Author Topic: Mathematics for Jailbirds (Read 11283 times)

Mad Dog Mike · « **on:** March 20, 2018, 05:50:57 pm »

I found an inspiring 4 page paper from Italy called, Through the bars. Learning and Teaching Mathematics in Jail.

I was thinking, I wouldn't mind teaching high school mathematics to adults in jail as long as I was not an inmate myself, that is.

Through the bars. Learning and Teaching Mathematics in Jail
Aldo Frezzotti, Federico G. Lastaria, Stefano Mortola
Dipartimento di Matematica, Politecnico di Milano, Italia

Abstract

Teaching and learning mathematics in jail is a dramatic special instance of dealing with social barriers and cultural differences. Teachers and students involved in such experience are confronted with problems ----- lack of readable mathematics textbooks, need to simplify, motivate and synthesize the subject, need to cope with psychological and environmental difficulties that go much beyond the specific situation and are a worthy challenge for the whole community of mathematics researchers and educators.

Mad Dog Mike · « **Reply #1 on:** July 24, 2019, 05:39:52 pm »

The full-fledged C++ program, irroots (Irrational Roots), which shows the technique for approximating a solitary root between two successive integers, that is, the solution to f(x) = 0 for decimal and even decimal representations of irrational numbers: irroots3-2.cpp (queries for required degree of precision).

Compile with GNU g++:

g++ -g -Wall -std=c++11 irroots3-2.cpp -o irroots

./irroots

enter details of polynomial, starting interval, and required significant figures (precision)

Ibra · « **Reply #2 on:** July 24, 2019, 06:03:10 pm »

As I have GNU (MinGW) installed on my windows machine, I compiled your cmd program. although I don't get the calculation technique, I messed around a bit and it is really interesting to have such programs at hand with detailed commentary and step-by-step guidance. much appreciated Mr Hentrich. If I develop an interest in mathematics again, I would pick this technique of yours.

Mad Dog Mike · « **Reply #3 on:** July 24, 2019, 06:51:14 pm »

I found it in a 1965 text we used in high school in 1984 when they were pushing the first Hewlett-Packard and Texas Instruments handheld calculators, so our generation just missed the tail end of when instructors actually taught or even knew these wonderous techniques.

By hand, I use the long version, but it is less complicated (and even relatively smooth) to format a display for the short-form table.

What you do (for the technique displayed by the programs) by hand is write the coefficients of the polynomial from highest power term down to constant.

In the example with f(x) = x^2 - 5 = 0, where x = sqrt(5), we write (for long form):

(remember, the first approximation is x1 = 2)

2 | 1 0 -5 You can see that f(2) = 2^2 - 5 = -1
2 4 What we did: 1*2 = 2; 0 + 2 = 2; 2*2 = 4; -5 + 4 = -1
----------------
1 2 -1 = f(2)

This is standard synthetic division that they used to teach before the mass-scale use of hand-held scientific graphing calculators. Since around 1996 or so, some of the models of these little contraptions even have Computer Algebra Systems in them. The methods I am preserving (ironically, with digital programming using modern languages) are meant to be done by hand, armed with at most, a basic calculator for the tougher arithmetic.

The "repeated synthetic division" (actually "synthetic substitution" seems a more appropriate term) is an extension of the normal synthetic substitution, where we replace P = [-5, 0, 1] with, say, R = [1, 2] (all but the last term in 1 2 -1, that is, 1 2).

So, long form again for the second part (this is the second row in short-form table):

2 | 1 2 You see the operations?
2 1*2 = 2; 2 + 2 = 4, and 4 is the value of the derivative
---------- f'(x) = 2x at x = 2.
1 4 = f'(2)

If you ever become interested in approximating the zero between two successive integers, and wanted to reckon with minimal dependence on sophisticated computer algebra systems, this program or the dirty little Python (SageMath) versions could help teach you the method with a little additional research. I prefer the more direct and simplified Python functions since I felt more relaxed in "playing with the format" so as to see in both rational and decimal form. It makes the entire process of grinding through the arithmetic less of a chore and more exciting, to be honest. Also, I want my results to be correct. By the time I am done with this, doing these by hand might become second nature and I could easily forget when and how I picked this up. This C++ version does everything from scratch. The python functions can be used on the fly.

Some people might suspect that I have entirely too much time on his hands.

Anyway, you would repeat the process a couple times to get about 4 to 8 significant figures.

The next value is x2 = x1 - f(x1)/f'(x1) = 2 - f(2)/f'(2) = 2 - (-1)/(4) = 2 + 1/4
= 2.25

So, we then use 2.25

2.25 | 1 0 -5 2.25 | 1 2.25
2.25 5.0625 2.25
------------------------- ---------------
1 2.25 0.0625 = f(2.25) 1 4.5 = f'(2.25)

Then x3 = x2 - f(x2)/f'(x2) = 2.25 - f(2.25)/f'(2.25)

= 2.25 - 0.0625/4.5 = 2.25 - 0.01389 …. or you can think 9/4 - 1/72

The result is about 2.236, or exactly 2.23611111111111 ….. well, can you see we can never have an exact decimal representation and must be satisfied with VERY CLOSE approximations?

Isn't great to be able to apply Newton's Method and "The Calculus" without actually "doing" any calculus. Working through the synthetic division twice, placing in appropriate places in the table, provide the value of f(x) at x and f'(x) at x in a mechanical manner.

Take care, Ibra.

I hope I was able to clear things up with the longer form.

The short form would only list this:

2 | 1 0 -5
1 2 1 = f(2)
1 4 = f'(2)

At which point you have the necessary values to apply the Newton-Raphson method:

nextX = x - f(x)/f'(x)

For the C++ version, I used f(x) = 2*x^3 - 6*x^2 + 6*x - 3 = 0

So you enter 2 -6 6 then -3 (constant).

This creates P = [-3, 6, -6, 2]

I choose 0 and 3 as first interval. This allows the program to find zero between 1 and 2 and then from there it continues, with each next approximation doubling the degree of precision. The thing with this C++ version is that it does everything, and I want something lighter that I just use for each repeated synthetic division and subsequent application of "Newton's method."

Mad Dog Mike · « **Reply #4 on:** July 24, 2019, 10:12:37 pm »

For the example used with the Python (sage) code:

mwh@coyote2:[~]:
$ ----> irroots2

What is the degree of your polynomial? 2
coefficient of x^2: 1
coefficient of x^1: 0

constant term: -5

endpoints of interval to be searched: 1 4

Enter the LEAST number of significant figures required: 4

x | | f(x)
---------------------------------------------------
0 | 1 0 -5
1 1 1 -4
2 1 2 -1
3 1 3 4

There is a zero between 2 and 3.

x = 3 | | f(3)
---------------------------------------------------
1 3 4

x = 3 | f'(3)
---------------------------------------------------
| 1 6

row2.at(1) = 6 = dy = 6

Next X =
x - f(x)/f'(x) = 3 - [f(3)/f'(3)]
= 3 - (4/6) = 2.33

CHOOSE X: 2.33

With precision 1, x = nextX = 2.3 = 2.3

The zero is approximately 2.3

Is this approximation satisfactory <Y/N> ? : n

Our two consecutive integers have been found.

Previous value of x = 3.0 nextX = 2.3
We've updated x = nextX = x - f(x)/f'(x).

current value of x = 2.3
Since dx = 1.0 and x2 = 4.0 we're done
applying the Location Principle, and now proceed to
apply Newton's method using repeated synthetic division.

Press ENTER [ <-----||| ] to continue.

x = 2.3 | | f(2.3)
---------------------------------------------------
1.000 2.300 0.290

x = 2.3 | f'(2.3)
---------------------------------------------------
| 1.000 4.600

row2.at(1) = 4.600 = dy = 4.600

Next X =
x - f(x)/f'(x) = 2.3 - [f(2.3)/f'(2.3)]
= 2.3 - (0.290/4.600) = 2.23696

CHOOSE X: 2.2370

Approximate X: With precision (2^RSD) = (4), nextX = 2.2370
The zero is approximately 2.2370

Is this approximation satisfactory <Y/N> ? : n

x = 2.2370 | | f(2.2370)
---------------------------------------------------
1.00000000 2.23700000 0.00416900

x = 2.2370 | f'(2.2370)
---------------------------------------------------
| 1.00000000 4.47400000

row2.at(1) = 4.47400000 = dy = 4.47400000

Next X =
x - f(x)/f'(x) = 2.2370 - [f(2.2370)/f'(2.2370)]
= 2.2370 - (0.00416900/4.47400000) = 2.2360681717
CHOOSE X: 2.23606817

Approximate X: With precision (2^RSD-1) = (7), nextX = 2.2360682
The zero is approximately 2.2360682

Is this approximation satisfactory <Y/N> ? : n

This approximation is already good to 8 figures.

There were 3 repeated synthetic divisions.

Z E R O S:

There is a zero between 2 and 3.

Our efforts yield: x = 2.2360682

________________________________________________________
For f(x) = 2*x^3 - 6*x^2 + 6*x - 3 = 0

mwh@coyote2:[~]:
$ ----> irroots2

What is the degree of your polynomial? 3
coefficient of x^3: 2
coefficient of x^2: -6
coefficient of x^1: 6

constant term: -3

endpoints of interval to be searched: 0 3

Enter the LEAST number of significant figures required: 7

x | | f(x)
---------------------------------------------------
0 | 2 -6 6 -3
0 2 -6 6 -3
1 2 -4 2 -1
2 2 -2 2 1

There is a zero between 1 and 2.

x = 2 | | f(2)
---------------------------------------------------
2 -2 2 1

x = 2 | f'(2)
---------------------------------------------------
| 2 2 6

row2.at(1) = 6 = dy = 6

Next X =
x - f(x)/f'(x) = 2 - [f(2)/f'(2)]
= 2 - (1/6) = 1.83

CHOOSE X: 1.83

With precision 1, x = nextX = 1.8 = 1.8

The zero is approximately 1.8

Is this approximation satisfactory <Y/N> ? : n

Our two consecutive integers have been found.

Previous value of x = 2.0 nextX = 1.8
We've updated x = nextX = x - f(x)/f'(x).

current value of x = 1.8

Since dx = 1.0 and x2 = 3.0 we're done
applying the Location Principle, and now proceed to
apply Newton's method using repeated synthetic division.

Press ENTER [ <-----||| ] to continue.

x = 1.8 | | f(1.8 )
---------------------------------------------------
2.000 -2.400 1.680 0.024

x = 1.8 | f'(1.8 )
---------------------------------------------------
| 2.000 1.200 3.840

row2.at(1) = 3.840 = dy = 3.840

Next X =
x - f(x)/f'(x) = 1.8 - [f(1.8 )/f'(1.8 )]
= 1.8 - (0.024/3.840) = 1.79375
CHOOSE X: 1.7938

Approximate X: With precision (2^RSD) = (4), nextX = 1.7938
The zero is approximately 1.7938

Is this approximation satisfactory <Y/N> ? : n

x = 1.7938 | | f(1.7938 )
---------------------------------------------------
2.00000000 -2.41240000 1.67263690 0.00037600

x = 1.7938 | f'(1.7938 )
---------------------------------------------------
| 2.00000000 1.17520000 3.78071060

row2.at(1) = 3.78071060 = dy = 3.78071064

Next X =
x - f(x)/f'(x) = 1.7938 - [f(1.7938 )/f'(1.7938 )]
= 1.7938 - (0.00037604/3.78071064) = 1.7937005384
CHOOSE X: 1.79370054

Approximate X: With precision (2^RSD-1) = (7), nextX = 1.7937005
The zero is approximately 1.7937005

Is this approximation satisfactory <Y/N> ? : n

This approximation is already good to 8 figures.

There were 3 repeated synthetic divisions.

Z E R O S:

There is a zero between 1 and 2.

Our efforts yield: x = 1.7937005

____________________________________________
Now I look forward to spending a few days with pencil, scratch paper, and having the code as a back-up so I don't become too irritable with the arithmetic operations. The goal is to be able to this by hand with maybe just a basic calculator or with pencil (or with one of those toys Holden found that kids use for doodling ... that too would work, drawing tables then erasing them. just using them for the values f(x) and f'(x).

I ought to mention that this method works only if neither the first or second derivative is = 0. That is, the slope cannot be a horizontal line.

Also, including rational numbers is not at all related to this, and I only added that aspect "for a cheap thrill".

Messing around with say 16 figures of precision gave a fraction like this:

-[450*2^(218/245) * 3^(4/35) * 5^(9/49) * 7^(148/245)] / 343

Believe it or not, this is -12.000304

This has nothing to do with anything, really, but I find it interestingly weird, this different representations for the same "number". For the kind of precision we want (4 figures), the rational numbers remain "tame". When I increase the precision unnecessarily, the fractions get "weirder."

Breaking it down: 2^(216/245) = 1.84245917558579
2^(216/245) * 450 = 829.106629013606

3^(4/35) = 1.13377830746771, so [2^(216/245) * 450 * 3^(4/35)] = 940.023110553305

5^(9/49) = 1.34394732031821, so [2^(216/245) * 450 * 3^(4/35) * 5^(9/49)] = 1263.34154046530

7^(148/245) = 3.23972542174317, so [2^(216/245) * 450 * 3^(4/35) * 5^(9/49) * 7^(148/245)

= 4092.87970498962

Lastly, inserting the negative sign and dividing by 343: -4092.87970498962/343 = -11.9325938920980

and yet, (-450*2**(218/245)*3**(4/35)*5**(9/49)*7**(148/245)/343).n() =-12.0003040000000

0.0677101080000000 difference is considerable round-off error.

The Lesson? When programming machines, stick to decimals. When working by hand, suit yourself working with fractions until they become a burden, then switch to decimal format. Do not be ashamed to check work with calculator as arithmetic errors are quite common for us. No amount of study will make arithmetic any "easier." That's why I call it "work." So I am usually always "working on something," but not somebody's "code-monkey" to be handed a "to do list."

The world may be uninterested in this old stuff mixed with new stuff, but the feeling is mutual, for I have very little interest in what the masses seem to find so "thrilling".

I don't push this stuff on anyone, and I strongly suspect the math has made me even a little crazier than I would be otherwise. That is, I can become excited about gaining understanding of this material, just going with the flow, growing inwardly, but simultaneously reflecting on how rapidly each of us will vanish from this earth.

Ibra · « **Reply #5 on:** July 25, 2019, 03:59:26 am »

Thanks Hentrich for detailed explanation. I still need to dig the theory behind to get the hang of it.

Quote

So I am usually always "working on something," but not somebody's "code-monkey" to be handed a "to do list."

I like this sentiment against instrumentalism.

stay safe and keep glitching this system/matrix.

Mad Dog Mike · « **Reply #6 on:** July 25, 2019, 12:23:34 pm »

For the example used with the Python (sage) code, with the C++ program:

mwh@coyote2:[~]:
$ ----> irroots2

What is the degree of your polynomial? 2
coefficient of x^2: 1
coefficient of x^1: 0

constant term: -5

endpoints of interval to be searched: 1 4

Enter the LEAST number of significant figures required: 4

x | | f(x)
---------------------------------------------------
0 | 1 0 -5
1 1 1 -4
2 1 2 -1
3 1 3 4

There is a zero between 2 and 3.

x = 3 | | f(3)
---------------------------------------------------
1 3 4

x = 3 | f'(3)
---------------------------------------------------
| 1 6

row2.at(1) = 6 = dy = 6

Next X =
x - f(x)/f'(x) = 3 - [f(3)/f'(3)]
= 3 - (4/6) = 2.33

CHOOSE X: 2.33

With precision 1, x = nextX = 2.3 = 2.3

The zero is approximately 2.3

Is this approximation satisfactory <Y/N> ? : n

Our two consecutive integers have been found.

Previous value of x = 3.0 nextX = 2.3
We've updated x = nextX = x - f(x)/f'(x).

current value of x = 2.3
Since dx = 1.0 and x2 = 4.0 we're done
applying the Location Principle, and now proceed to
apply Newton's method using repeated synthetic division.

Press ENTER [ <-----||| ] to continue.

x = 2.3 | | f(2.3)
---------------------------------------------------
1.000 2.300 0.290

x = 2.3 | f'(2.3)
---------------------------------------------------
| 1.000 4.600

row2.at(1) = 4.600 = dy = 4.600

Next X =
x - f(x)/f'(x) = 2.3 - [f(2.3)/f'(2.3)]
= 2.3 - (0.290/4.600) = 2.23696

CHOOSE X: 2.2370

Approximate X: With precision (2^RSD) = (4), nextX = 2.2370
The zero is approximately 2.2370

Is this approximation satisfactory <Y/N> ? : n

x = 2.2370 | | f(2.2370)
---------------------------------------------------
1.00000000 2.23700000 0.00416900

x = 2.2370 | f'(2.2370)
---------------------------------------------------
| 1.00000000 4.47400000

row2.at(1) = 4.47400000 = dy = 4.47400000

Next X =
x - f(x)/f'(x) = 2.2370 - [f(2.2370)/f'(2.2370)]
= 2.2370 - (0.00416900/4.47400000) = 2.2360681717
CHOOSE X: 2.23606817

Approximate X: With precision (2^RSD-1) = (7), nextX = 2.2360682
The zero is approximately 2.2360682

Is this approximation satisfactory <Y/N> ? : n

This approximation is already good to 8 figures.

There were 3 repeated synthetic divisions.

Z E R O S:

There is a zero between 2 and 3.

Our efforts yield: x = 2.2360682

________________________________________________________
For f(x) = 2*x^3 - 6*x^2 + 6*x - 3 = 0

mwh@coyote2:[~]:
$ ----> irroots2

What is the degree of your polynomial? 3
coefficient of x^3: 2
coefficient of x^2: -6
coefficient of x^1: 6

constant term: -3

endpoints of interval to be searched: 0 3

Enter the LEAST number of significant figures required: 7

x | | f(x)
---------------------------------------------------
0 | 2 -6 6 -3
0 2 -6 6 -3
1 2 -4 2 -1
2 2 -2 2 1

There is a zero between 1 and 2.

x = 2 | | f(2)
---------------------------------------------------
2 -2 2 1

x = 2 | f'(2)
---------------------------------------------------
| 2 2 6

row2.at(1) = 6 = dy = 6

Next X =
x - f(x)/f'(x) = 2 - [f(2)/f'(2)]
= 2 - (1/6) = 1.83

CHOOSE X: 1.83

With precision 1, x = nextX = 1.8 = 1.8

The zero is approximately 1.8

Is this approximation satisfactory <Y/N> ? : n

Our two consecutive integers have been found.

Previous value of x = 2.0 nextX = 1.8
We've updated x = nextX = x - f(x)/f'(x).

current value of x = 1.8

Since dx = 1.0 and x2 = 3.0 we're done
applying the Location Principle, and now proceed to
apply Newton's method using repeated synthetic division.

Press ENTER [ <-----||| ] to continue.

x = 1.8 | | f(1.8 )
---------------------------------------------------
2.000 -2.400 1.680 0.024

x = 1.8 | f'(1.8 )
---------------------------------------------------
| 2.000 1.200 3.840

row2.at(1) = 3.840 = dy = 3.840

Next X =
x - f(x)/f'(x) = 1.8 - [f(1.8 )/f'(1.8 )]
= 1.8 - (0.024/3.840) = 1.79375
CHOOSE X: 1.7938

Approximate X: With precision (2^RSD) = (4), nextX = 1.7938
The zero is approximately 1.7938

Is this approximation satisfactory <Y/N> ? : n

x = 1.7938 | | f(1.7938 )
---------------------------------------------------
2.00000000 -2.41240000 1.67263690 0.00037600

x = 1.7938 | f'(1.7938 )
---------------------------------------------------
| 2.00000000 1.17520000 3.78071060

row2.at(1) = 3.78071060 = dy = 3.78071064

Next X =
x - f(x)/f'(x) = 1.7938 - [f(1.7938 )/f'(1.7938 )]
= 1.7938 - (0.00037604/3.78071064) = 1.7937005384
CHOOSE X: 1.79370054

Approximate X: With precision (2^RSD-1) = (7), nextX = 1.7937005
The zero is approximately 1.7937005

Is this approximation satisfactory <Y/N> ? : n

This approximation is already good to 8 figures.

There were 3 repeated synthetic divisions.

Z E R O S:

There is a zero between 1 and 2.

Our efforts yield: x = 1.7937005

____________________________________________
Now I look forward to spending a few days with pencil, scratch paper, and having the code as a back-up so I don't become too irritable with the arithmetic operations. The goal is to be able to this by hand with maybe just a basic calculator or with pencil (or with one of those toys Holden found that kids use for doodling ... that too would work, drawing tables then erasing them. just using them for the values f(x) and f'(x).

I ought to mention that this method works only if neither the first or second derivative is = 0. That is, the slope cannot be a horizontal line.

Also, including rational numbers is not at all related to this, and I only added that aspect "for a cheap thrill".

I don't know what compelled me to do that, but I can see that, after a certain point, the decimals may be easiest.

I can see myself "fuucking around with" (tinkering with) the rational version just to explore options, but the decimals are far easier to manipulate as f(x)/f'(x) gets smaller and smaller. On the other hand, the fractions become uglier and uglier, especially when rounding off to higher and higher levels of precision.

Mad Dog Mike · « **Reply #7 on:** July 25, 2019, 05:34:09 pm »

Which would you rather calculate by hand, (1.167)^2 or (7/6)^2 ?

In approximating "irrational" numbers, we are always dealing with an approximation, no? And yet, what I have to remind myself of constantly is that, while rational numbers are EXACT, like 7/6 or 1/3, their decimal representation is also only an approximation.

7/6 does not equal 1.167

7/6 = 1.166666666666666 ... I would write 1.16 and draw a little horizontal hat over the 6. The name of the flat extendable horizontal line over the "repeating digits" must be the same as the name for the horizontal line over a conjugate or the horizontal line going over the radicand after the radical symbol. That word is vinculum. I learned this word while researching to build the Old School Square Root Algorithm Teacher. I imagine I can call the line a vinculum. Whatever it is, it is a useful little symbol for showing repeating decimals, and I do use it.

I also miss having the '=' symbol with a dot over it. This means the same thing as the "wavy" equals sign. I could make one up just for here, one I use in some of my programs to mean "is approximately": ~=~

Well, then I was thinking: hmmmmm ... With the radical symbol and the number 2 as radicand, we also have an EXACT representation of the irrational number sqrt(2). The decimal representation is an approximation. We never forget that.

The thing is, neither sqrt(2) or 7/6, one irrational, the other rational, can be exactly represented with a decimal number. Only decimal fractions can be represented as such, that is, fractions where the denominator is a power of 10. Every terminating decimal representation can be written as a decimal fraction. The thing with the irrational numbers is that the digits that go on and on do not have repeating digits. If they did, the number can be expressed as a ratio of two integers. Since every approximation we can produce has terminating digits (we cut them off at a specific number of figures), all our approximations to irrational numbers are RATIONAL.

Even without the programming technicalities, the very nature of the difficulties can be seen when working with paper and pencil. It's just more in your face on the programming level. .

Any result we produce with the Newton-Raphson method, whether with computer programs or with paper and pencil, x2 = x1 - f(x1)/f'(x1), to whatever number of significant figures, is technically a decimal fraction. No matter how large the decimal number you show me, I can represent it as a rational number, the ratio of one integer to another. The key idea is stated at the end of the Wiki article [I had forgotten to link to]: The decimal representation [of IRRATIONAL numbers] neither terminates nor infinitely repeats but extends forever without regular repetition.

When I say sqrt(17) is 4.123, I have to follow that with "to four significant figures."

Otherwise 4.123 is an extremely close approximation to rational number 771 / 187 = 4.1229999999999, with repeating digit 9, but 4.123 = 4123/1000, exactly, un punto. This is a decimal fraction.

sqrt(17) is closer to 4.12311111111111111111111111111111... = 37109 / 9000
than it is to 4.123 = 4123/1000 = (4123*9)/(1000*9) = 37107/9000 < 37109/9000

WAIT! Do you see? There is a number between them (there always is!): 37108 / 9000 = 9277 / 2250, which is also 4.12311111111111

I have built several programs to help me transform back and forth between the representations. The cool thing is that, I need not refrain from using any of my programs when exploring an exercise. I work it out by hand and then some!

sra-cl 17 20 ------> sqrt(17) ~=~ 4.12310562561766054982 ...

The digits go on and on without repetition.

But the number before the ..., the number 4.12310562561766054982, can be written as a decimal fraction since it does in fact terminate, hence the ~=~ or the ...

How many have gone mad thinking about these things? And yet, consider how connected this is to infinite series ... The entire number line is this Great Abstraction where there are an infinite amount of fractions. You keep zeroing in and zeroing in ...

Once again, just thinking on the keyboard.

Mad Dog Mike · « **Reply #8 on:** July 25, 2019, 07:23:05 pm »

Ibra,

If you were to use the C++ program, understand that, at the end, when it declares, "already good to such and such figures," I would change the code eventually to compare the last two results and only report number of significant digits as the number of digits which are the same. It's a technicality, but one worth mentioning. The way Theoretically, the figures should double, so I proceeded with 1, 2, 4, 8, 16, ...

But, it appears that when 16 figures are displayed, only 6 are the Real McCoy.

So, if you cut the code some slack in that respect, I can correct such things later.

For the exercises, rather than solving each one way, I am offering different versions of each exercise (for this section) in my own personal solution key (a long series of composition notebooks). All the while I have these passionate philosophical emotions about the "fuzziness" at the edges of the structures on which these methods are based.

As for the ideas leading up to this method, they may have evolved from the method of "false position," or interpolation.

I will not ramble on at this time since I have difficulty (frustrations) typing about math. It's like working in a straight-jacket. I'm going outside to scribble with a clipboard and scrap paper, working out the exercises. I then wish to transcribe the solutions in a notebook, but not in the manner of the Solution Key. I want to add commentary and show where students might exclaim, "Fuuck this."

as in multiplying 1.167 times itself. ).

I want to assure them that others have been as frustrated with "approximating irrationals numbers". You always end up with a decimal fraction and no irrational number.

I know you understand these frustrations as I recall you mentioning them here.

So, we approximate the value of an irrational number with a rational number.

It's like pulling a hat out of a hat and calling it a rabbit. $:-\$
_________________________________________________________

I will explore starting off with the rational form which, for numbers with fewer digits at least, are a little more elegant (to me, personally, at least). At the point where the by hand divisions and multiplications get too ugly, a return to the table style with decimals can be worked out.

Trust me, I understand that for quite a long time now calculating contraptions have been used to assist, not only the great masses, but the students as well. In 1965, I suppose the students would have had access to something capable of helping with the arithmetic. We were encouraged (in the early 1980's) to use any scientific calculator we could afford. There were no graphing calculators, but I find the current ones remarkable, like powerful computers.

Still, though, there is a certain aesthetic pleasure derived in scribbling with pencil, creating tables of data to construct of graph, learning some of the old ways for the pure enjoyment. It's one of the few fun things for the human animal to do to amuse its otherwise bored, disgruntled, and anxiety-stricken soul.

You can visualize the roots and see them. Schopenhauer would have chuckled at the visualizations. That's how he saw things, visually. I learned the most about the Pythagorean Theorem in a couple pages of Schopenhauer's visual explanation. That's how I try to see things too, but often I am caught up in the whirl and tumble of actual numbers. It's how I am wired as a mental creature. Like Holden, I learn a great deal witnessing specific numbers flowing through rules and methods.

Maybe The Buddha of Berlin would have sternly but tenderly advised me against too much arithmetic calculation and stress over not only real-time limits in human mental capacity in general, but just the abuse of the brain for such strenuous activity ... But - to me, I sometimes catch myself actually having something that can be called "fun" - that is, I am enjoying my own company, enjoying the brain's "activity." This is what Schopenhauer himself advises us to do. He might see that this is my way of enjoying my own company, and that I am using what I have been picking up throughout my life, studying in between decade long drinking bouts/downward-spirals.

I am learning ways to take some of the drudgery of hand computations by approaching exercises in various ways. Rather than solving directly in a robotic manner, taking my time, building code, not merely to serve as an assistant/guide/reminder-of-method, but, having forced myself to write the code, I had to first understand what was going on.

Mad Dog Mike · « **Reply #9 on:** July 27, 2019, 04:50:35 pm »

For jailbirds without a ton of scrap paper, while the repeated synthetic division technique was a great guide for programming a machine to do our dirty work (the unavoidable arithmetic), it is most likely best to use direct substitution directly.

Nowadays, a calculator might even be painstakingly annoying to use, and one would lazily but understandably prefer to write a quick and dirty function in SageMath or some other computer algebra system, such as:

f(x) = x^5 + 3

def nextX(x1):
return x1 - f(x1)/diff(f(x), x).subs({x:x1})

So, if we need to approximate the 5th root of -3: f(x) = x^5 + 3 = 0, f'(x) = 5*x^4

x2 = x1 - f(x1)/f'(x1) = x1 - (x1^5 + 3)/(5*x1^4), which can be simplified to

(1/5)*[ 4*x1 - (3)/(5*x1^4)

Then just substitute directly a few times. By hand you would create this formula for a few approximations and be done. With a CAS:

First think: 5th root of -3 < 5th root of -1 = -1
and 5th root of -32 = -2 < 5th root of -3, so we know the zero x is -2 < x < -1

Starting with, say, somewhere in between, like -3/2 = -1.5:

sage: nextX(-3/2)
-178/135

sage: nextX(-3/2).n()
-1.31851851851852

sage: nextX(-178/135)
-849280612597/677616202800
sage: nextX(-178/135).n()
-1.25333575125220

sage: nextX(-5/4)
-3893/3125
sage: nextX(-5/4).n()
-1.24576000000000

sage: nextX(-1.24).n()
-1.24578415867100

So, we would say that the 5th root of -3 is about -1.246 (to 4 significant figures)

The thing is, my brothers, that, while this answer agrees with the Solution Key, I will note in my own personal solution notebook that the 5th root of -3 has 5 roots, four of which are complex numbers with a real part and an imaginary part. There is one "real" root, and that is "negative 5th root of 3".

By Descartes Rule (sign changes), you see there are no sign changes in f, so there are no positive real zeros (zeros = roots ----> f(x) = 0, also called a "solution").

Also by Descartes Rule, f(-x) = (-x)^5 + 3 = -x^5 + 3 has one sign change, and hence 1 negative real root (real zero).

Holden · « **Reply #10 on:** July 27, 2019, 06:46:08 pm »

I learned the most about the Pythagorean Theorem in a couple pages of Schopenhauer's visual explanation.-Herr Kaspar

I would like to say that Schopenhauer has had a similar effect on me too.I mean,his description of the world as representation and four fold roots of the principle of sufficient reason,rightly viewed, could only have been written by someone who understands what's going on in the field of mathematics.

In fact,I have been working on something for close to an year now which is inspired ,at least partly, by Schopenhauer's views on logic and mathematics.
It may also be noted that Kant,whom Schopenhauer admired, was a very capable logician and a physicist.

But for now,I would like to continue to quietly work on this philosophico-mathematical project on my own,I have a gut feeling that I am heading in the right direction with Schopenhauer's philosophy as the gold standard.Frankly, I don't possess the requisite comprehension of mathematics/programming to have any meaningful discussions with you.Yet.
If the project works out,and it would take me at least 2 years more under the best of circumstances,I might be in a position to have something meaningful to say to you on these matters.
(It all depends on my continuing to being alive, which I hope is not the case😃)

Also, let's keep in mind that symbolic logic is just like algebra and Russell came up with this the whole idea of analytic philosophy (I mean Principa Mathematica particularly,I remember you spent many hours pouring over it)partly because of calculus rationator of Leibniz.
Frege, primarily a mathematician,knew Kant like the back of his hand.

I find it fascinating how Russell's and Frege's interest in the foundations of mathematics bleeds into philosophy.

Let's remember that quite apart from their philosophical interests,both of them, particularly,Frege, was quite capable tinkerer of mathematics too.

Mad Dog Mike · « **Reply #11 on:** July 27, 2019, 09:56:34 pm »

(It all depends on my continuing to being alive, which I hope is not the case😃)
--------------------------------------------------------------------------------------------
I was just logging in to tell you that today I hardly felt like eating. I said to myself, I think I am getting tired of eating, shiiTTing, and computing.

I've always wanted to investigate the connection between Husserl and Frege.

I do not mean to alienate anyone with this talk of math and code. I am just being myself.

It's how I get through these days without succumbing to alcoholic inebriation, although, there are some days … Holden … there are some days I might like to "take the edge off," but I suppose I have learned the valuable lesson that I must lack the gene for processing the stuff. With my tendency to go without eating anyway, if you add alcohol to the mix, I am afraid my life might fall to pieces rather rapidly were I to practice the religion of Alcoholism again.

Begging for change so as to get the keys to the Church … (liquor store).

Do you remember Of Mice and Men when the big guy is daydreaming of a stable place to live and call his home? Well, I like to daydream about a little patch of woods where I might dwell in a small solar-powered hut growing pot plants and letting myself, well, just go to pot. But it's not like that, you see. No, there will be winters and hunger and the same old existential predicament. There's no escape, Brother Holden of Northern India.

I'm not for taking on any great philosophical project, but I understand how this might appeal to you. I am content to deal with the structures and methods as I find them. I am experimenting with ways to make the arithmetic inherent in much of mathematics less of a burden. There are intellectual snobs who will tell you that arithmetic is not "mathematics." They are full of shiit, and I suspect many professors in universities would be hard-pressed to show their work were they handed some run of the mill "pre-calculus" problems to work out. I used to read some works in critical theory, and it all started to sound, well, like total bullshiit.

Like I said, I am more content challenging myself with undergraduate mathematics, going on many tangents, learning what I can on the fly to be able to create supportive mathematical software for my own personal amusement.

Anyway, my own mother is sick of hearing Schopenhauer's name. The native (Blackfoot "American" Indian), JR Chiefstick, who I met in a tent city out in the Seattle area like 10 years ago, had told me that, when I was drunk, besides singing loudly, I would go on and on about Schopenhauer. Hell, I guess you could say I even love the man, just because he has been the most honest (with us all) about how he really sees this world. Like Silenus said in another post, even his eruptions of anger are steeped in humor. It's just so hilarious to witness a genius like that defiantly "not keeping his eye on the ball."

They're all chasing balls around like it matters. No wonder some athletes make efficient soldiers. The team, the platoon, the company, the herd … serving abstract notions such as "the State" or "the Almighty God".

I would not make a good soldier since I might be tempted to shoot the Boss, or the bullies in my own company. Hell no, I would not want to be "trained" or "disciplined." Isn't this a consequence of so-called bloody civilization? A huge hallucinated plantation where technological advancement requires ARMIES and PRISONERS, dare I say, etc... ?

The military needs servile scientists as well. They would like to trump up charges to put a bright youth in a jam, threatening he will get a bronze diick up his a-- if he doesn't join the ARMY (their exact words, not mine [1987]). Rather than join the military to get out of jail, I returned to my cell to continue reading Alexandr Solzhenitsyn.

One of my favorite lines by Schopenhauer is when he says that it is amazing how "the genius" will be abused (ganged up on) by a half a dozen blockheads. All that shiit cracks me up about this most radical of troublemakers from the land of some of my most recent ancestors.

an aside: Two Types of Mathematician

Quote

Broadly speaking I want to suggest that geometry is that part of mathematics in which visual thought is dominant whereas algebra is that part in which sequential thought is dominant. This dichotomy is perhaps better conveyed by the words “insight” versus “rigour” and both play an essential role in real mathematical problems.

There’s also his famous quote:

Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.’

I suppose I could not resist the Devil's offer, but this is why I go over Analytic Geometry so often … it is where algebra and geometry merge.

another quote from the aside:

Quote

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

We each live inside our own heads, and so it is best we become accustomed to being alone, and, most importantly, even thinking alone. Nothing that is so, is so - is now cliché. Nothing is what it seems. Who is a "mathematician"? Don't you get so tired of "professionalism?" It's like a fuucking priesthood, for God's sake.

I'm for celebrating the capacity to be alone, and to honor my bitterness and resentment by stating in no uncertain terms that the conspicuous consumption of the alphas, the celebrities and celebrity worshippers, does not impress me in the least.

It's not just sour grapes. I think we ought to give voice to our bitterness.

When an aunt mentions her granddaughter had graduated with a computer science degree and has a "programming job" with some company (ABC or some crap) [note the Holden-Caufieldesque angst], I proclaim, "she's not messing with the kinds of ideas I'm into. I'm nobody's code-monkey."

Any chance I have to state the unpleasant facts, I just can't control myself.

I like going through Stroustrup's C++ text, but I strongly resent the deference he pays to professionalism and "working on large projects with a team" - the whole idea that corporate industrial strength commercial software is somehow superior simply because it is more [GUI] sophisticated in appearance. Screw that. These computers can be programmed by C++, the language Stroustrup "created" out of "C with Classes," and that's why I study it, so I might become more creative in the command-line mathematics oriented "scientific hacking around" JUST FOR MY OWN PERSONAL USE. It's my right if I took the time to study, to use it, and I don't have to feel like a failure just because I'm not answering to some wealthy businessman. I do not feel inferior to some wealthy businessman, and I think my days of kissing up to an employer are long gone.

I'm simply DISGUSTED.

raul · « **Reply #12 on:** July 28, 2019, 09:38:09 am »

Hentrich,

When I see all these mathematical notations I indeed realize that being in the land of numbers is another universe. In my case I can only say that the only number I take into account is zero because I have zero expectations, zero health, zero in my head, zero everywhere. But that is life.

Drive safely and keep studying.

Mad Dog Mike · « **Reply #13 on:** July 28, 2019, 02:20:44 pm »

Hola Raul,

My quote for the "high school year book" (age 17 when we had to submit whatever goofy or relgious or "deep" -- even corny if you so please -- saying/lyric/aphorism which was supposed to be a reflection of your personality at the time): The one under my face in 1984?

"zero = infinity" or "zero equals infinity"

I remember how disappointed I felt when I saw how they wrote it.
I wanted them to have in purely mathematical form with the actual 0 and the actual oo.

I would have preferred: 0 = oo

where I just type 'o' two times to stand for infinity

I thought it profound, even mystical. Like some ancient pre-agricultural symbol-language (Kabbalistic? Hebrew or Scandanavian Runelore), these symbols may say different things to different people.

With the statement, 0 = oo (which is not true, by the way), my subconscious mind might have been attempting to articulate a relation not necessarily the "same" as EQUALS, that 0 and oo, zero and infinity, seem to be somehow on equal footing, not so much opposites as common sense night dictate. Inifinity is definitely NOT a number, but zero is a number, just not like any other goddang number in the universe of numbers. So, in this way, it kind of can be considered as living in the world of numbers and non numbers at the same time. For, wherever 0 appears, there is no number to be found, and it can often suck every number near it into it like a black hole. If 0 shows up in the denominator, well, that's not defined and is a show-stopper. We have the older cultures of the "eastern" hemisphere to thank for introducing this concept to the rest of the peoples of the earth.

What I am trying to say, Senor Raul, is that you have chosen the most powerful number of all.

As I said, infinity is just a point.

As far as points go, the infinity of one dimension just might coincide with the origin (0 point) of the next dimension.

So, as points, 0 and oo, seem to share an identity.

Mad Dog Mike · « **Reply #14 on:** July 28, 2019, 09:31:24 pm »

Quote from: Hentrich (1984)

0 = oo

I feel I must point out, just to be "responsible," that this statement is logically FALSE.

Obviously (?) zero does not equal infinity.

Ask a Computer Algebra System such as Sage (as defined my mathematicians transformed into programmers and so most likley even more meticulous about their definitions):

sage: 0 == oo
False

sage: 0/oo
0

sage: solve(0*x - oo == 0, x)
[ ] (means { }.the empty set, the void is NULL: no solution exists)

0 divided into however many parts is still 0. (Kind of mystical and logical at the same time, if that's possible).

Furthermore, if 0 = oo then, if f(x) = 0*x, f(1) == oo, but 0 times any number is 0, so f(x) = 0*x = oo has no solution, or, to be ultra-technical, the solution is { }, the empty set.

In Sympy [CAS]:

In [14]: 0 == oo
Out[14]: False

In [15]: solve(0*x - oo, x)
Out[15]: [ ]

The above means 0*x - oo == 0 |----> 0*x == oo |-----> x == oo/0 (which is undefined since, for any n, n/0 is not defined.

By the definition of what it means to "divide into parts," since you can not divide something into 0 parts, anything divided by zero makes no sense (nonsense).

And yet! In the short thread, Leaning Into Your Own Madness, in the last sentence of the seed post, I will quote in full, I mention Dostoevsky's CRIME AND PUNSIHEMENT - specifically, the comment about one's own "nonsense."

Quote from: gorticide

Just beyond the first glimpses of wretched misery, there is an Imaginary Dimension, a Secret Realm that most of us knew quite intimately as children. Just as I feel a serious bout of depression forming, suddenly it vanishes, or I should say, "I" vanish ... the ego vanishes ... the persona vanishes.

What is left is the Imaginary Protagonist who is apparently some kind of monk in his own private monastery of one. His "religion"? School Mathematics.

That's right. In this imaginary world of "make believe," exploring the properties of complex numbers, the mysterious square root of negative one, becomes some kind of religious rite ...

Is it important to watch the evening news?

Do I really need to know about all the tragic events?

There may have been a time not so very long ago when the acquisition of knowledge was viewed as "evil" - in some of the Western world religions ... you know, eating from the Tree of Knowledge Between Good and Evil ... or simply protecting oneself against hubris.

But I have found that the pursuit of knowledge actually does protect one against hubris, for when you become serious about focusing on one little area at a time, paying close attention to parts where you might have always been confused about for the longest time, well, the entire process is one of patience and humility - if anything, ego-destroying, not ego-inflating.

And is it so imaginary after all is said and done? Is my private inner world of mathematics somehow less real than the World Soccer Play-Offs or a Baseball World Series or Football Superbowl?

Really. I'm serious. Is it all the gambling taking place and the selling of products on the television commercials that make the world of professional sports somehow more legitimate than the world of mathematics that a lifelong student approaches daily?

Which world is more legitimate, the imaginary world of numbers and mathematical concepts in my head, or the world of blood, sweat, tears, television, and money that represents the world of professional sports?

Put another way, which is more real, the world of the crowd, the world of the stadium, or the world we carry around on our shoulders, the world of the unconscious when we sleep?

In Dostoyevsky's Crime and Punishment, in a talk in the prison, one of the characters insisted that one's own nonsense was more precious than another person's sense.

Maybe then, we might say that one's own madness is more precious than another person's sanity or common sense.

WARNING: possible run-on sentence (sorry Signore Fiddlefamiliare).

So, for what it's worth, 0 = oo was my own kind of nonsense. With an explanation of where I was coming from, one of the religious-order "brothers" told me, "That was Jesus's entire message, wasn't it? In death we merge with all."

Wow, so he got my nonsense. Even as he knew it was logically more than a little fuzzy, he was able to accept a mystical interpretation. By the way, Catholic "brothers" are not "priests," but more like teaching-order nuns. Also, it was significant to me that the brother who humored my madness was the "AP Calculus" instructor, Brother Bob (the one who was not able to reach me when we were supposed to be learning what was in Dolcinai's Modern Introductory Analysis, one of the texts which, 35 years later, I am spontaneously creating commandline oriented supportive math programs for). I was too precoccupied with a so-called nervous breakdown, possibly triggered by suppressd feelings about parents' divorce (in combination with mesculine and more than occassional alcohol abuse).

Another aside: The creators of the computer algebra systems SageMath and SymPy have defined infinity as a type of SYMPY: number / SAGE: ring.

Let's see:
SymPy

In [16]: type(oo)
Out[16]: sympy.core.numbers.Infinity

In [17]: type(0)
Out[17]: int

In [18]: type(-oo)
Out[18]: sympy.core.numbers.NegativeInfinity

SageMath:

sage: type(0)
<type 'sage.rings.integer.Integer'>

sage: type(oo)
<class 'sage.rings.infinity.PlusInfinity'

sage: type(-oo)
<class 'sage.rings.infinity.MinusInfinity'>

The bottom line is that 0 != oo. Maybe I was just trying to stir up trouble. Maybe I was out of my fuucking mind.

Philosophers in Rags :: We dive deeper, and come up muddier.

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Author Topic: Mathematics for Jailbirds (Read 11283 times)

Mad Dog Mike

Mathematics for Jailbirds

Mad Dog Mike

Re: Mathematics for Jailbirds

Ibra

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Mad Dog Mike

using the C++ program

Ibra

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Mad Dog Mike

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Mad Dog Mike

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Holden

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raul

Zero is my favorite number

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