "The right answer is the rather inelegant, "for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |f(x) − L| < ε.""

Also, I think you should have clarified that the above statement can be represented as
Lim f(x) = Lim f(x) = L
x->c+ x->c-
since x can tend to c from the RHS or the LHS, because the norm |x - c| > 0 , either from the LHS or the RHS

Quote: "it is undeniably true.
"

And don't go giving everybody the wrong impression. Real Analysis isn't an exact science. Anyone is allowed to punch holes in it. Zeno did it. http://en.wikipedia.org/wiki/Zeno_of_Elea

There I go punching holes in your proofs again.
Your welcome to do the same with mine.

And where in Heaven's name did you find the symbols for delta and epsilon or are you embedding Unicode?
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Green Gandalf
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Posted: 26th Aug 2008 19:22
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Quote: "And where in Heaven's name did you find the symbols for delta and epsilon or are you embedding Unicode? "


First sane thing you've said in this entire thread.
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mamaji4
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Posted: 26th Aug 2008 19:36
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Quote: "First sane thing you've said in this entire thread."


And its probably going to be the last one too.
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Green Gandalf
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Posted: 26th Aug 2008 20:59
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Quote: "And its probably going to be the last one too."


Is that an example of Russell's Paradox?

Russell's_paradox
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Mahoney
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Posted: 26th Aug 2008 21:10
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Quote: "'2' doesn't exist, but it is a logic I apply to the universe - for example, I have 2 speakers, I'm just applying a number from my head and we'd agree that I have 2 speakers because it is logical. But it is all from reason."


That's sort of what I meant. I guess I worded it badly.

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mamaji4
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Posted: 26th Aug 2008 22:23 Edited at: 26th Aug 2008 22:48
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Quote: "Is that an example of Russell's Paradox?"


Hell no! I'm not as smart or as insane as Russell or Einstein, but I'm working at it, as is evident.
It would be an honour if someone labeled the 10 > 10 as the 'mamaji4 parodox'

Regarding the flat earth thingy.
It all depends on your frame of refernce as Albert once said.
1) If you stand at the center of the Earth it is SHAPELESS(Probably because your eyeballs evaporated with all that heat)
2) If you stand at the surface, the Earth is FLAT in the immediate vicinity of where you are. The ant's eye view.
3) If you stand 10^n Light years away, n -> infinity the Earth is INVISIBLE.

CONCLUSION:
We can therefore assume that the Earth is SPHERICAL when we perceive it from the Exosphere.

Kind of like Mathematics. Depending on which Theory you want to use
you can manipulate facts, prove and disprove things, all in the same breath. And until someone smarter than you comes along telling the world about the flaws in your theory, people assume that no flaws exist, and all and sundry swear by it.
'Concrete proofs' have holes in them at the microscopic level. Just because you can't see the holes in the concrete it appears to be solid. You need to develop the microscope to know that the holes exist.
If you develop your math skills sufficiently, who knows how many holes you can find in how many theories.
The point is, just because you can't see the holes in the concrete with the naked eye, doesn't mean they don't exist. And just because no one has cared to question math theories does not mean the flaws don't exist.

MORAL:
The next time you say the Earth is spherical you have to also mention the frame of reference, or your statement will be half baked.
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Omega gamer 89
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Posted: 26th Aug 2008 23:16
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Sorry, I had to.

If the good lord had intended us to go outside or have a social life, he wouldn't have invented the internet.
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Chris K
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Posted: 26th Aug 2008 23:52
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Quote: "I may have grown a bit rough round the edges since its been some time since I did my major in Pure Math"


Um, you mean major as in at College?? Because didn't you already admit before you were still at High School...?

---

I am not really sure what it is about Maths and Physics that makes people think they can just kind of wade in with a basic understanding and little or no actually rigorous, unglamorous hard work and think they are right.

Actually, it is the same with things like Philosophy and Literary Theory. These are not idle day dreams, this is people's actual, hard work.

There is a big difference between reading about stuff and actually doing it.

I mean, people don't just rock up and think they can perform open heart surgery. "Have you done it before?", "Well, no not really but I watch a lot of House, plus I read quite a lot about it on Wikipedia".

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mamaji4
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Posted: 26th Aug 2008 23:55 Edited at: 26th Aug 2008 23:58
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Quote: "Um, you mean major as in at College?? Because didn't you already admit before you were still at High School...? "


You got me there. You sure have a good memory. Now everyone's going to know I'm still in High School. Gee, you're always spoiling my fun Chris.

Quote: "Sorry, I had to."

Looks like a Black Hole to me, although I daresay nobody has seen one, what with the gravitational force being so great it bends light and high fangled stuff like that.
Anyway, that's what happens when you try to divide by zero.
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Chris K
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Posted: 27th Aug 2008 00:05
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I just remembered this website that you might find interesting:

http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html

It is a collection of essays written by one of my university lecturers. He is one of the best Mathematicians in the world (he won the Fields Medal), but he is also a good speaker and good at explaining things.

He also wrote the book 'A Very Short Introduction To Mathematics'. I don't know if anyone has read a book from that series they are little paper backs on a variety of academic subjects.

Some of the things on that page are complicated (I wouldn't recommend anyone plough through "Why study finite-dimensional vector spaces in the abstract when they are all isomorphic to Rn?" for example )

But these ones should be quite accessible...

http://www.dpmms.cam.ac.uk/~wtg10/commutative.html
http://www.dpmms.cam.ac.uk/~wtg10/roottwo.html
http://www.dpmms.cam.ac.uk/~wtg10/decimals.html <- !
http://www.dpmms.cam.ac.uk/~wtg10/reals.html
http://www.dpmms.cam.ac.uk/~wtg10/dodecahedron.html
http://www.dpmms.cam.ac.uk/~wtg10/metamathematics.html
http://www.dpmms.cam.ac.uk/~wtg10/biased.html
http://www.dpmms.cam.ac.uk/~wtg10/definition.html
http://www.dpmms.cam.ac.uk/~wtg10/philosophy.html
http://www.dpmms.cam.ac.uk/~wtg10/equations.html

Enjoy.

There is also a lecture floating round online somewhere that he gave that is quite good. I will try and find it.

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Chris K
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Posted: 27th Aug 2008 00:13 Edited at: 27th Aug 2008 00:14
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Ok here is a lecture: http://www.youtube.com/watch?v=BsIJN4YMZZo

The articles are actually more complicated than I remember, sorry about that. They should give people an idea of how complicated simple questions can become.

-----------

I think the lecture is from when they started the Clay Mathematics Prize (a series of $1,000,000 prizes for unsolved problems).

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mamaji4
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Posted: 27th Aug 2008 00:15 Edited at: 27th Aug 2008 00:21
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Well, I have studied tensors, divergence, curl, Greene's theorem, so it does take me down nostalgia lane a bit.

Hey Chris, what do you say we collaborate and solve the conjecture problem and stuff, and get paid for it. Are you game?

Quote: "He is one of the best Mathematicians in the world"

Hey, I just read that. I want bragging rights too. I studied under a perfectionist. The only person to have scored 100/100 in all eight papers for a grand total of 800/800, in the final year. The record still stands at the Univ.
His advice to me. Question everything. Or how will you know whether the theory stands the test of analysis.
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Chris K
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I don't think there's any way of asking this over the internet without sounding rude, but do you suffer from a mental condition such as Asperger's Syndrome?

I mean, we just established that you are still at high school, didn't we?

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mamaji4
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Posted: 27th Aug 2008 01:03 Edited at: 27th Aug 2008 01:19
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I do believe you are a wee bit peeved at me. Probably because of the number of times I pointed out your mathematical incapability, and discrepancies in earlier threads.

Hell, you could have just said no if you felt that you just weren't up to my standard, and I would have asked your world class mathematician lecturer instead.

Quote: "I don't think there's any way of asking this over the internet without sounding rude, but do you suffer from a mental condition such as Asperger's Syndrome?
"

And why on earth would you think that sounds rude. I think you are the epitome of courteousness.

And speaking about mental conditions. If anger is temporary insanity is it possible that since you are permanently peeved at me that you are permanently...
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RalphY
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Posted: 27th Aug 2008 01:17
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ok the maths with a bit of thinking I can understand, mamaji4 though...

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mamaji4
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Posted: 27th Aug 2008 01:21
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Quote: "ok the maths with a bit of thinking I can understand, mamaji4 though..."


Yep. I have that effect on people. But you'll get used to me after a bit.
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Chris K
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Posted: 27th Aug 2008 03:03 Edited at: 27th Aug 2008 03:05
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Sorry to press, but you didn't actually answer whether or not you suffer from an autism spectrum disorder...?

---

I am not annoyed with you by the way, I am not sure where you got that idea.

-= Out here in the fields, I fight for my meals =-
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mamaji4
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Posted: 29th Aug 2008 13:32
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Quote: "Sorry to press, but you didn't actually answer whether or not you suffer from an autism spectrum disorder...?
"


If I carry this conversation with you any further, I would be reducing myself to your level. I refuse to do that.
Instead I will answer your question as you would answer an aggravated infant by shoving a pacifier into its mouth, because that is the intellectual level you exhibit.

The answer to your question is, yes, if it makes you happy.
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mamaji4
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Posted: 29th Aug 2008 14:36
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Actually no amount of words can convince a person that the magnitude of the difference between the function and its limit actually is NEVER zero, according to what the theory of limits proclaims.
So we shall use a more formal proof
From the definition of a limit
for any E > 0 ... |f(x) - L| < E
i.e. z < E where z is a positive real number
i.e. z - E < 0 , E > 0
i.e. -(z - E) > 0
i.e. E - z > 0
This equation implies that for any E > 0
E - z > 0 --- EQUN. 1
But we know that for any E > 0
E = z <=> E - z = 0 where <=> represents 'if and only if'
i.e. E = z => E - z = 0
In symbolic logic terms we can say
A => B
Negating this statement
~(A => B)
~B => ~A
i.e. IF E - z is not equal to zero THEN E is not equal to zero --- RESULT 2

Using EQUN.1 with RESULT 2 we can see that E is NEVER equal to zero
i.e. |f(x) - L| is NEVER equal to zero
i.e. f(x) is NEVER equal to L
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Chris K
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Posted: 29th Aug 2008 14:57
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Um, I think most of what you said there was correct, but if you step back, it didn't really achieve that much.

The first thing the definition for a limit says, is "for any small positive number you give me" - "for any E > 0".

You then went on to show, in a rather round about way, that E =/= 0. But we knew this anyway, because E had to be bigger than 0 (E > 0).

It follows straight away; if it is bigger than zero, then it isn't equal to zero!

-----------

What you a kind of noticing here though, is a VERY important point (and quite an advanced one).

You realised (in a way) that the limit of function at a point, does not rely at all on the actual value of the function at that point.

(I am trying to say this clearly but it is actually quite hard...)

-----------

Quote: "Using EQUN.1 with RESULT 2 we can see that E is NEVER equal to zero
i.e. |f(x) - L| is NEVER equal to zero
i.e. f(x) is NEVER equal to L "


Um, I don't think the second points follow from the first. In fact, I know that they don't.

The definition says "for any E > 0 ... |f(x) - L| < E".
|f(x) - L| is smaller than E, so it can be zero, whatever E is.

------------

My one piece of advice to you would be just as enthusiastic as you are, but a little bit more cautious when you come to actually write your maths down.



-= Out here in the fields, I fight for my meals =-
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mamaji4
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Posted: 29th Aug 2008 15:51 Edited at: 30th Aug 2008 18:33
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Quote: "Um, I don't think the second points follow from the first. In fact, I know that they don't.
"


My advice to you is to study Logic first before making such a banal statement.

Quote: "|f(x) - L| is smaller than E, so it can be zero, whatever E is.
"


It can be zero, is not the same as saying it is zero. If you can show me any E such that the difference is zero, I can show you for that same E that the difference will not be zero, no matter how small you make that difference.
Go on try it.
Give me that E such that I cannot give you another number less than E that is non-zero.

You want to know the truth. Its the guys who wrote this theorem that are using "intuitive" reasoning, in that, finally the value of
E will be so small that it will HAVE to be equal to zero. We mere mortals know that is not true. Because there will always be infinitely many numbers between E and zero no matter how small you make E.
The ONLY way you cannot have infinitely many numbers between E and 0 is if E = 0.
But then that would nullify the theorem completely because E > 0 is the original assumption.

Quote: "this is people's actual, hard work.
"

And whatever makes you think I haven't worked hard to poke a hole in the theorem. It kept me awake all night. Isn't that hard enough

Quote: "You then went on to show, in a rather round about way, that E =/= 0. But we knew this anyway, because E had to be bigger than 0 (E > 0).

It follows straight away; if it is bigger than zero, then it isn't equal to zero!"

I can't find a single instance where I tried to show E =/= 0. Isn't that the assumption I started with, in the proof?
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Chris K
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Posted: 29th Aug 2008 16:15 Edited at: 29th Aug 2008 16:17
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Mamaji, please try and be a bit nicer. I promise I am not trying to annoy you. I don't think I have said anything nasty to you.

----

It seems at the moment that everything I say, you ignore or try to pick holes in. I with give you the benefit of the doubt and assume that you are just trying to be strict with your understanding, but at the moment you are being very antagonistic and unfriendly.

However, I believe you are a good person and that you really like Maths, and I am pretty patient so I will keep on explaining as long as you keep listening.

----

However, this is NOT a contest. I am not trying to score any points and I will not respond or acknowledge any ad hominum attacks.

==============================

OK, so with respect to the Maths. It seems as if you have changed what you are saying; as you originally said,

Quote: "i.e. |f(x) - L| is NEVER equal to zero""


and now you are saying,

Quote: "It can be zero"


-----

When you said that it was NEVER equal to zero, I said that it could be equal to zero.

I honestly can't tell how much of this you actually understand, and how much of it you are bluffing, but I would be happy to try and explain Limits from the beginning if there is anything you are not clear about.

It is one of my favourite topics, I could talk about it for years.

-= Out here in the fields, I fight for my meals =-
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mamaji4
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Posted: 29th Aug 2008 16:25 Edited at: 29th Aug 2008 16:29
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Quote: "It can be zero"


I can't find anywhere, where I said "It can be zero".
Are you inserted your own phrases into my quotes.

Sorry. Typo and semantic error. The sentence should read
"Are you inserting your own phrases and claiming I quoted them?"
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Chris K
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Posted: 29th Aug 2008 16:28 Edited at: 29th Aug 2008 16:29
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mamaji4
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Posted: 29th Aug 2008 16:32
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You said "It can be zero"

I said "It can be zero, is not the same as saying it is zero"

Are you trying to use my quotes out of context to establish I am wrong. Surely you can do better than that.
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Chris K
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Posted: 29th Aug 2008 16:34
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Um, OK then I must have misunderstood.

Are you suggesting that I said it is always zero?

-= Out here in the fields, I fight for my meals =-
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mamaji4
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Posted: 29th Aug 2008 16:41
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Quote: "Are you suggesting that I said it is always zero?
"


Words fail me. Who else said that "it tends to zero" and actually "is zero" (That's a mathematical contradiction if I ever heard one.
Refer to you first post.

I think the discussion has reached a level of tomfoolery so I'll end as I always do with you Chris.
You win. I lose.
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Chris K
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Posted: 29th Aug 2008 16:51
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OK, that's your choice I suppose.

I think that you have got confused over exactly what we are talking about.

Again, if you want me to explain limits from the very beginning then I will attempt to.

-= Out here in the fields, I fight for my meals =-
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mamaji4
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Posted: 29th Aug 2008 17:10
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Quote: "Again, if you want me to explain limits from the very beginning then I will attempt to.
"


I'll be sure to trouble you. All said and done, you are an excellent parrying partner when it comes to math.
In the meanwhile, I just saw on the MIT website about a 1 million dollar award to show that P = NP
If you ever feel like attempting it, let me know.
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