Sorry your browser is not supported!

You are using an outdated browser that does not support modern web technologies, in order to use this site please update to a new browser.

Browsers supported include Chrome, FireFox, Safari, Opera, Internet Explorer 10+ or Microsoft Edge.

Geek Culture / Calculus with hyperreal numbers

Author
Message
Xarshi
19
Years of Service
User Offline
Joined: 25th Dec 2005
Location: Ohio
Posted: 3rd Dec 2009 23:12
Link
Who has done this approach to calculus? It's very similar to the limit based approach, but slightly different as it introduces a new number system called "hyperreal" numbers. This is basically done to prove that infinitesimal calculus could be rigorous.

If you don't know what an infinitesimal number is, it is a number that is infinitely small. There are three main principles behind hyperreal numbers - The Extension Principle, the Transfer Principle, and the Standard Part Principle (as hyperreal numbers are part of nonstandard analysis). The Extension principle states that:
Quote: "A) The real line is a part of the hyperreal line.
B) There is a hyperreal number that is greater than zero but less than every positive real number.
C) For every real function f of one or more variables we are given a corresponding hyperreal function f* of the same number of variables called the natural extension of f."


After that, you have the transfer principle. That simply states that
Quote: "every real statement that holds for one or more particular real functions holds for the hyperreal natural extension of these functions."


Then we have the standard part principle which says:
Quote: "Every finite hyperreal number is infinitely close to exactly one real number. Let b be a finite hyperreal number. The standard part of b, denoted by st(b), is the real number which is infinitely close to b."


These three principles let us work with hyperreal numbers the same way we do with real numbers. So for instance, with f(x) = x^2, obviously the derivative of that is 2x. Using hyperreal numbers, you'd go about finding it like so:


So you can see that it works in a similar manner to a limit (as you let h approach 0 using the [f(x + h) - f(x)] / h formula).

Anyways, my point is that there is a very interesting free book over this written by Jerome H. Keisler. It's very interesting and I suggest downloading it if you are interested in mathematics at all!

Here's a link to the books page http://www.math.wisc.edu/~keisler/calc.html

I'm sure it can explain all this better than I did. It's based off of Abraham Robinson's work on hyperreal numbers.

If you like math like I do you may enjoy reading this. Even just the first chapter and where they go into detail over how hyperreal numbers were "discovered" using set theory. Comments would be cool too, I guess. n_n
Back to top
Profile PM Email
demons breath
21
Years of Service
User Offline
Joined: 4th Oct 2003
Location: Surrey, UK
Posted: 3rd Dec 2009 23:39 Edited at: 3rd Dec 2009 23:40
Link
I like the idea of hyperreal numbers being "discovered"

"My gosh look at that number."
"Oh I say... It looks almost... [pause for dramatic effect]... infinitely small"
"It can't by"
"By gum Arnold, I think we've made a most splendid discovery"

Obviously, any number-discoverers (as they're technically known) would naturally speak in a posh 19th century British accent. The facts are unarguable.

I'm downloading it now though. I really should be working on the maths I'm supposed to be doing - got one last piece of coursework for Monday, and a 4 hour exam on Tuesday, and I've barely attended lectures all year so I've basically got to learn everything from scratch. Therefore, I think my time would most valuably be spent on this.

EDIT: Won't open online, and says the file's damaged whenever I download it.

Back to top
Profile PM Website
lazerus
17
Years of Service
User Offline
Joined: 30th Apr 2008
Location:
Posted: 3rd Dec 2009 23:54
Link
My brain esploded from the title. Its too late to research am too tired to interpret lol...

Wuha? i looked at infinitesimal a while ago. Ill take another look tommorow.


http://lazerus-reborn.deviantart.com/
Back to top
Profile PM Email
Xarshi
19
Years of Service
User Offline
Joined: 25th Dec 2005
Location: Ohio
Posted: 4th Dec 2009 00:06
Link
Quote: "EDIT: Won't open online, and says the file's damaged whenever I download it."

Did you download the entire thing? If so, I recommend just downloading chapters. It is a very large book.

And yeah, you should probably focus on your real studies n_n What math are you taking?
Back to top
Profile PM Email
Neuro Fuzzy
17
Years of Service
User Offline
Joined: 11th Jun 2007
Location:
Posted: 4th Dec 2009 01:15
Link
Very cool! I love stuff like this!

Still... i really can't do much with this yet (as far as comparing goes), since i don't know calculus. I'm studying ahead pre-calc.

I'll bookmark that book, or save it to my hard drive, and maybe in a couple months i'll have some cool hyperreal-related math stuff to show

Still, i don't get all this "natural extention of f" stuff. So the f(x)=x^2, and f'(x)=2x, so f*(x)=?

Back to top
Profile PM Email
Xarshi
19
Years of Service
User Offline
Joined: 25th Dec 2005
Location: Ohio
Posted: 4th Dec 2009 04:22 Edited at: 4th Dec 2009 04:22
Link
Quote: "Still, i don't get all this "natural extention of f" stuff. So the f(x)=x^2, and f'(x)=2x, so f*(x)=?"

When you are taught calculus you will not need to know the natural extension stuff. You will be taught based upon limits.

The reasoning behind this is simple. When Leibniz was using infinitesimals, as well as Newton, they did not have a clear definition of what an infinitesimal number was. Then came Abraham Robinson in the 1900s (Isaac Newton and Gottfried Leibniz "invented" calculus around the same time in the 1600s (1666 and 1674 I believe, respectively)) who gave a rigorous treatment to the idea of infinitesimal values using hyperreal numbers. But before this, the idea of an infinitesimal value was argued, so instead, they tended to use a limit based approach. This ended up catching on and becoming what is the dominant form of calculus that is taught. The concept behind a limit is remotely difficult to grasp, and I'm not sure I could explain it myself in words, but it makes sense when you use it and if you read the definition.

Anyways, f(x) = x^2 and f'(x) = 2x is just simple differentiation. 2x is the derivative of x^2, aka the slope of the curve at x.

However, f*(x) is simply the natural extension of f(x), as stated by the extension principle. However, when dealing with equations, one can simply drop the asterisk.

Here's one way to think about it. If you understand object oriented programming and know what polymorphism is, you can understand this.

Suppose you have a class RealNumber. Then you have a class HyperrealNumber. HyperrealNumber inherits from RealNumber. Lets say you had a static method in RealNumber called "myFunction(RealNumber x)". Then lets say you have a HyperrealNumber instance defined. If you call "RealNumber::myFunction(myHyperrealNumber)", it would then be casted into a RealNumber. Aka, it would have the standard part taken from it. So if your hyperreal number was 4 + e where e was an infinitesimal number, the standard part would just be 4.

I feel as if that was a terrible explanation. But in simple terms, f*(x) = x^2 still. It's just the natural extension where x is hyperreal as opposed to real. Make sense?
Back to top
Profile PM Email
Neuro Fuzzy
17
Years of Service
User Offline
Joined: 11th Jun 2007
Location:
Posted: 4th Dec 2009 05:08
Link
Quote: "It's just the natural extension where x is hyperreal as opposed to real."

Ahhh, now i get it

Back to top
Profile PM Email
Diggsey
19
Years of Service
User Offline
Joined: 24th Apr 2006
Location: On this web page.
Posted: 4th Dec 2009 20:49
Link
@Xarshi
Surely the hyperreal number class would be inherited by the real class. (Since the hyperreal line contains the real line according to your definition)

It makes you wonder what the very top class would be.

Here is an inheritance diagram of some types of number I know:



New DBPro IDE + integrated debugger!
Back to top
Profile PM Email Website
NeX the Fairly Fast Ferret
20
Years of Service
User Offline
Joined: 10th Apr 2005
Location: The Fifth Plane of Oblivion
Posted: 4th Dec 2009 21:23
Link
Can we speak English please?

Athlon64 2.7gHz->OC 3.9gHz, 31C, MSi 9500GT->OC 1gHz core/2gHz memory, 48C, 4Gb DDR2 667, 500Gb Seagate + 80Gb Maxtor + 40Gb Maxtor = 620Gb, XP Home
Air cooled, total cost £160
Back to top
Profile PM Website
demons breath
21
Years of Service
User Offline
Joined: 4th Oct 2003
Location: Surrey, UK
Posted: 5th Dec 2009 03:33
Link
Quote: "
"

Is it just me or does that look like the woman symbol? Maybe there's something to this. Either all women are numbers, or all numbers are women... There's definitely something going on here they don't want us to know about

Back to top
Profile PM Website
Xarshi
19
Years of Service
User Offline
Joined: 25th Dec 2005
Location: Ohio
Posted: 5th Dec 2009 07:28 Edited at: 5th Dec 2009 07:32
Link
Quote: "Surely the hyperreal number class would be inherited by the real class. (Since the hyperreal line contains the real line according to your definition)
"

For the sake of describing it with polymorphism I did not have it work that way, hah. Technically yes, the hyperreal numbers are derived from the real numbers. It's the same concept with a square and a rectangle. A square is a rectangle, but a rectangle is not necessarily a square. So if you had a method that took a rectangle parameter, you could pass in a square. In my example you would have hyperreal numbers be the squares and real numbers be the rectangles. It was described that way only for intuition, and I should have stated that.

Anyways, you forgot irrational numbers in your diagram. And hyperintegers. Really, using model theory you could most probably define a "hyper" superset of each number type, as the order relation x < y is a subset of the order relation x <* y in the hyperreal numbers.



updated

I probably left some out.
Back to top
Profile PM Email
Sven B
20
Years of Service
User Offline
Joined: 5th Jan 2005
Location: Belgium
Posted: 5th Dec 2009 16:31
Link
Hyperintegers? How would you define an integer infinitesimally close to another?

Cheers!
Sven

Back to top
Profile PM Email
Xarshi
19
Years of Service
User Offline
Joined: 25th Dec 2005
Location: Ohio
Posted: 5th Dec 2009 16:44 Edited at: 5th Dec 2009 17:00
Link
Quote: "Hyperintegers? How would you define an integer infinitesimally close to another?"

Hyperreal numbers do not have to be infinitesimal. There are three types of hyperreal numbers:
infinitesimal numbers
finite numbers
and infinite numbers

The standard part of an infinitesimal number is 0. The standard part of a finite number is the closest real number to it, and the standard part of an infinite number is not defined.

And a bit more now that I went to read a little more on hyperintegers.

Their definition comes from the greatest integer function [x]. (which is the greatest integer n such that n <= x). It is what happens when you plug in a hyperreal number for x. So in definition:
A hyperinteger is a hyperreal number y such that y = [x] for some hyperreal x.

XenoPhysics - A Havok Wrapper for DarkBasic Pro. In progress.
Back to top
Profile PM Email
demons breath
21
Years of Service
User Offline
Joined: 4th Oct 2003
Location: Surrey, UK
Posted: 5th Dec 2009 17:57
Link
All irrational numbers are are real numbers which aren't rational. Do they really need to be included in the diagram? It would be like having inheritance between natural numbers and non-positive integers.. Technically true but not really worth mentioning.
Also why have you got integers leading to whole numbers?

Back to top
Profile PM Website
Xarshi
19
Years of Service
User Offline
Joined: 25th Dec 2005
Location: Ohio
Posted: 5th Dec 2009 18:33
Link
Quote: "All irrational numbers are are real numbers which aren't rational. Do they really need to be included in the diagram? It would be like having inheritance between natural numbers and non-positive integers.. Technically true but not really worth mentioning.
Also why have you got integers leading to whole numbers?"

I didn't bother to see how the map was leading out. I saw it as whole numbers leading to integers leading to hyperintegers and so on.

XenoPhysics - A Havok Wrapper for DarkBasic Pro. In progress.
Back to top
Profile PM Email
Back to top

Login to post a reply

Server time is: 2025-05-25 10:29:43
Your offset time is: 2025-05-25 10:29:43