A Revolutionary Indictment Against the Burden of Existence
General Category => Why Mathematics? => Topic started by: Holden on June 17, 2017, 03:26:40 am

In this thread I'd post mathematics which maybe anythingbut it will not be boring.

So many things are going through my mind, especially considering Raul's last post about Pythagoras and his Brotherhood. I wonder what they would have thought about Euler's Number e. I wonder what Wildberger thinks about this "real number".
The formula is like some wizard's magic potion, no?
Then there is the representation in a computer algebra system where the symbol "e" takes on two different meanings depending on the context which it is used. it is amazing that they are able to design such systems to know the difference.
56e9*e^(0.025*1.75) = 5.85043839493512e10
To the uninitiated, this might cause some confusion, since 56e9 means 56 x 10^9 = 56 billion = 56000000000, whereas the 'e' in e^(r*t) is approximated as
e.n() = 2.71828182845905.
It looks much better with paper and pencil, but I doubt anyone alive today would attempt to perform the arithmetic without the assistance of an electronic calculator of some sort.
Another thing that came to mind is the political agenda behind such calculations. In other words, there is a hidden message in such mathematical forecasts that sees an increase as "good" and a decrease as "bad". Many years ago when I was struggling with the reality of my somehow having become one of the countless superfluous deadbeats of the Industrialized World, I read a book called "The Joy of Not Working: A Book for the Retired, Unemployed and Overworked (http://libgen.io/search.php?req=The+Joy+of+Not+Working&lg_topic=libgen&open=0&view=simple&res=25&phrase=1&column=def). I searched my abandoned blog at wordpress, but my reference to this must have been posted on the borad that got zapped in 2013 or so. I won't mention it's url since it's name was just a strange coincidence.
I will reconstruct it here from Library Genesis (the gift that keeps on giving).
Found it: page 34...
What the G in GNP Really Stands For
Economists, businesspeople, and politicians tell us we will all be better off if our countries have substantial increases in the gross national product (GNP). Gross national product is the value of all services and products sold in a country during any given year. It is the measurement that tells us if we have been successful as a nation. The wise men and women of business and economics tell us that the goal in any country's economy is growth in GNP.
Another goal for the economy is to eliminate unemployment. The ability to generate new jobs is dependent on economic growth. A certain level of GNP is supposed to provide jobs for everyone able to work, whether they want to work or not.
Having taught economics courses at private vocational schools and universities, I have always had a problem with GNP as a yardstick of prosperity. GNP is improved by increases in such questionable activities as consumption of cigarettes and the production of weapons. A substantial increase in car accidents will favorably affect GNP because more funerals, hospital visits, car repairs, and new car purchases will result.
With the growth in GNP considered such an important yardstick, it surprises me that the skipper on the Exxon Valdez didn't receive a Nobel Prize for economics. The gross national product in the United States increased by $1.7 billion due to the Exxon oil spill. More such massive oil spills would do wonders for the GNP. Lots more people would also be employed.
Growth in GNP for the sake of growth doesn't necessarily reflect something beneficial to society. Growth for the sake of growth is also the philosophy of cancer cells. Instead of standing for the gross national product, GNP should stand for the grossness of the national product.
We in North America can probably get by with half the resources we use and still maintain a good standard of living. This can be accomplished, in part, by changing our values. We must eliminate frivolous work and consumption, such as the production of stupid trinkets and gadgets that people buy and use for a week or two before throwing away.
More than a hundred years ago, John Stuart Mill predicted that if the world continued on its path of economic growth, the environment would be totally destroyed.
Not boring, indeed. Who could know that a little mathematical exercise would spur such reflection?
One last thing that also came to mind in this flood of thoughts was the geometric definition of "e and the natural logarithm" I was introduced to while studying Sheldon Axler's "Algebra & Trigonometry (http://libgen.io/search.php?req=Sheldon+Axler+Algebra+Trigonometry&open=0&res=25&view=simple&phrase=1&column=def)".
(p349357)
e is one of those things Wildberger refers to as "The Transcendentals", and I assume he would like to do without them ... anything involving infinite series ...
Myself, I don't share the same prejudice. Even though I have a special respect for rational and natural numbers, I am not totally against e, pi, sqrt(1), sine, cosine, tangent, etc ...
I find I am still quite impressed with even the most elementary algebra and trigonometry. As you know, boredom and ennui is right up there with pain in Schopenhauer's conception of what it means to suffer.
The strange thing is that to many, mathematics has a reputation of being somehow boring.
I am not going to leave any links to any definitions of euler's number but prefer that you, if it is not too much trouble, download the Axler book from libgen.io and refer to pages 352 to 354.
e is the number such that area(1/x, 1, e) = 1.
Of course, that will lead back to page 350 to see what is meant by area(a,b,c)
All in all, it is interesting if you are drawn to it at the moment, but maybe a distraction if you were studying something else.
I have been making little breakthroughs trying to incorporate computer algebra systems and graphing calculators into my studies.
Yesterday I was kind of surprised that some trouble I was having representing (graphically) systems of inequalities using Sage were actually helped when I incorporated the oldfashioned method of constructing a graph using a table of values. This seems to take too much time, but it can be automated by using a TINspire. I won't go into details, but when I took the extreme ends of my values for x which I plotted by hand, and used these as the arguments in the parameters for the plot in SageMath notebook(), I made a breakthrough.
There was not much documentation for this simple task that can be accomplished by hand by shading in regions with a couple different colored pencils. It was more work to get Sage to do it, but it forced me to get a feel for my own way of making it happen, and, in the end I was happy with how the Pythonese code resembled the mathese. I will post a snippet to show you what I mean. I love when I can do something with computers where the code I write looks unmistakeningly familiar to its mathematical counterparts.
If I could just figure out a way to reproduce it here, you could witness how UNBORING (I know this is not a word) even some seemingly boring math can be.
It can be "technically challenging" to find a balance between the old ways and the socalled modern or technologicallyassisted ways.
If one does not first do these things by hand to get a feel for the whereabouts of the graph in the plane, then the code generated graph will not look right, if you can get the code to run at all. When you get it right, there is a sense of fulfillment, but it is on a very personal level and nothing to brag about since one is always taking baby steps.
Maybe in the future we could use sage math cloud (https://cocalc.com/settings) which has already been rebranded. I mean, this way you would not have to get Sage up and running on a mission or virtual machine. Eventually we will be able to copy from here and paste into a live Sage session, so I'll leave it here for now.
The gist: I was quietly thrilled with this little breakthrough which occurred rather spontaneously yesterday while "farting around"/"tinkering" with "sage notebook()" :
x = var('x')
f(x) = x^2  2*x 8
g(x) = 8  2*x  x^2
def h(x,y):
return (y > x^2  2*x  8) & (y <= 8  2*x  x^2) # goddamn emoticon is an 8 and )
# This quite evilly mocks my feeble attempt to communicate how mathlike the code looks ::)
# In "mathese" : {(x,y): y > x^2  2*x  8} ⋂ {(x,y): y <= 8  2*x  x^2}
G = Graphics()
G += plot(f, 4,6, linestyle="")
G += plot(g, 6, 4, linestyle="")
G += region_plot(h, (10, 10), (9, 12), plot_points = 300, incol='gold')
G.show()
f(x) = 3  2*x  x^2
def g(x,y):
return (y < 3  2*x  x^2) & (2 >= abs(y))
G = Graphics()
G += plot(f, 5, 3, linestyle="")
G += plot(2, 10, 10, linestyle="")
G += plot(2, 10, 10, linestyle="")
G += region_plot(g, (10,10),(10,10), plot_points=300, incol='green')
G.show()
_____________________________________
PS:
In SymPy, although the graph is not as elaborate, for the second plot, one line suffices, which is kind of "elegant":
p = plot_implicit(And(y<32*xx**2, 2>=abs(y)))
:\

You can copy the following and paste it into a cell at sagecell.sagemath.org (http://sagecell.sagemath.org/)! This might help us to communicate.
Don't worry about the emoticon. It will magically disappear when you paste the code into the cell. Do this for each code snippet separately.

x = var('x')
f(x) = x^2  2*x 8
g(x) = 8  2*x  x^2
def h(x,y):
return (y > x^2  2*x  8) & (y <= 8  2*x  x^2) # goddamn emoticon is an 8 and )
# In "mathese" : {(x,y): y > x^2  2*x  8} ⋂ {(x,y): y <= 8  2*x  x^2}
G = Graphics()
G += plot(f, 4,6, linestyle="")
G += plot(g, 6, 4, linestyle="")
G += region_plot(h, (10, 10), (9, 12), plot_points = 300, incol='gold')
G.show()
____________________________________________________________
f(x) = 3  2*x  x^2
def g(x,y):
return (y < 3  2*x  x^2) & (2 >= abs(y))
G = Graphics()
G += plot(f, 5, 3, linestyle="")
G += plot(2, 10, 10, linestyle="")
G += plot(2, 10, 10, linestyle="")
G += region_plot(g, (10,10),(10,10), plot_points=300, incol='green')
G.show()

Holden,
You once expressed interest in linear programming.
I wonder if this sage code (http://www.steinertriples.fr/ncohen/tut/LP/) would work in the sagecell (http://sagecell.sagemath.org/) referenced to above.
I placed the following in a sagecell, and the result was different than what it was on the resource page.
p=MixedIntegerLinearProgram( maximization=True )
p.set_objective( 2*p[1]+p[2] )
p.add_constraint( 3*p[1]+4*p[2], max = 2.5 )
p.add_constraint( 1.5*p[1]+0.5*p[2], min = 0.5, max = 4 )
p.solve()
print "The optimal values are x_1 = "+str(p.get_values(p[1]))+", x_2 = "+str(p.get_values(p[2]))
I get 4.722222222222221 while they said it would be 1.6666666666666665.
Also, whereas they show: The optimal values are x_1 = 0.833333333333, x_2 = 0.0
My result was: The optimal values are x_1 = 3.27777777778, x_2 = 1.83333333333
buggy?
The next one we were in agreement with:
g=graphs.PetersenGraph()
p=MixedIntegerLinearProgram(maximization=False)
b=p.new_variable()
for (u,v) in g.edges(labels=None):
p.add_constraint(b[u]+b[v],min=1)
p.set_objective(sum([b[v] for v in g]))
p.set_binary(b)
p.solve()
b = p.get_values(b)
print b
print [v for v in g if b[v]==1]
[/u]
Anyway, just a reminder that, even though I have been obsessed with installing SageMath in every operating system, you can always experiment a little with the online sagecell.
Consider this "future reference". I understand if this is not your thing at the moment.
If you ever decide to check out Sage (or anyone else who passes through here by accident), there is a good essay on Sage on a CUNY Math Blog (https://cunymathblog.commons.gc.cuny.edu/2012/04/09/sage/)
Sage Tutorial for linear programming (http://doc.sagemath.org/html/en/thematic_tutorials/linear_programming.html)
Also see Sapien Games: History, Politics, & Other Heresies (http://sapiengames.com/2014/05/18/installsagemathematicsarchlinuxdistros/)

I understand that you are not currently interested in anything related to programming. I also realize that Linear Programming does not necessarily have anything to do with computer programming, and that it finding minimum and maximum values given constraints on a system of linear equations.
I only placed these links here like someone placing a pair of warm socks in the back of a drawer for some distant season.

Herr Hentrich,
I am very sorry for my late response.I am travelling. I do greatly appreciate your posts about maths.
Once again I must say that if not for our preordained meeting in 2014 I might still be drifting away aimlessly.
It is only because of you that I possess some sort of sense of direction.I am studying mathematics but it is of very fundamental kind.If my foundations are weak that I can never expect to study math of more advance variety. I am studying things like :the infinite sum of 1+4/7+9/7^2+16/7^3+25/7^4+.......
I am taking it very slowly but I am STUDYING maths.I am not saying this just to please you.I am genuinely interested in maths.
By the way,did you know that Kant was a very competent mathematician and was very much interested in Newtonian mathematics?

That's good to hear, Holden. I share your views about the importance of the fundamentals, which is why I have temporarily put aside the texts on multivariable calculus, linear algebra, differential equations, physics, etc, and am challenging myself to take seriously a couple high school geometry texts. I thought I would breeze through them over the summer, but I suspect I will be "stuck" in Geometry Land for a good year before I am able to continue with my "Algebra/Trigonometry/Analysis" coding.
I had wanted to go through the Dolciani series, translating the programming exercises from BASIC and Pascal to C/C++ and even SAGE/Python, but I found the Dolciani series weak in the subject of Geometry so I am incorporating a high school (honors) textbook from 2011 that I found on ebay for $13 (to supplement the more formal and traditional Jurgensen/Brown/Jurgensen text).
While it does contain some goofy looking characters that at first I found annoying, some of the "challenging" exercises are exactly that. I wake up in the morning and try not to classify myself by age or (lack of) occupation. Fortunately I am not under anyone's microscope so I am able to engage with the material very HONESTLY. This is only possible in solitude. There is no pressure to delude myself. Nor is there any reason to feel ashamed if I actually have to think or if I find I am learning.
You had once told me that this is a courageous decision. I now understand what you meant. It takes courage to face the fact that I could use an overhaul in my understanding. It doesn't matter that I graduated from a university 15 years ago. I feel this need to revisit high school mathematics in order to remain true to myself.
I have to say that I appreciate your focusing on the fundamentals for this shows me that you sincerely respect my decision to do the same.
We, Holden, are not fuucking around. We do not aim to impress others, but are actually genuinely interested in developing our understanding.
I am studying things like :the infinite sum of 1+4/7+9/7^2+16/7^3+25/7^4+.......
Ah ... strangely enough, I spent a good part of this past winter going over Infinite Sequences and Series.
I regret that we are not able to exchange notes with paper and pencil. This message board makes for a very clumsy medium.
There is a way to work around the limitations of our present "editor".
If we agree on some way of making certain details clear ...
For instance, with 1+4/7+9/7^2+16/7^3+25/7^4+ ...
Would it be wrong for me to assume the following?
1 + (4/7) + 9/(7^2) + 16/(7^3) + 25/(7^4) + ....
=
1 + (2^2)*(1/7) + (3^2)*(1/7)^2 + (4^2)*(1/7)^3 + (5^2)*(1/7)^4 + ...
So, I recognize this as a geometric series, that is, a geometric sequence of partial sums {n^2 * (1/7)^(n1)}.
When n = 1, it looks as though (1/7) is being raised to the 0 power, which gives (1/7)^0 = 1; hence the first term is 1. We now see that (1/7) [= r] is being raised to the power (n1).
Hence, the second term, where n = 2, is (n^2)*[ (1/7)^(n1) ], or
(2^2)*[ (1/7)^1 ] = 4/7
It helps to break it down like this just to make sense of each term (before considering the infinite sequence).
So the third term has n = 3 and n1 = 2:
(3^2)*[ (1/7)^(31) ] = 9*[(1/7)^2] = 9/(7^2)
The fourth term: (4^2)*[ (1/7)^3 ] = 16/(7^3)
and so on.
The main thing is to recognize and extract the form.
You want to be able to explain this to a computer algebra system, something like, where "sum" represents sigma or summation:
sum(n^2 * (1/7)^(n1), n, 1, oo)
"The sum of n squared times (1/7) to the (n1) power for n from 1 to infinity."
Input n^2*(1/7)^(n1) over here (http://calculator.tutorvista.com/math/608/infiniteseriescalculator.html) for n = 1 to infinity.
The sum is 49/27.
To understand why this is, think about the limit. The terms of this series converge to 1/7, which is less than 1. When considering the limit, notice that {n^2 * (1/7)^(n1)} will be called a_n, where n is the subscript.
a_subscript_(n+1) is just {(n+1)^2 * (1/7)^n}
In the following, they check the limit (as n approaches infinity) of the ratio a_(n+1) / a_n:
See here (https://www.symbolab.com/solver/seriescalculator/%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%20%7D%20n%5E%7B2%20%7D%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B7%7D%5Cright)%5E%7Bn1%7D)
The reason I am utilizing this "calculator" sites is because of the mathematical notation they are able to employ.
This helps us communicate and have some confidence that we are on the same page, so to speak.
For now, you probably don't need to think about limits (the second link) and are more concerned with finding the sum of the series.
This brings attention to a little dilemma.
Of course, there must be some kind of formula, right?
If it were not for the n^2 part, it would be straightforward:
1/(1(1/7)) = 1/(6/7) = 7/6
If you ask a CAS to solve: sum((1/7)^(n1), n, 1, oo), you will also get 7/6.
The sum of n^2 from 1 to infinity is infinity, but the combination of the two gives a sum of 49/27 ... so there is a factor of (7*2)/9 in there somewhere.
It's amazing that I was covering similar material throughout the winter, and I have to refer to my "notes" or else risk adding to the confusion rather than clarifying anything.
This is both funny and sad at the same time, no?
I would suggest looking at a problem of the form a + a*r + a*r^2 + a*r^3 + ...
The n^2 term is adding a level of complexity to the problem.
Start with 1 + 1/7 + (1/7)^2 + (1/7)^3 + ... first.
That's my suggestion. Anything involving limits and infinity is begging to use calculus. At least with the standard geometric series, you can apply some formulas.
Please do not hesitate to also check results with a CAS as this will help verify the results you get with a formula.
The reason for the formulas is that without them, you would require calculus.
Don't get discouraged. Meanwhile I have to eat some food before my brain becomes totally useless.
That's why the most important thing in all the schools on the planet would be to offer breakfast to all the students. First class: HOME GROUP, then some kind of breakfast! Without some nutrition, all the teaching is in vain.
I have been up since morning time, it is after 2PM, and I have not eaten yet. So, it is no surprise that my brain is fuzzy and I'm a little dizzy.
:P
It will be interesting to see if it turns out that we both have a low frustration tolerance when it comes to mathematics. I have found that, no matter what the "level" of mathematics I am looking at, the threshold for when I lose patience with myself is small. This is not the case in a classroom situation where I become engrossed in the task at hand, but when I return to anything outside that particular type of math, it is as though I am starting all over again. This leads me to suspect that I may delude myself into thinking I have comprehended something.
I do not say this to discourage you, but quite the opposite. I don't want to project an image of someone who has a deep understanding of things, but rather, I want you to feel more comfortable with your own frustrations upon witnessing how much concentration I have to harness just to get my bearings.
In fact, I am moving in a direction where I will either embrace the total disintegration of my ego or be hanging from the end of a rope. :D
I have to give my brain permission to be itself and not to bombard it with demands about "what it ought to understand". It is what it is, and I have to just get over the fact that my brain is not a storage bin for masses amount of knowledge, but rather a chaotic bundle of nerves which can only focus on the task at hand.
Since I do have a rather sensitive ego, isolation and privacy are crucial to such undertakings. I seek guidance from others (textbooks and other sources), but it has to be at my own slow pace.
Maybe we are unwittingly engaged in what could fall under the title "The Politics of Mathematical Empowerment".
Too much of formal education tends to have the opposite effect of mathematical disempowerment, hence, in her suicide note Kriti Tripathi has urged the government of India and human resource development (HRD) ministry to shut coaching institutes as soon as possible. “They suck,” she wrote.
Footnote (http://www.hindustantimes.com/education/kotasuicidehelpedmanycomeoutofdepressionbutnotmyself/storyu1IRTdSFJoGuaIjvN7igSK.html) (originally linked to by Holden)
There has to be a way to make this learning process more individualized so that we might appreciate the little we can cover in one lifetime rather than feeling that there is no point in it.
As you say, we are concerned with developing a solid foundation. I would add that the foundation and fundamentals are a worthy area of study in and of itself, and that there need not be any distant destination.
We are, after all, merely doing time in the penal colony of existence. This may be a good way to get through the days and years and decades.

Herr Hentrich thank you so much for solving the problem.Your solution is correct.
I have a little algorithm which I make use of to solve such problems. Here is how it goes:
S=1+4/7+9/(7^2)+16/(7^3)+25/(7^4)+....
Now what I do is ,I multiply both sides with1/7:
1/7 S=1/7+4/(7^2)+9/(7^3)+16/(7^4)+...
Now,I substract the 2nd equation with the first:
S (6/7)=1+3/7+5/(7^2)+7/ (7^3)+....
Again,I multiply both sides with 1/7:
S* 6/(7^2)=1/7+3/(7^2)+5/(7^3)+.....
Now ,I substract the 4th equation from the third:
S (36/49)=1+2/7+2/(7^2)+....
This is a geometric progression with first term=1 and common ratio=1/7
=1+(2/7)(1/(11/7)
S=49/27.

Let me see if I am following your logic.
Subtracting the 4th equation from the third:
S (6/7)=1+3/7+5/(7^2)+7/ (7^3)+....
S* 6/(7^2)=1/7+3/(7^2)+5/(7^3)+.....

S*[ (6*7)/(7^2) ]=1+3/7+5/(7^2)+7/ (7^3)+....
S* 6/(7^2)=1/7+3/(7^2)+5/(7^3)+.....

S*[ (426)/7^2 ] =1 + [3/7  1/7] + [ (52)/(7^2) ] + [ (75)/(7^3) ] + ...

S (36/49)=1+2/7+2/(7^2)+....
_____________________________________________
OK, then what? Let's see ... Do you then factor out the 2? or are you factoring out 2/7?
S*(36/49) = 1 + (2/7)*[1 + 1/7 + 1/7^2 + 1/7^3 + ...]
ah, and that is your geometric progression with first term=1 and common ratio=1/7 ... where you use the formula 1/(1r) > 1/(1(1/7)) > 1/(6/7) > 7/6
So, S*(36/49) = 1 + (2/7)*(7/6)
S*(36/49) = 4/3
S = (4/3)*(49/36) = (1/3)*(49/9) = 49/27
Very clever. Kind of Gaussian. And no computers, you rebel! Sorry if I doubted you for a minute. I had to prove it to myself. Wow. That's very cool, Holden.
I like that. I was just shutting down the computer for the night, so I checked the board first. I'm pecking away at Infinite Jest again.
That is a very cool method you used, a real confidence builder where you attack it with brute force.
Your method reminds me a little of how one finds the sum of numbers between number x and number y, where you have
S = x + (x+1) + (x+2) + ... + (y2) + (y1) + y
Then you add the same sum (in reverse order):
S = x + (x+1) + (x+2) + ... + (y2) + (y1) + y
S = y + (y1) + (y2) + ... + (x+2) + (x+1) + x

2S = (x+y) + (x+y) + (x+y) + ... + (x+y) + (x+y) + (x+y)
There are n terms, so 2S = n(x+y)
Then, dividing both sides by 2, S = [ n (x+y) ]/2
Thanks for the inspiration. It's good to be able to work such things out by hand without a full scale 8 gigabyte computer algebra system! :D
Post Scriptum:
If you ever want to check your results, and you can extract {a(n)} from the given terms of the series, go to http://sagecell.sagemath.org/. For example, copy the following into the cell and press evaluate:
n = var('n')
h(n) = (n^2)*(1/7)^(n1)
sum(h(n), n, 1, oo)
Note that oo represents "infinity", that is, two lowercase o.

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

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

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.

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: (https://sagecell.sagemath.org/)
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 (https://www.youtube.com/watch?v=D_qKkMvwvdM)
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(nr)
I like the way he divides twice instead of typing the denominator with parentheses and using multiplication: factorial(n)/(factorial(r)*factorial(nr))
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(nr)
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(nr))
r = var('r')
sum(combo(4,r) * (3*x)^(4  r) * 1^r, r, 0, 4)
Copy this into a sagecell (https://sagecell.sagemath.org/)

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.

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.

Continuing from Holden's Inquiry about Binomial Theorem, and here (http://whybother.freeboards.org/mathdiary/theoneunforgivablesinistobeboring/msg3973/#msg3973) 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 buildcalculatorfromscratch 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.

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 selfpity 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.

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 freeflowing 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] (https://gmplib.org/) 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, nonterminating or terminating decimals (https://github.com/Gorticide/Rational_Numbers/blob/master/d2r_fix2.cpp).
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.

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/(10.1) = 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" 

UPDATE: all inlcuded in one PDF file if that is more convenient:
(see this post) (http://whybother.freeboards.org/mathdiary/theoneunforgivablesinistobeboring/msg8265/#msg8265)
______________________________________________________________
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)/(11/10) = (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 handwritten notes, printed from device to actual hard copy print might trigger a sense of selfrespect and dignity in yourself to validate your quest for understanding.
Prelude to Binomial Theorem ... 1 ... (https://i.ibb.co/12nJj7D/A3prelude0.jpg)
prelude page 2 (https://i.ibb.co/d51b8rB/A3ampprelude1.jpg)
____________________________________________________________
TESTING 1, 2 (http://whybother.freeboards.org/mathdiary/testingimgbbforlegibility/msg8259/#msg8259) ...
(will direct to another thread with door to jump into next post in this thread)

Binomial Theorem Notes #001 (https://i.ibb.co/PQtsxRG/A3binomialthm1.jpg)
I also have lived for understanding, and so am your true brother.
BT Notes #002 (https://i.ibb.co/PDhchHN/A37binthm2.jpg),
#003 (https://i.ibb.co/1sytzk1/A37binomialthm3.jpg),
#004 (https://i.ibb.co/3ynpRN3/A37binomialthm4.jpg),
#005 (https://i.ibb.co/r47F6Tr/A37binomialthm5.jpg)

More About the Binary Theorem (https://i.ibb.co/52RytHp/A37morebinthm1.jpg),
More About the Binary Theorem [2] (https://i.ibb.co/s5HTgZf/A37morebinthm2.jpg)
More About the Binary Theorem [3] (https://i.ibb.co/pKynTjG/A37morebinthm3.jpg)
More About the Binary Theorem [4] (https://i.ibb.co/Byv5CNp/A37morebinthm4.jpg)
More About the Binary Theorem [5] (https://i.ibb.co/17Hpx1n/A37morebinthm5.jpg)
[6] (https://i.ibb.co/87L0Q0Q/A37morebinthm6.jpg)
[7] (https://i.ibb.co/N6wfH0L/A37morebinthm7.jpg)

Prelude:
Infinite Geometric Series [1] (https://i.ibb.co/BCq5FFk/A37misc001.jpg)
Misc. [2] (https://i.ibb.co/Kw47KCC/A37misc002.jpg)
Misc. [3] (https://i.ibb.co/Z8SSpdK/A37misc003.jpg)
Misc. [4] (https://i.ibb.co/gTjWxR3/A37misc004.jpg)
Misc. [5] (https://i.ibb.co/Htj17d2/A37misc005.jpg)

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 (p17: number system) plus Binomial Theorem (p834) (https://archive.org/download/binThm_set1/binThm_set1.pdf) (in correct order)
The PDF files print easiest, and, at 33 pages, that's about 17 sheets if you print on both sides ...

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.

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 simpleminded) 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.

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 (https://archive.org/download/pascal_triangle_one_page/pascal_triangle_one_page.pdf) 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 reawakening ... 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_xAbtak

https://www.google.com/amp/s/www.wsj.com/amp/articles/frankramseyreviewthemostgenialgenius11585322801

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 ...
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.