Convergence (math please, forgive me)

This is probably the whole idea that made math simply beautiful. i'm not qualified enough to give a discourse on this topic, but it's just so brimming with possibilities that i cannot stop myself...
'disclaimer' tipe:
fine even if you listen or not, i like to keep rambling on, so this is not for morons who just can't appreciate the significance of maths. and neither is this for those who just can't keep up with my extremely irritating style of writing..(oops, i just lost my readership!)..

so what's with this convergence anyway huh??well, the first glimpse that i got of this function, was back in the days when i was preparing for the great IIT-JEE, when the class was introduced to Taylor's and McLaren's series... actually, i think, it's way before that... Remember mathematical induction? remember all those expressions for the nth term and stuff, sums of series types? well all those formulae are i think some kind of convergence... i vaguely remember there was a polynomial expansion for pi too. dunno. but definitely, there is an expression that evaluates to the value of 'e', the base of the natural logarithm, and closer the value for higher the degree of the polynomial you evaluate. so you see, the point of concern here is that the polynomial 'converges' onto the value of e for some n.. another instance, one you might relate with, is that of functions.. say f(x)=sin(x)/x. we know:
-1/x <= sin(x)/x <= 1/x; (since -1 <= sin(x) <= 1) we also know that (1/x) -> 0 as x->inf; and -1/x -> 0 as x -> inf; (the symbol '->' being approaches to ).. so we can clearly see, that
sin(x)/x -> 0 as x -> inf.

until now, if you've thought either "mamma" or "tty" or "abbeeyy" you really shouldn't be reading this, because this is all high school math till now. the reason i put up this post, beside the negative publicity that i'm sure it will create, is that this is probably the first that time that convergence knocked on our head and we didn't recognise. here, the function is "bounded" on both sides by convergent functions. 1/x cannever be zero. but we can make it as arbitrarily close as we want to. this is the idea of convergence. that in the near future, the hope that stability can be achieved. ganga and yamuna converge on the banks of allahabad. 1/x and -1/x can be thought of two rivers that are coming ever closer as we go to larger x, in search of the elusive Prayag. since sin(x)/x keeps bouncing between these two rivers, it also is bound to meet at prayag, namely, zero. but since nobody can plot out all the values of x (remember, the real number set is uncountably infinite) no one can say that it cannot reach 0.

these are only instances of convergence. like i said before, to me the word means the hope that stability will be achieved sometime later. the reason i chose this topic, is that this semester, especially after mla last sem, really feels converging to a point. although irregular, like sin(x)/x in the beginning, maybe the theory i read will someday achieve a closure of some kind. the hope that all different branches of mathematics or physical sciences will converge, is one kind of convergence, and such a Unification Theorem will achieve closure over knowledge gained from observations and experiments alone. i may not be blessed to live long enough to see this happen, but atleast the subjects that i read, if they converge on a topic, or on a set of ideas, that will more than be enough for me. my holy grail will have been achieved. already, there are a few basic ideas that are common to all the courses i've taken this semester. the idea that everything is a set, introduced first in mla, is now quite evident in moc. moreover, sets can have such deeper meanings that there are whole branches of mathematics dedicated to the study of one particular types of sets. Topology is a spinoff from Group Theory, i think, and Group Theory is but the study of specialised sets. a mathematician would have killed me at this point, for the sheer callousness with which I brush aside these holy and revered subjects.

point to note: Inner Direct Product and External Direct Product of groups is very similar to the internal and external products of two vectors. and since groups and vector spaces are both Algebraic Structures (note the capitals), probably there is some unifying theory for these Structures.

We were taught (a debatable point, i agree...) right in the first semester that matrices and systems of linear equations weren't all that different. now i see that even groups and vectors aren't all that different, and everybody knows vectors can be represented by matrices. you see, everything is coming closer... or did it start off like that? were matrices first discovered or Algebraic Structures? is math creation of useful tools to work with, like integrals and derivatives, or is it understanding the true nature (think fibonacci and golden rectangle) of Nature? i think these doubts can be classified as consequences of three essential 'dichotomies' - a dichotomy is a split - discussed in a journal in the library. infact i suppose this urge to post a blog on math is the direct effect of a rare occurence, namely my reading a journal in the library. this article discusses three fundamental dichotomies that can be found in math, and also in languages in general. well, more on that later...