Author Topic: SAGE MATH  (Read 2622 times)

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Holden

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Re: SAGE MATH
« Reply #30 on: March 01, 2019, 12:33:20 pm »
Something so fundamental as the fact that mathematics is an axiomatic-deductive system  was never  taught to me with  any degree of clarity in the high  school and even back then I almost reached that  conclusion on my own.There were no books, no teachers who could have helped me.

I am sick and tired of the human world. It does not interest me. Political positions and women hold not attraction for me.Another  thing which  attracts me to math is that it is about  the only thing I can do all  on my own  ,without anyone's interference.

Like you say, my brain is made of flesh  and blood and does get tired after a while but as long as it is not tired there is very little,in principle, which I find incomprehensible if I really put my mind to it.

Programming? Maybe someday. For now, I am focussing only on mathematics. I must first  develop strong foundations.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Holden

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Re: SAGE MATH
« Reply #31 on: March 01, 2019, 10:06:00 pm »
What the poet laments then holds for the mathematician, that his works are written with the blood of his heart -Boltzmann
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Kaspar Hauser

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Re: SAGE MATH
« Reply #32 on: March 02, 2019, 12:12:40 pm »
I found that factoring [quadratic] trinomials, especially using the method of completing the square, was strangley inspirational for me.

I only wished you might become more comfortable compiling programs from the command line since I have a few programs that you could use in conjunction with paper and pencil to kind of check your work.  The completing the square program shows each step and can even handle irrational and complex roots.  I may have mentioned this already, but quite often, after I have gotten a program to produce correct results, I return to it and build into it "educational" components.  This often requires far more work than just producing sensible output.  In a real way, such effort is like a gift to my future self should I get a little rusty and need a crash course on these fundamental procedures.

Are you still using the Mac machine?   I'm certain that you could install the GNU g++ compilers since Mac uses the UNIX system.  I'm not pushing this on you, just leaving links for some rainy day, or a series of rainy days, if you ever catch the codemode bug.  Also see How to Install g++ on the Mac.

Of course, if this seems daunting and unnecessary, then I take no offense if you prefer not to bother with it.

Still, I am curious how you feel about factoring trinomials.   What methods do you use?  Trial and Error?   Grouping?  Completing the square?

For some reason, this was one of the fundamental procedures which I found very satisfying to go over thoroughly.

Take care.  You sound as though you have embraced the reality that this will be a very long process, and I am pleased to witness you keeping your feet on the ground and getting into whatever you study at the moment.  Years pass like months when it comes to studying math.   It's good that you are not racing, since we are always living only in the present moment.
« Last Edit: March 02, 2019, 12:25:44 pm by Miserable Mike »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Holden

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Re: SAGE MATH
« Reply #33 on: March 03, 2019, 11:20:19 am »
I have only  studied Number systems,Arithmetic Progression,Geometric Progression,Harmonic  Progression,Averages,Alligations,Percentages   recently so far.So kindly keep your   questions  confined  to  these   topics. Thanks.

https://www.dailymotion.com/video/xdoj7y
« Last Edit: March 03, 2019, 11:25:22 am by Holden »
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Kaspar Hauser

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Re: SAGE MATH
« Reply #34 on: March 03, 2019, 12:31:11 pm »
Quote from: olden
I have only  studied Number systems,Arithmetic Progression,Geometric Progression,Harmonic  Progression,Averages,Alligations,Percentages   recently so far.So kindly keep your   questions  confined  to  these   topics. Thanks.

I was only suggesting "completing the square" and factoring quadratic trinomials as an area I think would give you some intellectual stimulation, or even a strong dose of satisfaction.   I only asked if you were familiar with the method, but did not pose any specific question.

Your request is noted. 

As for ideas and concepts related to programming, forgive me if I use this message board as a kind of notebook to highlight some revelations.   These posts may be of interest to you or not.   I will be posting such things not necessarily directed to anyone in particular, but simply as a means of highlighting certain "aha" moments I may experience, where philosophical terminology is used to describe certain technicalities in programming.

A computer language such as C++ provides a programmer with some built-in types, such as char, int, or double (char representing an alpha-numeric character or symbol, int representing an integer, and double representing a "real number".

Types that are not built-in are called user-defined types, such as classes (data structures).

What are user-defined types good for?

Types are good for directly representing our ideas directly into code.

We want to represent an "idea" or "concept" in code as a data structure plus a set of functions.

Of course, the devil is in the details, the technicalities, but the reason for mentioning this here is to breathe life into these key words:  representation and idea.

These are two English words, both used in translating the German concept of Vertretung.

die Vorstellung : imagination, perception, presentation, notion

to represent :   darstellen ?

Anyway, I'm in an in between zone (between chapters in two different math textbooks), and I am trying to force myself to focus a little on some technicalities in C++.

« Last Edit: March 03, 2019, 01:32:00 pm by Miserable Mike »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Kaspar Hauser

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Re: SAGE MATH
« Reply #35 on: March 03, 2019, 07:31:02 pm »
Quote from: Holden
Another  thing which  attracts me to math is that it is about  the only thing I can do all  on my own, without anyone's interference.

I understand this.  I would never study the way I do if some other person were insisting that I must.

I can be easily irritated. annoyed, and even overwhelmed unless I am engaging with something by my own volition.   As Henry Fool told Simon Grimm, you can't put a fence around a man's soul.  We need time to think, and we must be genuinely interested in what we choose to give our attention to.

In fact, when I find I am becoming a little sickened by the complexity of some code I am trying to understand, often the only thing that can calm my anxiety is to work through some math problems with paper and pencil.   So, I do not blame you, Holden, not in the least, if your instincts tell you to block out everything other than what is essential to whatever it is you are working on at the moment.   

- Brother Mike
« Last Edit: March 03, 2019, 10:37:16 pm by Miserable Mike »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: SAGE MATH
« Reply #36 on: March 04, 2019, 12:01:02 am »
Quote
I understand this.  I would never study the way I do if some other person were insisting that I must.
-Herr Kaspar

I  am sorry but that is not what I meant primarily. I mean that mathematics does not need  "team work", you do not have to  be a part  of  a group  to do it-its not  like  one guy solves  the first  5 steps  and the next    guy the following 3 steps. Generally, one guy solves  the whole of the question all on  his own.Or if  he  fails to  do so then its  again all on this own.



La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
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Kaspar Hauser

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footnotes: (1):

table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/12]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)

+----------+------------+--------------+
| $i$      | $cos(2*i)$ | $rational$   |
+==========+============+==============+
| 0        | 1.000      | 1            |
+----------+------------+--------------+
| 1/12*pi  | 0.8660     | 1/2*sqrt(3)  |
+----------+------------+--------------+
| 1/6*pi   | 0.5000     | 1/2          |
+----------+------------+--------------+
| 1/4*pi   | 0.0000     | 0            |
+----------+------------+--------------+
| 1/3*pi   | -0.5000    | -1/2         |
+----------+------------+--------------+
| 5/12*pi  | -0.8660    | -1/2*sqrt(3) |
+----------+------------+--------------+
| 1/2*pi   | -1.000     | -1           |
+----------+------------+--------------+
| 7/12*pi  | -0.8660    | -1/2*sqrt(3) |
+----------+------------+--------------+
| 2/3*pi   | -0.5000    | -1/2         |
+----------+------------+--------------+
| 3/4*pi   | 0.0000     | 0            |
+----------+------------+--------------+
| 5/6*pi   | 0.5000     | 1/2          |
+----------+------------+--------------+
| 11/12*pi | 0.8660     | 1/2*sqrt(3)  |
+----------+------------+--------------+
| pi       | 1.000      | 1            |
+----------+------------+--------------+
| 13/12*pi | 0.8660     | 1/2*sqrt(3)  |
+----------+------------+--------------+
| 7/6*pi   | 0.5000     | 1/2          |
+----------+------------+--------------+
| 5/4*pi   | 0.0000     | 0            |
+----------+------------+--------------+
| 4/3*pi   | -0.5000    | -1/2         |
+----------+------------+--------------+
| 17/12*pi | -0.8660    | -1/2*sqrt(3) |
+----------+------------+--------------+
| 3/2*pi   | -1.000     | -1           |
+----------+------------+--------------+
| 19/12*pi | -0.8660    | -1/2*sqrt(3) |
+----------+------------+--------------+
| 5/3*pi   | -0.5000    | -1/2         |
+----------+------------+--------------+
| 7/4*pi   | 0.0000     | 0            |
+----------+------------+--------------+
| 11/6*pi  | 0.5000     | 1/2          |
+----------+------------+--------------+
| 23/12*pi | 0.8660     | 1/2*sqrt(3)  |
+----------+------------+--------------+
| 2*pi     | 1.000      | 1            |
+----------+------------+--------------+


To plot:

 t = var('t')
polar_plot(cos(2*t), t, 0, 2*pi)


sage:  table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)
+---------+------------+--------------+
| $i$     | $cos(2*i)$ | $rational$   |
+=========+============+==============+
| 0       | 1.000      | 1            |
+---------+------------+--------------+
| 1/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 1/4*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 3/8*pi  | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 1/2*pi  | -1.000     | -1           |
+---------+------------+--------------+
| 5/8*pi  | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 3/4*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 7/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| pi      | 1.000      | 1            |
+---------+------------+--------------+
| 9/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 5/4*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 11/8*pi | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 3/2*pi  | -1.000     | -1           |
+---------+------------+--------------+
| 13/8*pi | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 7/4*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 15/8*pi | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 2*pi    | 1.000      | 1            |
+---------+------------+--------------+


sage:  table([(i,n(sin(2*i), digits=4), sin(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$sin(2*i)$", "$rational$"], frame=True)
+---------+------------+--------------+
| $i$     | $sin(2*i)$ | $rational$   |
+=========+============+==============+
| 0       | 0.0000     | 0            |
+---------+------------+--------------+
| 1/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 1/4*pi  | 1.000      | 1            |
+---------+------------+--------------+
| 3/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 1/2*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 5/8*pi  | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 3/4*pi  | -1.000     | -1           |
+---------+------------+--------------+
| 7/8*pi  | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| pi      | 0.0000     | 0            |
+---------+------------+--------------+
| 9/8*pi  | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 5/4*pi  | 1.000      | 1            |
+---------+------------+--------------+
| 11/8*pi | 0.7071     | 1/2*sqrt(2)  |
+---------+------------+--------------+
| 3/2*pi  | 0.0000     | 0            |
+---------+------------+--------------+
| 13/8*pi | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 7/4*pi  | -1.000     | -1           |
+---------+------------+--------------+
| 15/8*pi | -0.7071    | -1/2*sqrt(2) |
+---------+------------+--------------+
| 2*pi    | 0.0000     | 0            |
+---------+------------+--------------+
___________________________________________________________
(2):

How tables can be generated in Sage to help plot polar graphs.   Now you can copy and paste it into http://sagecell.sagemath.org to see what I mean:

table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/12]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)

___________________________________________________
t = var('t')
p1 = polar_plot(cos(2*t), t, 0, 2*pi)
p2 = polar_plot(sin(2*t), t, 0, 2*pi, linestyle="--")
show(p1 + p2)
_____________________________________________
table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)

« Last Edit: October 25, 2020, 11:32:50 am by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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