Infinity gaps in curves.

[BigMoosie]

Does it make sense to write:

 

[math]

y = \prod_{n=-\infty^2}^{\infty^2}\frac{x+{\tfrac{n}{\infty}}}{x+{\tfrac{n}{\infty}}}

[/math]

 

Would this create a line with infinity gaps in it? Or does it not make sense to write [math]\infty^2[/math] ?

 

I also was curios about the curve created by this psuedo-exponential:

 

[math]y = (-2)^x[/math]

 

It seems to create a regular exponential above and below the axis, both with infinity gaps, and both with infinity points, this interested me, is there a name for this type of graph? Are these two equations I have posted related in any way?

 

Thanks.

- BigMoosie

[matt grime]

no, that product makes no sense: remember that an infinity appearing in an index of in a sum or product merely tells you to no stop at any finite point, and it cannot appear in the terms of the product as it is not a real number.

 

What is an "infinity gap"?

 

k^x is the same as exp{xlogk} log of minus 2 is log(2)+ipi, so you get a real number when xpi is an even integer, the "gaps" are just the complex answers that cannot be plotted on a real curve.

[BigMoosie]

What I meant was infinity number of gaps.

 

With my second equation, wouldnt y be defined in the realm of real numbers when x=1/3 and when x=2/3 and when x=1/5 and when x=8/5 etc... It appears that any x that can be expressed as a ratio in its lowest form with an odd number as the denominator will return y as a real number, there are infinity number of gaps.

[matt grime]

As a matter of english infinity is a noun, not an adjective - you mean "infinite number of gaps"

 

"any x can be expressed as a ratio..." no, there are irrational numbers too you know.

 

the second part would depend upon whjat bracnch of log you were using - taking powers like 1/n gives n answers - you need to pick one ie pick a branch. the branch i picked is a uniqe branch - you are changing it for 1/3 and 1/5 and so on.

[BigMoosie]

I dont see why we need to use log, and I know of only one log mind you.

 

You quoted me wrong, I said: "any x that can be expressed as a ratio"

 

Surely (-2)^(1/5) is the same as -(2^(1/5) therefore real? But since we can find infinity x's that create real y's and there are an infinite x's that dont then it is like a very finely dotted line?

 

I know that 1/n indices create n solutions but if it is odd and the base is negative then surely one of them will be real?

[matt grime]

you don't need to use log, and you are free to pick a different branch of each root for your fucntion - i was explaining why my answer was different from yours ie why my answer didn't give a real root to (-2)^{1/5}. I was simply picking the first root anticlockwise round the origin from the real axis (the principal branch of log) and using it if it was real and ignoring it otherwise. you were picking one that is real if possible and ignoring the rest.

 

 

however, how would you decide what something to the power sqrt(3) was? it isn't important, though, but might explain where the log is useful.

 

 

apologies for misquoting you.

[BigMoosie]

I don't know how to solve radical exponents and was only presuming that they would create real y's when x was a special radical but it doesnt really matter...

 

Do these kinds of curves with infinite spaces in them have a specific name so I can do a search for them?

[matt grime]

no - they aren't really special since it is a badly defined function of a real variable (ie not defined at all the points you are claiming are in the domain), and not an interesting one of a complex variable - at least that is my opinion.

[BigMoosie]

Would I be correct to presume that if the line were followed through the complex plane that the distance from the origin will continually increase like a spiral cutting through the real line every time a dot appears? From my understanding this is a spiral that is infinitedly dense which is quite hard to get your head around, or am I way off?

[BigMoosie]

Back to my first equation, what I was after was something like:

 

[math]y = \frac{(x-\infty) ... (x-2)(x-1)(x)(x+1)(x+2) ... (x+\infty)}{(x-\infty) ... (x-2)(x-1)(x)(x+1)(x+2) ... (x+\infty)}[/math]

 

This would create a line that is one unit from the x-axis but not defined at any intefer. But what I was trying to create was somethin that would, instead of incrementing by 1 in each braket but increment by an infitessimal value so that we would have a line that has had infinte sections removed until it no longer exist, could you write the equation for this?

[matt grime]

you can't put the infinities in at the end!

 

there are several such functions, indeed uncountably many. why not take the product over all rational numbers, suitably ordered, or just define it to be 1 at each rational and not defined at each irrational? you need to stop thinking of functions as things that have a nice formula in terms of x.

[Dave]

Why don't you try something like:

 

[math]f : [0,1] \to \R, \ f(x) = \begin{cases}

1 & x \in \Q\\

0 & x \notin \Q

\end{cases}[/math]

 

That's rather a nasty little function.

[matt grime]

to be honest this horrible abuse of divergent products and improperly defined functions is making me queasy, can we do something more mathematical please?

[BigMoosie]

But what I am after is a nice formula in terms of x

 

Also note that what I am after is one such that is not defined at any real value be it rational or not, I was so certain that my first post was that

[matt grime]

how about sqrt(-|x|)... but this is all very very bad mathematics. I really cannot emphasize that enough. so bad in fact that i believe the entire thread ought to be removed. sorry, but it is just a horrible abuse of the subject.

[BigMoosie]

I am surprised to hear that this could make you quesy! We are just chatting here and I am gaining from this, I hope it doesnt cause you too much distress...

 

Perhaps what I wanted was

 

[math]

y = \lim_{a\to\infty}

\prod_{n=-\infty}^{\infty}\frac{x+{\tfrac{n}{\sqrt{a}}}}{x +{\tfrac{n}{\sqrt{a}}}}

[/math]

[matt grime]

What makes you think that pointwise limit is ever defined? At last you're using limits, but the point is why do you want a "function" on R that isn;t defined anywhere? this isn't even a fucntion, you know since it has no domain or range. that is why i am queasy - you are waving divergent objects around and completely misusng the word function

[BigMoosie]

I want to use it and study it for philisophical as well as mathematical reasons, kinda hard to explain but... anyway, we no longer need to discuss it if that is how you feel.

 

I would like to know however if this statement is true:

 

0 * undefined = 0

[matt grime]

what? it is undefined. what is *? what is 0? (i mean that most seriosly, for the expression is nothing in any part of mathematics, is it?) what is an "undefined"? 0 is usuallly an additive identity, but in what ring? whatoperation is *? on what objects? as written it makes no sense.

[BigMoosie]

Sorry for the confusion, the * means multiplication, I being a programmer feel this is pretty universally known but forget that it isnt outside of programming circles. I tried to use LaTeX but putting the word "undefined" in it caused it to think I was hacking or something...

 

Oh an 0 I mean zero, thought that was pretty obvious though

[matt grime]

in what ring? there is no such element in the real numbers as "undefined" so it makes no sense to ask what happens if you multiply a non-existent object by zero when said non-existent object is not something that can be multiplied. i know what you think they mean, and what they should be read as, but it makes no sense to write that. it as meaningful to ask 'is 0*hedgehog=cricket bat' true?

[BigMoosie]

it as meaningful to ask 'is 0*hedgehog=cricket bat' true?

 

I would expect it to be as meaningful as

 

0 * hedgehog = hedgehog

 

That to me seems like a correct statement.

[matt grime]

then you aren't doing maths whatever maths is, and surely 0* anything is zero by your own beliefs?

 

 

this isn't maths. it is nonsense; i don't mean that in a derogatory way, merely an observation of fact. maths deals with deductions from premises, reasoning from definitions. you are doing no such thing.

[Dave]

Oh come on guys, there's been about 20 million threads on this already. I hate to say it, but the subject has really been worn out. I don't want to have to close this thread, but if it continues on the current path I will.

[BigMoosie]

Ooops made an error, meant:

 

0 * hedgehog = 0

 

But never mind, must change topic like lil dave sayed.