modest Posted December 13, 2010 Report Share Posted December 13, 2010 Sure, except that as usual, in the limit of T approaching n, you get 1 raised to an infinite exponent instead of a different base and exponent with a finite limit (as in Don's version). Yes, I'm sorry—I had completely forgotten you were doing that. ~modest :doh: Quote Link to comment Share on other sites More sharing options...
Don Blazys Posted December 15, 2010 Author Report Share Posted December 15, 2010 By the way, what are the domains of the variables in all of those recently introduced terms? Quote Link to comment Share on other sites More sharing options...
Don Blazys Posted December 19, 2010 Author Report Share Posted December 19, 2010 Quoting myself:By the way, what are the domains of the variables in all of those recently introduced terms? This should not be a difficult question, especially for those who posted those terms. Why is it so difficult to admit that present day mathematics "defines" independent variables in a way that isextremely "loosey goosey" and "liberal", using domains that are impermanent, (and therefore inconsistent)absolutely unstable and entirely subject to change, much like the features of this fellow: http://www.google.com/imgres?imgurl=http://www.toy-tma.com/wp-content/uploads/2010/07/Potato_23.gif&imgrefurl=http://www.toy-tma.com/vintage-toys/mr-potato-head-retrospective/&h=407&w=338&sz=52&tbnid=Ff19MxjbYH0RaM:&tbnh=246&tbnw=205&prev=/images%3Fq%3Dmr%2Bpotato%2Bhead&zoom=1&q=mr+potato+head&hl=en&usg=__q5M7wqMHoZ60gyXRXXBJbuA2E2c=&sa=X&ei=LF8NTZJbgv7wBozx7fMN&ved=0CDEQ9QEwAA :QuestionM :QuestionM :QuestionM :QuestionM :QuestionM . Shouldn't we, at the very least, question, consider and perhaps investigate whether or not it might be possible to develop a new form of algebra that defines its variables perfectly and permanently, using permanent (and therefore consistent) domains? :QuestionM :QuestionM :QuestionM :QuestionM :QuestionM . Don. Quote Link to comment Share on other sites More sharing options...
Don Blazys Posted December 23, 2010 Author Report Share Posted December 23, 2010 A dozen years ago, [math]Ta^x[/math] divided by [math]T[/math] was always written as [math]\frac{Ta^x}{T}=a^x[/math]. A dozen years ago, nobody even suspected that [math]Ta^x[/math] divided by [math]T[/math] could be alternately expressed as [math]\frac{Ta^x}{T}=T\left(\frac{a}{T}\right)^{\frac{\frac{x*\ln(a)}{\ln(T)}-1}{\frac{\ln(a)}{\ln(T)}-1}} [/math]. Mathematics is more than just a science. It is also, an art. Thus having more choices, options and alternatives is a good thing! Having the freedom to manipulate the domains of variables (and thus define them) is a most useful tool, and a most precious gift. Most folks don't even know that they have this freedom! Anyway, after New Years, I will start a new thread with the purpose of compiling a list or table of well known terms written in "cohesive" form... that is... with all variables perfectly defined by their own unique domains. It will be an exiting "new game" with many interesting challenges, I promise!!! Merry Christmas to all! Don. Quote Link to comment Share on other sites More sharing options...
sanctus Posted December 29, 2010 Report Share Posted December 29, 2010 Don, just read their posts, the domains are there, written out, not to deduce ;-) Quote Link to comment Share on other sites More sharing options...
Nootropic Posted January 30, 2011 Report Share Posted January 30, 2011 It's almost nauseating how long this has gone on for. No one (and the entire mathematical community is included in this) is asserting that having "1" is an axiom of the real numbers. It's a cold, hard fact. Perhaps you might pick up a book on the foundations of analysis? You can construct the natural numbers from set theory, and then the integers and of course the rational numbers from the integers. Then using the slightly more complicated notion of Dedekind cuts, we obtain the real numbers. This has all been done rigorously and verified over the last 200 years (or possibly more). If you want to shake up mathematics, consider going back to school and learning the basics. Maybe you're what some mathematicians may take as axioms? For example, I work in commutative algebra and to me a "ring" is a commutative ring with identity. Some mathematicians, for example those who work in noncommutative ring theory take a ring to be a ring that is not necessarily commutative or containing an identity. The same goes for areas of analysis. For an undergraduate, an "integral" may mean a Riemann integral and for a graduate student of mathematics it may mean a Lebesgue integral. Quote Link to comment Share on other sites More sharing options...
Don Blazys Posted January 31, 2011 Author Report Share Posted January 31, 2011 Quoting Nootropic:It's almost nauseating how long this has gone on for.Well, I think it's an interesting (and extraordinarily important) topic, and it will go on until the matters herein are resolved to everyone's satisfaction. Quoting Nootropic:No one (and the entire mathematical community is included in this) is asserting that having "1" is an axiom of the real numbers. It's a cold, hard fact. Well, "having [math]1[/math]" (whatever you mean by that) is not the issue in this thread. The issue here is "multiplication by [math]1[/math]". You see, in present day mathematical literature, the number [math]1[/math], is called the "identity element of multiplication" which in turn gives rise to the "axiom" that says "Multiplying any number by [math]1[/math] leaves that number unchanged". (First axiom) Now, an "axiom" is supposed to be a "self evident truth", but unfortunately, present day mathematical literature never mentions that "other" axiom that says: "Not multiplying any number by [math]1[/math] leaves that number unchanged"! (Second axiom) Combining the above two axioms, we now have: "To multiply any number by [math]1[/math] is to not multiply any number by [math]1[/math]", which clearly means that the first axiom, (you know, the one in present day mathematical literature) can't possibly existbecause it has been logically negated by the second axiom, which is, in actuality, far more self evident than the first axiom! Consider this... If we multiply [math]1[/math] by a factor of [math]3[/math], then we actually do something because we increase [math]1[/math] by a factor of [math]3[/math].However, if we "multiply" [math]3[/math] by a "factor" of [math]1[/math] then we actually do nothing because we neither increase , nor decrease [math]3[/math].Thus, to me, (and hopefully, to any other mathematician who is guided by logic rather than the "status quo") it makes no sense whatsoever to "perform an operation" that doesn't do anything and is therefore "ineffectual". I invented the "cohesive term", in part, to remedy the rather convoluted logic inherent in the first axiomand to reflect the far more logical notion that multiplication by [math]1[/math] is, in fact, "non-commutative". Quoting Nootropic:Perhaps you might pick up a book on the foundations of analysis?If you want to shake up mathematics, consider going back to school and learning the basics. I study math only as a hobby, and learned most of what I know between fares while working as a taxi driver. I have plenty to read because when my grandfather (who was, at one time, a bridge engineer in Russia) passed away, he left me his collection of math books, (many of them from the late 1800's) along with his notebooks and several shoeboxes full of letters from that other great amateur mathematician, Srinivasa Ramanujan, who he met while in England. I find that material more informative and far more entertaining than most of the stuff on the internet, and certainly more profound than any recent book on analysis. Moreover, (and most importantly) I am not forcing you or anyone else to accept my views, and I'm certainly not trying to "shake up" mathematics. Quoting Nootropic:Maybe you're what some mathematicians may take as axioms?Huh? Well, that is certainly the wierdest sounding complement that I ever got... so thank you! Honestly Nootropic, it appears as if you were "not quite focused" while posting the above,but when you are more lucid, you might want to try answering the following simple yes or no question: Given the identity: [math] T a^x= \left(T a\right)^{\frac{\frac{x \ln(a)}{\ln(T)}+1}{\frac{\ln(a)}{\ln(T)}+1}} , [/math] can we let [math]T=1[/math] and thereby multiply [math]a^x[/math] by [math]1[/math] if we cover the right hand side with our fingers so that all we can see is [math] Ta^x [/math]? In other words, can we let [math]T=1[/math] and thereby have a "unit coefficient"or an "identity element" if we can't see the right hand side of the above identity? Don't forget, this is a very simple yes or no question that doesn't require any commentary whatsoever.My answer, in one word, is "no". So, what is your one word answer ? Don. Quote Link to comment Share on other sites More sharing options...
Nootropic Posted February 1, 2011 Report Share Posted February 1, 2011 The fact that multiplying by "1" leaves a number unchanged is NOT an axiom and is certainly provable. And I recommend you pick up a book on the foundations of analysis so that you can see the nonsense you're talking about. It's ridiculous when you try to assert something that lies on absolutely solid foundations is wrong. It is in no way mathematically interesting and would be tantamount to me starting a thread about how air doesn't really exist. Moreover, (and most importantly) I am not forcing you or anyone else to accept my views, and I'm certainly not trying to "shake up" mathematics. Mathematics is not about anyone's "view". You're either right or you're wrong. And you are wrong. Feel free to send your work to ANY mathematical journal and let me know when they accept it for publication and the entire world finally sees that the real numbers are in fact, not commutative. Huh? Well, that is certainly the wierdest sounding complement that I ever got... so thank you
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