Thread: On the positional representation of real "fractions" as a closed set.

Consider the set of real "fractions", F: Let b (/b/>1), a_i (i = 1,n) be a finite set of reals and F = {x| x = Sum (j = -1, -2, ...,) of a_j X b^j }. Take as a hypothesis that every real may be represented uniquely by such a sum if we include "whole" numbers (terms with non-negative exponents of b) in the above sum.

I wish to show that F is a closed set. My efforts to date are to consider that y is an accumulation point (ap) of F . Thus, any interval that includes y, no matter how small, includes a point z of F, by definition of an ap. Thus, the distance between y and z may be smaller than any chosen positive epsilon. I try to take an esimate of that distance in order to show that y is a member of F and, therefore, F is closed. By hypothesis, y has a unique positional representation, but it must include the "whole" numbers as well as the "fractions". My estimate, does not seem to lend itself simply to dropping the terms with non-negative exponents of b (the radix). After all b may be a positive or negative real, though greater than 1 in absolute value.

If what I have written is clear enough, perhaps I could get a hint or two.

Is my question too unclear, difficult, simple, or uninteresting to merit a reply? If there were a way I could remove it, I should do so. Perhaps, I am too impatient. If not, then I prevail upon a moderator to remove it from the forum.

Thanks for viewing! I have a proof that F is closed based on a positional term that is less than epsilon in absolute value. For terms sufficiently far out the difference between the ac point and x will be less than epsilon. Problem solved!

Originally Posted by goedelite

Is my question too unclear, difficult, simple, or uninteresting to merit a reply? If there were a way I could remove it, I should do so. Perhaps, I am too impatient. If not, then I prevail upon a moderator to remove it from the forum.

Originally Posted by goedelite

Thanks for viewing! I have a proof that F is closed based on a positional term that is less than epsilon in absolute value. For terms sufficiently far out the difference between the ac point and x will be less than epsilon. Problem solved!

Thank you for those two replies.
But quite frankly your OP post is incomprehensible.
If you want to be taken seriously, learn to post in LaTex.
Make your post clear and understandable.
Your OP is anything but that.

I do not think it incomprehensible in the sense that it cannot be understood, however difficult to read the avoidance of LaTex has made it, admittedly. LaTex is wonderful for type-setting a lengthy mathematical work, I am sure. I find I can solve my own problems in far less time than it would take me to learn LaTex or to refresh my recollection of it from time to time. LaTex presents a moral hazard when used in a help forum: my becoming skilled at LaTex would encourage me to seek help more often instead of working harder.

Originally Posted by goedelite

I do not think it incomprehensible in the sense that it cannot be understood, however difficult to read the avoidance of LaTex has made it, admittedly. LaTex is wonderful for type-setting a lengthy mathematical work, I am sure. I find I can solve my own problems in far less time than it would take me to learn LaTex or to refresh my recollection of it from time to time. LaTex presents a moral hazard when used in a help forum: my becoming skilled at LaTex would encourage me to seek help more often instead of working harder.

Then don't post questions that are unreadable then post a bump which is strictly against the rules. I say to you, how dare you post something that is unreadable and then take offence that it is not answered. Oh yes, if it is not in LaTeX it is probably unreadable. The fact that you do not think so is totally beside the point. Do you want help or not?

You say it is unreadable! Perhaps you did not try hard enough. "How dare I?" Who are YOU, the Queen of England? You are really a bit much, Sir! I do not need help from your sort. Please ban me from your site. I shall be grateful!

Originally Posted by goedelite

You say it is unreadable! Perhaps you did not try hard enough. "How dare I?" Who are YOU, the Queen of England? You are really a bit much, Sir! I do not need help from your sort. Please ban me from your site. I shall be grateful!

What the...

Your OP is unreadable. And I agree with Plato: " how dare you post something that is unreadable and then take offence that it is not answered" addresses the situation perfectly.

Besides, why ask a question on the forum when "my becoming skilled at LaTex would encourage me to seek help more often instead of working harder."

Should you have posted this or not? Paradox or frustration with a problem you need to solve? Leave if you wish. But if you eventually do want help then make sure you post your question clearly.

And for the record Plato is a quite valued member here and does his best to answer questions. And he does his job well. Stop trying to irritate the natives.

-Dan