{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

Very interesting …

A Recipe for a Common Logarithm Table

I may try to contact the person who made the video, Anshul Kumar, to inquire about the significance of the constant 2.42.

2.3 gets a closer result than 2.42, but I found that tweaking the constant to 2.33 got it a couple more digits of accuracy when compared to calculator and pretty much exactly what one gets when using a 4-place table with rightmost Proportional Part as well.

So I will make a note above to this effect.

Just for the record:



Let's get down and dirty.  I, for one, find the method described in this video, while fascinating, not realistic for me.  I won't remember all the details.

Let's try this:



I'm not too crazy about this technique either as it gives little insight into what is going on.  I will once again consult Dead Reckoning: Calculating Without Instruments, Ronald W. Doerfler, c.1993; this time, pp 133-144 (Approximate Logarithmic Inverses).

I just found the sources on line for some of the material I was reading from a book, so the interested reader need not wait for this ape-man to type up from his notes what they can print out with a printer.

Main site: 

Area of Current Interest:  Dead Reckoning, Chapter 4: Logarithms and Their Inverses

Bemer's technique

It is not out of laziness that I point out this resource.  It is valuable, and the author even provides links to hard to track down sources.

Very fascinating stuff ...

A link at the bottom of the page on Bemer's technique for finding "exponential" - that is, antilogarithms, points to John McIntosh, aka urticator.

 I'm not bragging when I say that my hand-written notes showing the work in pencil could prove invaluable to someone, and especially invaluable to me.  What this implies is that the best way to learn these techniques is to find an old book with problems meant to be solved using tables, and then, instead of using the tables, solve from scratch using the formulas and techniques.  Then compare results and notice that your results may actually be more accurate than those listed as "solutions" in the text which thinks you are using 4-place tables.

This can be a very satisfying experience.

Of course, I realize that it is certainly not the instant gratification we "moderns" are used to.  It entails work and concentration, and, of course, undisturbed leisure where one is not plagued with anxiety and worry.

I've seen it before, but I haven't found a need to use it.  It's interesting in that it does not use any trigonometric functions.  Even though the usual formula for area of a triangle does not explicitly include any trigonometric functions, as it is Area = (1/2)*base*height, in finding the height, the sine or cosine is used to help find the lengths of height or base.

I suppose if you were able to measure each side of a triangle, supposing you could actually find a triangle in Nature, then you might approximate the area with this trick.

I think I may start referring to mathematical formulas as "tricks".

As you know, I am currently under the spell of logarithmic trickery ... Trichery.

It is frustrating discussing mathematics, don't you think?

Well, it is certainly less frustrating than discussing unknowable unknowns such as whatever the word "God" or "the gods" means to anyone in particular, but consider discussing something like the trigonometric functions with someone who has just learned how to "use" them.

They may insist that they know all about the sine and the cosine, the tangent, and their inverses.   In reality, these are transcendental functions that require the notion of limits to be defined ... infinite series which converge; but learning about these is a prerequisite to studying Calculus.   The trigonometric functions are transcendental, depending on the idea of approaching a limit, depending on the idea of infinite, in short, depending on Calculus to be defined; and yet they are presented as a prerequisite to studying the very thing on which their identities depend.

It's rather circular, isn't it?  It's no wonder we are initially mystified by them.

Now, the problem with discussing something like "God" is that so many people think they know what they are talking about, and - as such, it's not such a great topic of conversation, except for blasphemy and heresy.

Now, when it comes to math, I don't mind so much discussing it as I do appreciate the challenge it presents in trying to communicate.  How does one communicate unknowing?  It is said that a little knowledge is dangerous.

I don't think so.  A little knowledge is ok as long as we have the confidence to stand up to those who wish to intimidate us.  After all, as long as we are willing to confess our general ignorance, or the possibility that not a great deal of knowledge and understanding can be retained in such matters, that it is quite possible that mathematics education may be somewhat of a sham, where so many people do a lot of posturing and pretending.

Do you get the gist of the direction in which I am leaning?

We are capable monkeys who can be trained to pass tests and to focus on the task at hand, and the to answer certain questions and solve certain problems based upon the material we may have recently been exposed to.

Have you seen all the texts available at Library Genesis?

Have you gathered more than you will be able to fathom in 100 lifetimes?

One of the reasons I like to focus on a handful of old high school math books is that it tends to keep me grounded.  Sure I have the books on Computational Physics as well as the verbose and systematic works of Kant; but when i want to feel authemtic, which is most of the time, I like to continually go over the fundamentals.

I hate pretensious snobs, and I would never wish to be one of them.

I am an honest man, and it is for this reason that employment is problematic.

One is required to lie to oneself in order to function is society.

The builders of the houses cannot confess any ignorance.   People are trained to hide their ignorance, and not to show weakness and vulnerability.

I like to be honest.  Unfortunately, honesty is not always polite and kind.  When put into practice, honesty may come off sounding hostile.  It may be offensive.

I like to say what I mean and mean what I say, and I am hoping one day to be able to write about mathematics in such a way that does not cause more confusion than already exusts in the world.

Most likely, I may only add to the confusion, unless I can muster enough honesty to force the reader to join me in this quest to uncover the truth that we will forever be in the dark and may only get glimpses here and their of a limited understanding.

Most importantly, we may suspect our entire civilization depends upon the lot of us deceiving ourselves and allowing ourselves to be deceived.

Most people do not really want to know the truths we are trying to uncover.

I vow to study math for the rest of my days even if the main thing I learn is that we are not machines that can program ourselves to understand.  We can learn how to compute and to calculate, but undertanding seems to be one of these hocus pocus mystical voodoo phenemenon based on delusion, self-deception, and pretension.

When an honest thinker comes along and confesses to knowing nothing, the world will take him at his word and hand him a bucket and a mop and some tools for cleaning toilets.

Meanwhile, the gorilla who thinks he knows everything pops open the hood off an automobile and proceeds to play mad scientist with the engine, investing money into tools and such.  He fancies himself a mechanic.

Another ape fancies himself an engineer who "designs Volvos".

And yet .... how much of this Colony of Industrial Civilizations runds as a kind of mysterious hive that not one individual ape-man can comprehend?

Each individual has enough trouble just getting through the day and eating meals, just like the creepy crawling giant mammalian insects that we actually are.

What do you find interesting about convex and concave quadrilaterals?

Is there something specific you want to mention?

I know writing in plain honest language can be a challenge when we are so used to these verbose philosophers who fill hundreds of pages explaining their "system of philosophy" as if it were necessary for others to validate the system they have come up with for describing our situation.

If only we could discover a way of communicating where we might be able to mean what we say and say what we mean.  If I have nothing to say, it might take 500 words to express this.  So be it!

One man says he knows how to build a house.  Can he do this without a shovel?

Who makes the shovel?  Another man designs an operating system to be loaded onto a hard-drive of a computer.  Who places the chips on the circuit board?  Who designs the circuit board?

When I install a web browser in Funtoo, a branch off of Gentoo created by the founder of Gentoo, I witness "it," the operating system that is, compiling all the code into a "program" which will be launched.  It takes several, if not many, hours.  It's nothing like watching a setup.exe program in Windows install a precompiled loader. 

Do you know how I feel when I issue the command to install such software?

It does not make me feel "smart".

It is all very humbling since I am only expected to know the command, "sudo emerge www-client/chromium" or whatever, and then I sit back and "pray" (to the gods of programming?) that all goes well, that these mother-scratchers know what they're doing.  I am amazed that there are so many super-intelligent monkeys who can pull off such feats as designing operating systems and motorcycles.

I am proud of myself if I can grow zucchini, squash, tomatoes, and cucumbers.

 

... and yet i can calculate and compute and follow some algebraic proofs.   I don't respect people who think having a Masters in Business Administration is a big deal.  I have more respect for, well, someone like myself, who has absolutely no "business-sense" and does not care to design bridges either.

I do like mathematics, a very limited amount that will keep the gears in my head spinning for the rest of my life, but I do not care for "NASA" and am totally not impressed at all with astronauts or people who study "rocket science".

In actuality, when the kids study trigonometry in high school, they are studying "rocket science" even if they won't even use that knowledge to earn a living constructing buildings.

One can know how to read and write and do a little math, but we have no choice but to be mystified by the complicated artifice all around us, at our fingertips even.  We are truly apes on wheels.

When I mention my interest in mathematics to a nosy passerby who inquires as to what it is I am "studying" there with my book on a stand and notebook on my lap, an immediate followup question is, "What are you studying for?  Why don't you attend a school or try one of those on-line courses?"

I become more than a little annoyed and irritated, and I don't feel like giving my life stroy, so I simply say that I prefer to study what I want, when I want, and how I want, and don't really think that there exists any "programmed" course of training.

Holden, like your namesake, I think there are many PHONIES in this world.

People that don't know what the hell they are talking about may be getting paid to tell others what to do with their lives.  They advise someone to attend community college or "upgrade their skills".

What if I want to learn how to calculate logarithms without a calculator?   Same with the arctangent.  I want to calculate that by hand as well?  Does the teacher know how?  Well, then why "upgrade my skills"?   What does that mean, anyway, be trained how to design "apps" for smartphones and smart televsions?  Build apps for the gorts?  Are they fuucking kidding me???

No, I want to build console programs in C/C++ and explore the computer algebra systtems Sage and SymPy created by the "scientific open source community".  They are the mysterious priests in my tribe, a tribe in which I have been playing the role of village idiot in.   

I'm "touched" in the head?   No.  I'm just an a-s-s-h-o-l-e who doesn't give a shiit anymore.
Would you think less of me if I went from "being drunk on math" to "doing math drunk"?   

I am as Raul describes those who watch tragedies unfold on the television news.  I am only concerned with the math book under my nose.  The thing is, I want to reach a point where I one can be upfront about being immune to the propaganda machine.  I want to think like a non-human creature.



What does it mean to say, "I don't want to cry for somebody else's need."   ?

In the context of:

"My Country I & II"

Something horrid....
I would not fight for my country
I would not work to no rule
Don't have no truck with no power game
Won't be some other jerk's tool
Don't have no part of no partisans
Won't have no part 'cos one party and another's all the same
(Workin' for some other man)
All the same
Gonna play it
(Workin' for some other man)
At my own game

I would not fight for my country
I would not work to no rule
Don't have no truck with no power game
Won't be some other jerk's tool
Don't have no part of no partisans
Won't have no part 'cos one party and another's all the same
All the same
Gonna play it at my own game

Don't wanna die for some old man's crusade
Don't wanna hear what they feed
Don't wanna kill for some cause of the age
Don't wanna cry for somebody else's need
Don't want no piece of no flag in the breeze
Don't want no part 'cos one party and another's all the same
All the same
Gonna play it at my own game

Ooooooh not me
You don't get me
Oi you

Ooooooh not me
Oi you
You don't get me
Oi you

[Note:  Also called Heron's formula.]

(1) In geometry we learn that the area of a triangle is given by the formula

K = (1/2)(base)(height)

I won't draw any diagrams, obviously, although I wish I could.  We don't have that kind of message board.

If we let the length of the base = b, and the height of the triangle = a*sin(C), we obtain:

K = (1/2)*a*b*sin(C)

To prove Hero's formula, K = sqrt[s(s-a)(s-b)(s-c)],
we first show that K^2 = (1/4)*a^2*b^2*sin^2(C)

That's kind of trivial:  K^2 = [(1/2)*a*b*sin(C)]^2 = (1/4)*a^2*b^2*sin^2(C)

(2)  In Hero's formula, s = (1/2)*(a + b + c) so that 2s = a + b + c


Hence   2*(s - a) = 2s - 2a = a + b + c - 2a = -a + b + c

2*(s - b) = 2s - 2b = a + b + c - 2b = a - b + c

2*(s - c) = 2s - 2c = a + b + c - 2c = a + b - c



(3) This part has 9 parts lettered a through i.


(a) a^2 + b^ - c^2 = 2*a*b*cos(C)


Why?  By the law of cosines, c^2 = a^2 + b^2 - 2*a*b*cos(C)

 So, a^2 + b^2 - (a^2 + b^2 - 2*a*b*cos(C)) =  2*a*b*cos(C)

(b) (a^2 + b^2 - c^2)^2 = 4*a^2*b^2*cos^2(C) = 4*a^2*b^2*[1 - sin^2(C)]


since sin^2(C) + cos^2(C) = 1, cos^2(C) = 1 - sin^2(C)


(c)  (a^2 + b^2 - c^2)^2 = 4*a^2*b^2*[1 - sin^2(C)]
= 4*a^2*b^2* - 4*a^2*b^2*sin^2(C)] = 4*a^2*b^2 - 16*K^2

In part (1) we defined K = (1/2)*a*b*sin(C) and K^2 = (1/4)*a^2*b^2*sin^2(C)

16*K^2 = 16*[(1/4)*a^2*b^2*sin^2(C)] = 4*a^2*b^2*sin^2(C

(d) 16*K^2 = 4*a^2*b^2 - (a^2 + b^2 - c^2)^2


Since (a^2 + b^2 - c^2)^2 = 4*a^2*b^2 - 16*K^2,

by the Additive Property of Equality, 16*K^2 = 4*a^2*b^2 - (a^2 + b^2 - c^2)^2


(e) 16*K^2 = [2ab + (a^2 + b^2 - c^2)]*[2ab - (a^2 + b^2 - c^2)]


reason:  x^2 - y^2 (x + y)*(x - y)


Therefore 4*a^2*b^2 - (a^2 + b^2 - c^2)^2 = (2ab)^2 - (a^2 + b^2 - c^2)^2
where x = 2ab and y = a^2 + b^2 - c^2



(f) 16*K^2 = [(a + b)^2 - c^2]*[c^2 - (a - b)^2]


Additive Property for Addition:

x^2 + 2xy + y^2 = (x + y)^2  and  x^2 - 2xy + y^2 = (x - y)^2


(g)  16*K^2 = [(a+b) + c]*[(a+b) - c]*[c + (a-b)]*[c - (a-b)]

Since x^2 - y^2 = (x + y)*(x - y)


(h) 16*K^2 = [2s]*[2*(s - c)]*[2*(s - b)]*[2*(s - a)]


By part (2), recall: 

2s = a + b + c

2*(s - c) =  a + b - c

2*(s - b) = a - b + c

2*(s - a) = -a + b + c

(i)  K = sqrt[s*(s - a)*(s - b)*(s - c)]


since 2*2*2*2 = 16 and K^2 = s*(s - a)*(s - b)*(s - c)

__________________________________________________

  

Heron did not pull that one out of his hat.

The other 6 formulas for finding the area of a triangle are much easier to show.