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#1 2006-06-02 15:18:20

John E. Franklin
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Registered: 2005-08-29
Posts: 3,564

infinitesimal values questioned.

As usual, I haven't read up on this topic yet.
But anyway.  Are infinitesimal values supposed to be between zero and some real number larger than zero but really small.  So like is infinitesimal the reciprocal of what really big number??  Something bigger than real numbers, but less than infinity?  Maybe there are conflicting definitions depending on what axiomatic method you are using?


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#2 2006-06-02 20:02:15

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: infinitesimal values questioned.

I think infinitesimal can be expressed in some of this way:

"(0"

(which means close to null).
Here's some definition I figured out:
Let have a sequence

(for example
). If :
1. for all n: a_n>0;
2. for any given real
, there exists integer
, so that for all n>n_0: a_n < \epsilon.
Then as n goes to infty, a_n is becoming infty small.
(I'm not sure. There may be errrors here...)

Last edited by krassi_holmz (2006-06-02 20:03:24)


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#3 2006-06-02 20:09:49

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: infinitesimal values questioned.

That's what I thought at first too, krassi.  But upon some reading, I found that the english definition of infinitesimal and the mathimatical definition are quite different.

Here is a pretty good article:

http://en.wikipedia.org/wiki/Infinitesimal

The way I understand it, infinitesimals are part of hyperreal numbers.  They are numbers which are smaller than any real number.  Because of this, I don't believe there is any way to express them besides through the use of symbols.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2006-06-02 20:33:09

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: infinitesimal values questioned.

Interesting...
So an infinitesimal is a "number", which behaves as a real number in multiplication, and, as 0 in addition... smile


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#5 2006-06-02 20:46:18

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: infinitesimal values questioned.

Can't we create our own "objects" as infinitesimals?
Let p be 1 in multiplication and 2 in addition...
What happens???:
for a-real:
pa=a;
p+a=a+2;
p(p+a)=p^2+pa=a+2, so p^2 must be 2 smile (but only in addition smile)
(p+a)/p= (a+2)/p=a/p+2/p=a/p+2;
(p+a)/p=p/p+a/p, so p/p=2/p=2.
Others:
(p* means p is used in mult.
p+ means p is used in addit.)
2p-p=2(p*)-(p+)=2-2=0. it is different from:
2p-1p=2(p*)-1(p*)= (2-1)(p*)= (p*)=1.
1p=1=1(p*)=1.1=1.
1(p+0)=1((p+)+0)=1(2+0)=2.
The basic laws won't be true.

Last edited by krassi_holmz (2006-06-02 20:47:23)


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#6 2006-06-03 01:34:39

George,Y
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Registered: 2006-03-12
Posts: 1,306

Re: infinitesimal values questioned.

The way I understand it, infinitesimals are part of hyperreal numbers.  They are numbers which are smaller than any real number.  Because of this, I don't believe there is any way to express them besides through the use of symbols.
---------------------
It was Thomson's Idea, he created the Analysis on Infinitesmall in the midth of last century.(Sorry I can't tell the detail and I may have some errors)

However, his idea did not put a solution for dispute on infinity thing in mathematic society, and, not well accepted.

Here's perhaps a reason-
Since after any differentiation or integration, the result is derived from limit theory, every result can then be expressed as the original result plus a infinitesmall(number form), or minus a infinitesmall, or 2? or 3? so the result is subjective to a mathematician's favor.

Standard Defination of Infinitesmall is just as krassi said, a variable that can be as small as you want, given the endogenous variable it depends on can change in some way.

This defination is invented by French mathematician Cauchy- it's odd, but it never fails. The reason why it's so orthogonal is that it claims least assumptions but solves most limit problem. It has a very delicate logic inside, that it  doesn't claim any new defination, that it just said "IF" the endogenous variable can change in a certain way, which is left with doubt.

Whether the condition is finally satisfied is a big question, it involves whether dt in ds/dt can reach null, or a line can be cut into REAL infinite segments(points), or an area can be made of infinite line segments.

And that need pure mathematic assumption and defination.

I'm for the impossibility for the condition, and for approximation view, and also, for Cauchy's defination.


X'(y-Xβ)=0

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#7 2006-06-03 04:07:09

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: infinitesimal values questioned.

I have to aknowledge i can't really understand the infinitesimal (although I use it).
Here's my question-is it well-defined - does it REALLY exist?
How can a number be greater than infinity?


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#8 2006-06-03 04:31:31

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: infinitesimal values questioned.

krassi, you read my mind.  It would be interesting to read how they first came about.  Hopefully, it wasn't just some guy on too much opium who said, "Dude, you know, what about numbers greater than infinity?"


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#9 2006-06-03 05:14:53

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: infinitesimal values questioned.


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#10 2006-06-03 21:05:35

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: infinitesimal values questioned.

krassi_holmz wrote:

I have to aknowledge i can't really understand the infinitesimal (although I use it).
Here's my question-is it well-defined - does it REALLY exist?
How can a number be greater than infinity?

It's up to you, dude.

If you believe, it exists.

If you don't, it does not. tongue


X'(y-Xβ)=0

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#11 2006-06-03 21:34:16

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: infinitesimal values questioned.

Yeah...
dude.


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#12 2006-06-04 13:12:52

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: infinitesimal values questioned.

big_smile
My dictionary say dude means playman, but I find dude a common address.


X'(y-Xβ)=0

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#13 2006-06-04 21:00:31

gnitsuk
Member
Registered: 2006-02-09
Posts: 119

Re: infinitesimal values questioned.

For a very "natural" way of handling Infinitesimals (and Transfinites) see:

http://www.tondering.dk/claus/sur15.pdf

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#14 2006-06-05 02:56:12

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: infinitesimal values questioned.

What a great paper!  I'm only about 15 pages through it, but it is as interesting as it is easy to follow.  I just wish there were more resources like this.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#15 2006-06-05 10:58:41

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,552

Re: infinitesimal values questioned.

Well I am only at the start, and I must agree that his writing style is excellent. Instead of making himself look "clever" with complex statements, he has set about explaining things as clearly as possible (which is actually harder).

I know that when I write a page it takes a long time as I draft and re-draft. Reorganising paragraphs. Finding new ways to define things. Figuring out when to introduce certain concepts. Adding graphics etc.

And the whole purpose is that when it is all done, people can read it and go "ahhh, yes!" smile

Ricky wrote:

I just wish there were more resources like this.

Yes!

Maybe I could run a contest.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#16 2006-06-05 11:43:20

numen
Member
Registered: 2006-05-03
Posts: 115

Re: infinitesimal values questioned.

Just search for pdf documents typically, they usually contains more serious and well documented topics for some reason, I've experienced.

Don't have time to read it all right now, but I downloaded so I can read later. Looks good smile


Bang postponed. Not big enough. Reboot.

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#17 2006-06-05 12:14:10

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: infinitesimal values questioned.

In my experience, most pdf documents do a lot of hand waving and use concepts without defining them.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#18 2006-06-05 13:28:05

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: infinitesimal values questioned.

So as a summation of the article, surreal numbers are method of using sets to define real numbers.  But when you use infinite sets, you can wind up with a number that is non-real, such as infinity or an infinitesimal value.  So my ealier post:

They are numbers which are smaller than any real number.  Because of this, I don't believe there is any way to express them besides through the use of symbols.

Is dead wrong.  A simple way to describe it is:

Imagine that you have the set {1, 1/2, 1/4, 1/8,...} and imagine the number at the end of that sequence (even though it goes on to infinity).  That number is infinitesimally small.

The above is "incorrect", but it describes the basic idea behind it.

Like I said before, very interesting stuff.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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