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Posted by: mic check
« on: March 29, 2020, 11:51:44 am »

That link leads to Wall Street Journal teaser paragraph.

Ramsey died at the age of 26 probably from leptospirosis (bacteria from the feces of animals) contracted by swimming in the river Cam.  His death made his friends and family (including his brother Michael, who later became Archbishop of Canterbury) question the meaning of life.

 Something Ramsey wrote that might give you pause.  I can't blame him for wanting to "live well."  It is the best revenge, after all.  And yet ...

Quote from: Ramsey
My picture of the world is drawn in perspective, and not like a model to scale. The foreground is occupied by human beings and the stars are all as small as threepenny bits. … I apply my perspective not merely to space but also to time. In time the world will cool and everything will die; but that is a long time off still, and its present value at compound discount is almost nothing. Nor is the present less valuable because the future will be blank. Humanity, which fills the foreground of my picture, I find interesting and on the whole admirable. I find, just now at least, the world a pleasant and exciting place. You may find it depressing; I am sorry for you, and you despise me. But [the world] is not in itself good or bad; it is just that it thrills me but depresses you. On the other hand, I pity you with reason, because it is pleasanter to be thrilled than to be depressed, and not merely pleasanter but better for all one’s activities.

OK, so applying this logic, my being thrilled to witness the pointless misery of existence unravel is better than being depressed by it.   I don't believe tin this kind of happiness.  This happiness (being thrilled by unfathomable uncertainty) does not really exist.  It is a lie other people tell you in order to make you feel worse ... even if you are one of the few who actually embraces your depression as a sign of intellectual honesty and integrity.
Posted by: mic check
« on: February 09, 2020, 04:12:46 pm »

I wrote out Pascals Triangle - two different versions, one above the other; but it is on "art paper" so was difficult to copy.  I found it helpful.

I uploaded on Wayback Machine if you want to take a look.  I had to make two different copies to try to fit everything.  You'll be able to get something out of it, especially if you recreate by hand your own version.

It seems others depend on me for things, besides my mother, and I feel very taxed at the moment, where I become drained.  I would rather be getting into these deeper interests, but Life is pulling at all sides, it seems.  A friend I helped a week ago with a great physically challenging task now requires my attention with a letter to a judge.

Life is one big Prison Farm Zoo, as our literary compadre Raul is fond of saying; and I am what might be called a jailhouse lawyer … It is my place in the scheme of things … Wherever I may find myself, there I will be, with all that this entails.   

The part of me that was invalid in bed mocks this part which writes. 

Also, when run down, taking off a  flat tire on a 4 cyclinder vehicle is duck soup compared to typing/editing another person's statements to a judge.   People may not totally respect these abilities and not realize that you might be using your brain at the moment ... for other things, maybe?  [my appetite is re-awakening ... it's back ...]

But, we do what we do, and help those who have helped us in some way in the past.  Of course, the Mother gets peeved when someone else is using her personal 24/7/365 slave, so, that's a little disturbing to witness.   And yet, for now, mom and I are some kind of dynamic duo until one of us is hurled into the void ... at which time the great unraveling may begin for one of us, at least.

unrelated:

They declared me unfit to live
Said into that great void my soul'd be hurled
They wanted to know why I did what I did
Well sir I guess there's just a meanness in this world

https://www.youtube.com/watch?v=iir_xAbt-ak
Posted by: mic check
« on: February 03, 2020, 02:29:43 pm »

You are very welcome, Holden.  As you know, I had also been intrigued with the writings of, first Deleuze and Guattari (AntiOedipus [Schizoanalysis]) and even Badiou, but I had found that returning to fundamental theorems and working through the process of applying the theorems to be more satisfying than familiarizing myself with the jargon used by the critical theorists.  There is something about the writings of Husserl, Heidegger, and Srtre as well that have that difficult to follow jargon.  It is tempting to call it derogatory names, and yet I acknowledge that "postmoderrnism" can be interesting.   

You know I try to be as honest and clear as possible.  There is a certain kind of mathematics that keeps me grounded, while some kinds of philosophy seems to be a web of words.   I would prefer to at least be clear and coherent (and appear rather simple-minded) than to appear quite sophisticated but make no sense whatsoever to the man of the street.
 
I am no authority on critical theory, but I do know a thing or two about "school mathematics" --- and with that I have had some experience, although I have no social status as any kind of "credentials."

I just wanted to share some notes so you coould see how cumbersome it would be to write nCr or (n r) ... Writing on paper with pen or pencil is so much clearer (to me).

We both have different times when we check in here, so rather than upload all the notebooks, I can see which ones might be of interest to you.    I have gotten great satisfaction in studying such things.
Posted by: Holden
« on: February 03, 2020, 01:20:00 pm »

Thank you for the material regarding the Binomial Theorem. You are a noble soul,like the lotus which is pure despite of being surrounded by mud on all the sides.I have got some hang of the Binomial Theorem.
Posted by: mic check
« on: February 02, 2020, 11:04:38 am »

PLAN:  The image files will do for now, but I would prefer organizing these into PDF files elsewhere, such as archive DOT org, The Wayback Machine perhaps …

prelude (p1-7: number system) plus Binomial Theorem (p8-34)  (in correct order)

The PDF files print easiest, and, at 33 pages, that's about 17 sheets if you print on both sides ...
Posted by: mic check
« on: February 01, 2020, 11:07:17 pm »

Posted by: mic check
« on: February 01, 2020, 09:45:18 pm »

Binomial Theorem Notes #001

I also have lived for understanding, and so am your true brother.


BT Notes #002,
#003,
 #004,
#005
Posted by: mic check
« on: February 01, 2020, 09:43:49 pm »

UPDATE:  all inlcuded in one PDF file if that is more convenient:
(see this post)

______________________________________________________________

0.55 ... = 5/9

How?   The second 5 is repeating, so this is an infinite geometric series. 

0.55 ... = 0.5 + 0.05 + 0.005 + ...

first term is 1/2, and the ratio 0.05/0.5 = 5/50 = 1/10, so the sum of the terms of this infinite geometric progression (sequence), called an infinite geometric series [the Sum, that is the series], would be (1/2)/(1-1/10) = (1/2)/(9/10) = (1/2)*(10/9) = 5/9

The notes included allow the spontaneous use of the vinculum over the repeating digits, but here on this board I would be forced to use bold or something to indicate such.    I have to run.   I hope the notes give you some insight into the numbers system's relation to infinite series ....

just a prelude to the Binomial Theorem notes.   Maybe printing hard copy with hand-written notes, printed from device to actual hard copy print might trigger a sense of self-respect and dignity in yourself to validate your quest for understanding.


Prelude to Binomial Theorem ... 1 ...

prelude page 2

____________________________________________________________
TESTING 1, 2 ...


(will direct to another thread with door to jump into next post in this thread)

Posted by: mic check
« on: February 01, 2020, 05:36:52 pm »

Given 0.abcde.... ===> (implies) 0 + a/10 + b/100 + c/1000 + ...

The Limit as the number of decimal fraction elements/components of the Sum increases toward infinity, with each element/component becoming smaller by 1/10 can be represented as a decimal fraction a/b where a,b are Integers and b != 0.

example:  0.222 where 2 would have a bar over it ....

[to be continued after I cook The Mother's dinner].   :(

0.222  = 0 + 0.2 + 0.02 + 0.002 + 0.0002 + ...

This is a geometric progression with a = 0.2 and r = 0.1

Hence, the limit as n approaches infinity is SUM = 0.2/(1-0.1) = 0.2/0.9 = 2/9

I will find some pages about the observations on the binomial expansion (x + y)^n then will upload.

We each have different "down times" -----
Posted by: mic check
« on: February 01, 2020, 09:56:58 am »

Quote from: Holden
Well,I asked about Binomial Theorem as I could not understand it.I was reading about it in a chapter concerning the number system.

That second part of the inquiry makes all the diffeence in the world.  My reaction must have appeared too intense, but, well, you see, I had read that only about 10% of "professional programmers" today could, out of the blue, with just editor and debugger, write a binary search program in C++ from sratch, with the proper #includes on the fly.  The dependence upon libraries, while good (and compicated) for PRODUCTION, may not be that good for Lifelong Learning and deepening of understanding.

Your approach is your own, and for whatever energies were diverted from rigid technical education, you have invested in literature and philosophy and just trying to BE YOU.

I only wished to display how one might see the code as poetry.   That was that, and now THIS is this.

Sorry if I am so particular, but I am intent on leaving the other thread for showing how to explore such code with gdb debugger.

____________________________________
When you mentioned the binomial theorem,  my mind went back to code for Combinations which I wished to include in a customized major tweaking and alteration of a calculator project from the Stroustrup text.  The nCk [read "n choose k"] can actually represent each coefficient in the polynomial expansion of (a + b)^n

The Binomial Theorem is just a series of observations about the exact nature of this expansion.

Please understand that, when it comes to mathematics I may sound like a robot, but I assure you this is where the poetry hides, in the rigor, seriousness, and clarity of thought.

I was also tempted by the free-flowing AntiOedipus Schizoanalysis, and I can certainly see where your Lovercraftian tastes would be drawn to the oozing manifold of mulitplicities.

I, on the other hand, may have sold my soul to the Devil for the pure delight in algebraic structures, and the manipulations of such structures.

Forgive my rigor and my tendency to have more respect for high school level topics than the obfusificated tomes worshipped by critical theorists.   On the other hand, I also sympathize strongly with those repelled by the rigor of mathematical proofs.

I am in between, not quite with Artuad, no - and again no - certainly not trying to stand in any shadows of Kant, Knuth, nor Stroustrup.   I think my perspective might be valuable to YOU in particular, Holden.

My recent encounter with the nature of these expansions of (a + b)^n involved defining the factorial 0! = 1, 1! = 1, 2! = 1 * 2, etc...

The numbers get very big so the challenge for me was to use special libraries [GMP] for doing the Combinations (n choose k), which involve factorials ... the Binomial Theorem - that series of observations - are tied up in this, so it is difficult to pin it down as an object in the phenomenal world.  I am not being purposely obtuse in trying to extract a more detailed inquiry from you, Holden.

Anything having to do with our number systems can be intensely fascinating.   I was inspired to write some code about representing decimal fractions, with repeating decimals, non-terminating or terminating decimals.

The numbers to the right of the decimal point are a series, a sum of a sequence of decimal fractions in places 1/10, 1/100, 1/1000, ...

If we can narrow down where you are confused, whether about the nature of number systems in general, or where the expansions of (a + b)^n come into play, I would be able to assist in a more helpful manner.   The last thing I want to do is discourage you asking me a question or dampen your taste for such things.

I will eat Steel Cut Oats and keep an eye on this thread over the coming weeks.
In the meantime, I might also use that little site for some temporary links for our exploration of exactly where your interest and confusion is.

Keep in mind that my back aches from strain, and that I am no academic.  I trust that you respect my life as a lifelong Learner.  I have documumented my own explorations, and I will offer you this kind of support now or for however long we are able.  This way we might zero in on some topics for you to return to later when, say, one of us is preoccupied with the an engagement with their Lord or Destroyer (Grim Reaper).

Perspective:  Mass Pandemic Hysteria?  All we can do is groan.  Without health, these words do not exist.   The world is our idea.  Without health, the cosmos vanishes.
Posted by: Holden
« on: January 31, 2020, 01:38:24 pm »

Well,I asked about Binomial Theorem as I could not understand it.I was reading about it in a chapter concerning the number system.
My approach to mathematics,& some of the other things as well, is borrowed from Delezues idea of the rhizome.I jump from topic to topic.From one page of the book from another.Admittedly,it is not systematic. Its a fact that I have had no formal training in mathematics ,let alone programming.It is not something that I would like to cause the feeling of self-pity in me.It is what it is.Given the circumstances, the Rhizome approach is the best I can come up with.Perhaps.

There are problems which make me want to give up on mathematics altogether but I come back to it again and again.In Hindi they have a saying which implies that even the soft rope,which is used to draw water from a well, eventually manages to leave its impression on hard rock.

There are many things in mathematics which I don't understand.Binomial theorem is just one of them.But I don't want my incomprehension of any one topic in mathematics to stop me from learning any of it.Because there are topics like Descrates formula to find the number of roots of an equation which I do understand.

So,even if this rhizome is shattered at one place, it will not stop it from spreading it out in other places.Any traditional/formal approach has only caused me heart ache and pain.I understand that it would take me years to develop the kind of comprehension that I seek and I also know that there is no guarantee of anything,no guarantees of continued health,sanity,shelter,free time to study or even of ones being able to continue to draw breath.

But I solider on,there will at the very least be the satisfaction of having done ones best.Even if there is not good enough by some kind of objective standard.In India ,because of Hinduism, there is a very strong belief that if one dies without a certain strong wish having been fulfilled,he is reincarnated ,specially to fulfil that desire in the next birth.I have no desires,at least I like to think that, but if there were indeed such a desire strong enough to drag me back and mire me in existence ones again, it would be that of comprehending mathematics better.

But,I think ,in the last analysis, not even that.





Posted by: mic check
« on: January 31, 2020, 06:49:52 am »

Continuing from Holden's Inquiry about Binomial Theorem, and here in The Unforgivable Sin thread ...

I found at least 5 consecutive notebooks covering the subject matter in depth.   I might get to pecking on scanning some notes and exercises (with my solutions based on, but not solely on, official text answers/solutions).

an aside:  Friday the 31st
Is this a numerologically sound number, or does it not reek of danger?

The mechanical tedium of applying the Binomial Theorem with pencil and paper might make it a bore (and prone to accident) for humans, but the process can be automated - the binomial expansions ...

The funny part is that these numbers get BIG fast, and they are great at exposing the limitations of the hardware and (invisible brains = software) of the Interface.   The rational calculator (a major tweak to a build-calculator-from-scratch project in the Stroustrup C++ text (chapter 7)) can handle very large factorials, up to 107! rather than 20!

This can have quite an impact on the range of actual cases exceeding the 20! limit of most current calculator implementations.

I think the notes of the exercises from exotic 1960's texts would be appreciated by the Inner Lifelong Learner trapped in the bones of one trapped in economic bondage to serving an economic caste, and consider yourself lucky.   May your interests in such things as this sustain your inner life so that you will always cherish solitude rather then sun or dread it.

The key ingredients for exploration of these ideas are time to think and reflect and contemplate.

It is a very easy world to get distracted by, sucked into even.   "Spirits" get eaten.  Brains get hijacked.
Posted by: Holden
« on: August 06, 2017, 10:35:17 pm »

Thanks a lot for solving the book page numbers question.I could not comprehend how it was solved in the text book.You depicted it in a very clear fashion and I could get it immediately.  :)
I never thought that I should try adding up the numbers so that the sum could be as close to 1000 as possible.
I am working on the Binomial theorem in a number of ways and would share the details with you.Thanks for your input.It would help a great deal.
Posted by: mic check
« on: August 06, 2017, 01:12:39 pm »

Quote from: Holden
Some questions are very badly written. But I am trying to sort out these things. Keep well.

Yes, that can be very frustrating, and it takes a great deal of confidence to determine just when there is such a mistake.

Now, "All the page numbers from a book are added,beginning at page 1 .However, one page number was mistakenly added twice.The sum obtained was 1000."

I would not know where to begin except brute force.

The sum of the numbers 1 to 44 is 990.

(1 + 44)/2 * 44 = 990

Page 10 may have been counted twice, which would result in 1000.

Posted by: mic check
« on: August 06, 2017, 01:01:36 pm »

We are in synch, Holden.

In the video above, the guy stops short just before he had the opportunity to introduce a formula based on Pascal's Triangle.

In the following video, you will have to forgive the instructor for his sloppy hand writing and sometimes quirky way of speaking (as in, using the acronym "FOIL" as a verb and other idiosyncratic annoyances).  At first, the voice may conjure up images of the overbearing Hasidic Jew in the black and white film, PI, but you can tell he really loves math.

Unlike that character in PI, this guy eventually grows on you, becoming even amiable.  I especially like his genuine appreciation of Sage.  He is refreshingly enthusiastic about it.  This reinvigorates my interest in computer algebra systems.

What attracted me to his videos in the first place is I was curious to see how he would present Sage to high school students.

Strangely, I was viewing this video when you sent the above link.

Can I use the word "uncanny" here, or is the term "coincidence" more appropriate?

Skip through to about 7 minutes.

You are interested in patterns, and there is a cool pattern which includes the row and column of Pascal's triangle without actually having to write the entire triangle of numbers.  Probably in around 11 or 12 minutes he gets to the good shiit, Combinatorics:  (nCr).

Each row is identified by n, where the first row of Pascal's Triangle is n = 0, the second is n =1, etc, and each column "r" is r = 0 for first column, r =1 for second column, etc .

For example, 3C2 would correspond to the fourth row, third column.

You can find each coefficient in this manner.

All in all, there is no getting around the tedious drudgery involved ... which is why one can't help but appreciate Sage (free, as in free beer).  Observe how elegantly it constructs the following:

Copy this into a sagecell:

for n in range(10):
    show(expand((x+1)^n))

Note that, if the SageCell complains about a syntax error, it is very picky about spaces.  Backspace from show until it is just before the colon, :, and then hit enter so show(expand((x+1)^n)) is indented precisely with the s under the n.  That is, four spaces indented.

shadowfax

Maybe skip into 26 minutes at first.  You have to patient as the video is in real time, which means the instructor is assisting students with "technical difficulties" throughout the video.  The example is (3*x + 1)^4, and he shows the coefficients in terms of nCr, where nCr --->  (n   r)  [but vertical instead of horizontal, obviously]

nCr means (n!)/[r! * (n - r)!]

So, 5C2 means (5!)/[2! * 3!] = (5*4)/2 = 10

Notice the 3! = 3*2*1 in the denominator cancels out the 3*2*1 in 5! leaving 5*4 in the numerator.


----------------------------------------------------------------------------------
PS:  for (3*x + 1)^4
In Sage:

for n in range(4):
    show(expand((3*x+1)^n))

def combo(n,r):
    return factorial(n)/factorial(r)/factorial(n-r)

I like the way he divides twice instead of typing the denominator with parentheses and using multiplication:  factorial(n)/(factorial(r)*factorial(n-r))

As he points out, multiplication by a reciprocal is division, so it is cooler, I think to write it as double division ...

Cheap thrills, I know.   8)
---------------------------------------------------------------------------------------

On paper, how would you show the work?  What's the pattern?
Let nCr be represented by the defined function, combo(n, r)
To find binomial expansion of (3*x + 1)^4:

    combo(4,0) * (3*x)^4 * (1)^0
 + combo(4,1) * (3*x)^3 * (1)^1
 + combo(4,2) * (3*x)^2 * (1)^2
 + combo(4,3) * (3*x)^1 * (1)^3
 + combo(4,4) * (3*x)^0 * (1)^4

 = [4!/(0!*4!)] * 81*x^4 * 1
 + [4!/(1!*3!)] * 27*x^3 * 1
 + [4!/(2!*2!)] * 9*x^2 * 1
 + [4!/(3!*1!)] * 3*x * 1
 + [4!/(4!*0!)] * 1 * 1
 = 81*x^4 + 4 * 27*x^3 + 6 * 9 * x^2 + 4 * 3*x + 1
 = 81*x^4 + 108*x^3 + 54*x^2 + 12*x + 1

In sage:
combo(4,0) * (3*x)^4 * (1)^0 + combo(4,1) * (3*x)^3 * (1)^1 + combo(4,2) * (3*x)^2 * (1)^2 + combo(4,3) * (3*x)^1 * (1)^3 + combo(4,4) * (3*x)^0 * (1)^4

In a sagecell, copy and paste the following:

def combo(n,r):
    return factorial(n)/factorial(r)/factorial(n-r)

show( combo(4,0) * (3*x)^4 * (1)^0 + combo(4,1) * (3*x)^3 * (1)^1 + combo(4,2) * (3*x)^2 * (1)^2 + combo(4,3) * (3*x)^1 * (1)^3 + combo(4,4) * (3*x)^0 * (1)^4)


The pattern is

for r in range(5):
    combo(4,r) * (3*x)^(4 - r) * (1)^r

So this is a series (a + b)^n = Sum( combo(n,r) * a^(n - r) * b^r, r, 0, n)

The sum of combo(n, r) * a^(n - r) * b^r from r = 0 to r = n

I will take it one step further than MrG in the above video, and write the summation as sage code. Remember we have to define combo(n, r) and the variable r in the sagecell.

In Sage (for the binomial expansion of (3*x + 1)^4), where a = 3*x and b = 1 [n = 4]:

def combo(n,r):
    return factorial(n)/(factorial(r)*factorial(n-r))

r = var('r')
sum(combo(4,r) * (3*x)^(4 - r) * 1^r, r, 0, 4)


Copy this into a sagecell
Posted by: Holden
« on: August 06, 2017, 12:56:41 pm »

Herr Hentrich, I have found that here the maths books have a lot of misprints,much more than any run of the mill book. Also, that it is very important to identify when a maths problem has been wrongly elucidated otherwise one ends up losing confidence in one's understanding because of some lazy book printer.Well, any way just to give you the idea about the kind of things I am grappling with at the moment-"All the page numbers from a book are added,beginning at page 1 .However, one page number was mistakenly added twice.The sum obtained was 1000. Which page number was added twice? Some questions are very badly written. But I am trying to sort out these things. Keep well.
Posted by: Holden
« on: August 04, 2017, 01:46:47 pm »

Herr Hentrich,very sorry about the late reply.I have been reading a bit about the Binomial Theorem. Its a tricky one. Here is a video you might like to check out sometime:    https://youtu.be/OMr9ZF1jgNc
Posted by: Holden
« on: August 02, 2017, 10:25:30 pm »

Thank you for the response. I have just reached home from the tour and am tired.Will post again in the evening.