strange function

zarniwoop

i'm not much of a mathematician, but i was dabbling a little and i found something interesting. i'm not going to get into all the details yet but i'll give an outline:

i have a function f(s,t), where s and t are functions of z where z is an interger, who's values are always rational, because of the form of the functions s(z) and t(z).

now what sort of implications does it have that:


the limit as z -> infinity, f(s,t) = sqrt(2)

?

to me, it just says that there exist q,r such that q/r=sqrt(2) iff q and r are large enough.

Capt'n Midnight

Have a look at the maths forum http://www.boards.ie/vbulletin/forumdisplay.php?s=&forumid=380

but without knowing more of the function it could be as trivial as
s=z
t =z
f(s,t) = s /( t +cos(pi/n))

[edit] f(s,t) = s /( t *cos(pi/n)) n being a constant.

zarniwoop

Originally posted by Capt'n Midnight
Have a look at the maths forum http://www.boards.ie/vbulletin/forumdisplay.php?s=&forumid=380

hmm.. thats where i meant to post it. oh well. i figure if there are mods here, it will eventually get moved.

Originally posted by Capt'n Midnight
but without knowing more of the function it could be as trivial as
s=z
t =z
f(s,t) = s /( t +cos(pi/n))

the value of that limit is 1.

but regardless, the thing with my formula is that it gives a ratio of integers as being equal to the square root of 2, which isnt supposed to be able to be represented by a ratio of integers.

zarniwoop

your function still doesnt have a limit to sqrt(2), but whatever. :dunno:

edit: unless n = 4. then you have a point...

BUT cos(pi/4) isnt an integer, therefore 1/(cos(pi/4)) doesnt really count as a ratio of integers representing sqrt(2)

ecksor

Originally posted by zarniwoop
i have a function f(s,t), where s and t are functions of z where z is an interger, who's values are always rational, because of the form of the functions s(z) and t(z).

Please post the function?

now what sort of implications does it have that:


the limit as z -> infinity, f(s,t) = sqrt(2)

?

to me, it just says that there exist q,r such that q/r=sqrt(2) iff q and r are large enough.

Nope! Careful here. Analysis courses will show how to approximate irrationals to arbitrary degrees of accuracy by rational numbers, and you may have discovered this for youself (and well done if you have!). However, the implication of the above is that if I say "Pick a rational number that is within e of sqrt(2)" then you can pick a high enough value of z such that f will be within e of sqrt(2) (i.e, z > N => f(s,t) - sqrt(2) < e ). The function converges.

The limit itself is a different thing. You can't actually pick a real number that is infinity, it is essentially just a nonsense concept that is often used to tie up the loose ends when working with certain sets (in the case of the real numbers, it serves as an upper bound).

zarniwoop

Originally posted by ecksor
Please post the function?



Nope! Careful here. Analysis courses will show how to approximate irrationals to arbitrary degrees of accuracy by rational numbers, and you may have discovered this for youself (and well done if you have!). However, the implication of the above is that if I say "Pick a rational number that is within e of sqrt(2)" then you can pick a high enough value of z such that f will be within e of sqrt(2) (i.e, z > N => f(s,t) - sqrt(2) < e ). The function converges.

The limit itself is a different thing. You can't actually pick a real number that is infinity, it is essentially just a nonsense concept that is often used to tie up the loose ends when working with certain sets (in the case of the real numbers, it serves as an upper bound).

alright, here's the "function":

take the following set of numbers:

S: {s | s = q^2, q element of Z}

and

T: {t | t = (p^2 + p)/2, q element of Z}

now take the intersection of these two sets and call it K. for each element k of K, solve for its respective p and q.

after all that take the ratio p/q of each k. as k gets bigger, p/q "converges" pretty rapidly to sqrt(2).

i havent done the delta-epsilon proof of this limit or proved it in any way shape or form. i figured i would see if it actually meant anything before challenging myself that much.


so there you have it. but yeah, maybe it doesnt really have any sort of greater implication. but it set off alarm bells in my head when i found that there was some sort of weird ratio of integers out there, even if we dont know what they are, that equal to the square root of two.

Capt'n Midnight

I believe the greeks got fairly upset over finding out that root two was irrational and tried to hush up the whole thing. (Men in Black chariots etc.)

So you won't find any ratio of two integers that will produce root two. And unfortunately root two is one of those numbers that keeps popping up from so many places so the signifigance of could come from many different areas, angles , ratio of products etc.

Never could get my head around functions where you select only some of the elements ..

now take the intersection of these two sets and call it K*. for each element k of K, solve for its respective p and q.

zarniwoop

Originally posted by Capt'n Midnight
I believe the greeks got fairly upset over finding out that root two was irrational and tried to hush up the whole thing. (Men in Black chariots etc.)

So you won't find any ratio of two integers that will produce root two. And unfortunately root two is one of those numbers that keeps popping up from so many places so the signifigance of could come from many different areas, angles , ratio of products etc.

Never could get my head around functions where you select only some of the elements ..

yeah.. i'm familiar with the proof that there doesnt exist a ratio of integers that equals the square root of 2.

i just thought it was funny that this function that i came across which shows that a particular ratio when one of the variables is pushed towards infinity (but not AT infinity, as the definition of the limit implies) is equal to root of 2.

its just something that made me wonder...

ecksor

I've briefly tried to find a proof that there's an infinite number of elements in K* (why the * btw? are you leaving out 0?) but can't find one. Have you got one or does anyone else see one?

A brief poke about with trying to find squares that are adjacent to squares multiplied by 2 or 8 finds
p = 1, q = 1
p = 6, q = 8
p = 35, q = 49
p = 204, q = 288 ...

zarniwoop

Originally posted by ecksor
I've briefly tried to find a proof that there's an infinite number of elements in K* (why the * btw? are you leaving out 0?) but can't find one. Have you got one or does anyone else see one?

A brief poke about with trying to find squares that are adjacent to squares multiplied by 2 or 8 finds
p = 1, q = 1
p = 6, q = 8
p = 35, q = 49
p = 204, q = 288 ...

yeah.. sorry, the * was put in there as an asterisk because i wanted to talk more about K later on in the post but forgot. sorry about the confusion.

okay, the elements of K can be found by the following fourmula (found by one of my friend through some doodling):

[((k1)-1)^2]/(k0)

let me explain this with an example, since its not easy to look at with this post:

ex: with some trial and error you can figure out that the first 2 elements of K are 1 and 36 (for k=1: p=1, q=1 and for k=36: p=8, q=6). so, to find the next element of K, we use the formula:

[(36-1)^2]/1 = 1225

for 1225: p= 49, q= 36

now one thing that needs to be proven is that this formula gives the elements of K in order, but whatever.. it suffices to show that there are an infinite number of elements k in K.

(the proof that the solution to this equation is in fact an element of K has been done already by me and my friend about a year ago. i seem to have misplaced it, but it wasnt that hard to do. if i can find the paper somewhere i will post it. if not, i'll try to remember it and sketch it out at some point this week while i'm picking my nose at work.)

zarniwoop

^