The basic pi generation algorithm (this is a pi generation formula and not a formula giving pi) is pasted below:-
1. (n!)/((n!)^n) factor was used. Extrovertedness. Jealousy.
2. comb(2n,n) factor was used. Measuring behaviour. Doubling.
3. 6/85. A rational number.
4. Iteration from 0 to 100.
2. comb(2n,n) factor was used. Measuring behaviour. Doubling.
3. ((n+2)^0.5)/((n+1)^0.5). Belief in irrational numbers. Staircase. Simple Machine.
4. 100000/860762. A rational number.
5. Iteration from 1 to 100.
1. (n!)/((n!)^n) factor was used. Extrovertedness. Jealousy.
2. comb(2n,n) factor was used. Measuring behaviour. Doubling.
3. 6/85. A rational number.
4. Iteration from 0 to 100.
Pi equation using staircase as simple machine is as below:-
1. (n!)/((n!)^n) factor was used. Extrovertedness. Jealousy.2. comb(2n,n) factor was used. Measuring behaviour. Doubling.
3. ((n+2)^0.5)/((n+1)^0.5). Belief in irrational numbers. Staircase. Simple Machine.
4. 100000/860762. A rational number.
5. Iteration from 1 to 100.
In the first case 1/pi is generated to 5 places. In the second case 1/pi is generated to 8 places.
All I have to do is to take care of transcendental property of pi (I am not sure what I mean though !!). I need to develop that block and attach it to second case and generate further places of pi.
33 comments:
I am currently learning j-invariant and Eisenstein series.
These are functions of complex numbers. They have modular forms.
While j-invariant works with Weierstrass elliptic functions; Eisenstein series expresses the complex function as series.
Concepts of numbers 1.
We go 1,2,3 etc when we look at similar things. Our brain is conditioned thus to say 1,2,3 etc. And to say things are similar. But in reality there are no similar things to minutest level. So there is no 2.
What surely happens is we shift our eyes or our fingers when we go 1 to 2.
This shift is pi.
Also 1 to 2 may be due to greed, temptation etc.
Concepts of numbers 2.
I wrote eating = food + i.happiness. i is complex number.
It is common that food can make us unhappy too. Fear of putting on weight.
So eating = food + i. unhappiness is also true.
We observe that eating has conjugates. Happiness and unhappiness. Also the moment we eat from happy state we become unhappy. Possibly because we have eaten more. So happiness and unhappiness alternates with food.
These are concepts of complex numbers.
Madhava of Sangamagrama did pi by 4 with alternating + and -.
Madhava gave us pi by 4 by complex analysis.
However pi is a real number. It exists.
The formula by Neelkanth Samyaji of pi =3 + 4/(2×3×4)+4/(4×5×6) . . . shows pi as being irrational. Irrationally long.
Then Ramanujan too gave pi formulas.
I have downloaded "Theory of complex functions" by Reinhold Remmert from internet.
I can possibly learn in pseudo-mathematical ways some of the concepts. I am confident because I experience neutrality, negatives, Taylor series and Reimann series naturally.
Reinhold Remmert says there are three things for calculating complex functions:-
1. Calculus approach - Cauchy method
2. Mapping approach - Reimann method
3. Power series approach - Weierstrass method
I feel Weierstrass method to be most appropriate.
There are two reasons :-
1. Eating = Food + i.happiness or its conjugate Eating = Food + i.unhappiness has a part Food. Now, humans are so advanced and stand on two legs that Food is also complex.
Food = Ingredients + i.Goodmood or its conjugate Food = Ingredients + i.Badmood.
Now, again, humans are so advanced and stand on two legs that Ingredients is also complex.
So it is a power series.
2. e^(i.theta) the formula by Euler is a complex function and is a power series.
For me, Weirstrass methos is most suitable as I want to restrict myself to pi and series only.
If e^itheta = cos theta + i sin theta, then
d/dx e^itheta = cos (theta+pi/2) + i sin (theta+pi/2)and
integral e^itheta = cos (theta+3pi/2) + i sin (theta+3pi/2)and
The above is like <-^->. In other words, the upper half of the axes system.
You may recall that Felix Klein's j-invariant is of top quadrants only. So the j-invariant complex function is suitable for my study as I am interested in e^itheta. And its study is in upper quadrant only.
Moreover, j(i) = 1728 is a crucial mathematical history. Ramanujan told Prof Hardy that 1729 can be expressed as 10 cubed + 9 cubed and also 12 cubed + 1 cubed.
So, currently j-invariant complex function study is a must for my progress in pi.
Also, I have physical copy of the book "Complex Analysis" by Purna Chandra Biswal.
So Wikipedia's j-invariant and the above book for now.
This finishes.
Some crazy stuff
To humans doubling comes easy. Folding a paper, symmetrical limbs, four fingers etc.
There is strong 1 too. One head, thumb, one hand being stronger etc.
Putting 1 and doubling a human may wonder about 3. And more numbers. He especially wants it to count cows, bags of grain etc.
But does a human have any organ in him which manifests numbers apart from strong 1 and doubling?
I doubt. Only doctors can say.
Crazy stuff continues. . .
So 3 and above are imagination. Possibly the term imaginary numbers foundation is this.
There's one thing that a human sees and wonders about. No he does not see 3. Or other higher numbers.
What he sees is his thumb. It seems to make 90 degrees. And Pythagoras theorem comes to our mind.
But in reality there is no hypotenuse. Only x and y axes. Thumb and pointing finger.
Crazy politics . . .
We Indians may think we gave bags of grain but got nothing. Or we allowed you to enter our region for tourism and we got nothing.
Europeans may say Indians see strong 1, doubling, x and y axes only. No further numbers or hypotenuse.
Americans may say we made advances with human doubling phenomenon. Digital technology.
What do you think?
Crazy politics continues . . .
Indians should not get disheartened. We gave bhakti to the world.
My desires . . .
Learn complex numbers and/or be with my family. Being with my family is more important.
My Reality
It is tough to explain what I want to learn in complex numbers. Perhaps only j-invariant. And at the most Eisenstein series.
And then European thoughts are so precise. It may be tough.
Philosophy of an office order
Why should a person obey an office order?
Bread. True.
Order from seniors. False. When a senior does not consider a junior as his junior. Why should I treat seniors as a senior?
English system. True.
Salem steel plant. True
As a challenge. True.
I have started to becoming a Hitler.
But naturally I am built for fame. And perhaps money. But not power.
Praising Anandji (of Kalyanji Anandji) should work. As he is famous. And Gujju. And Modiji is also a Gujju.
My response to world politics. Europeans hypotenuse and other contribution. Americans digital (doubling) contribution.
1. My dafli playing teaches me 1.(1+x=x squared);2.(1+x=x squared);3.(1+x=x squared); . . . . Fibonacci series.
2. 1,2,3,4,5,9,10. UK flag. e and pi expressions.
3. 2,3,4,5 and 3,2,4,5. Complex e series, Felix Klein theory.
I can visualise Laplace and Fourier transform.
My fundamentals in calculus are strong. Practice is less.
Moreover, I was unlucky. I cleared jamia engineering test in 12th. But failed interview.
I claim that my merit is good. But unlucky.
I have made about 7 rational expressions which are original.
I have made modified expressions for constants.
I have 1 or 2 original constant expressions.
I have been involved with measuring behaviour and Becoming extroverted for 8 to 9 years.
Peace and happiness
Thinking of Jesus and praying Shiva gives me peace and happiness.
This concludes
The problem as I see is in India people born to be famous are not able to make it big.
People gossip and most say and have bad opinions.
"Kys faydaa cartoon banake. Biwi to gurcharre uda rahi hai"
"Yeh dono bhai tho dhun achchi banate hain. Lekin suna hai in dono ke beech problem hai".
"Maths kar raha hai. Apne aap girega".
Amd power and money go hand in hand in India.
Argument with Europeans and Americans continue
I claim that I could have got i to 50 percent and tan theta to 60 percent. But perhaps not sin theta and cos theta.
Now the modern world and books have changed me.
What is money? If asked by European or American I would have told them you tell me. We are associated with marwaris.
Now in this blog post of "The last Pi", there are so many irrelevant comments. There are comments on mathematical concepts. But then there are comments on unrelated things as well. These happen due to frustration and unrelatedness 8n events.
As I wrote, a man is conditioned by education to expect 2,3,4 etc. after 1. When 2,3,4 etc. do not happen after 1 frustration creeps in. And when things happen 1,2,3,4,5 etc. to some people and to others it does not; then it causes pain, frustration. So some of the above comments.
Concepts of numbers 3
There are two things in this world as I see it. Fairness and unfairness. Measuring behavior.
If you look at a quadratic equations, it leads to complex number solution too. In fact, complex conjugates. Just as we have fair and unfair behaviour, we have a+ib and a-ib.
Concepts of numbers 4
I wtote 1.(1+x=x squared);2.(1+x=x squared);3.(1+x=x squared) . . . as Fibonacci series.
1+x = x squared is a fibonacci series equation and gives golden ratio.
Then there is another part to it. Suppose we write 1+x = x squared and in the next step 1+x squared = x cubed and then 1+x cubed = x to the power 4 etc.
What do we get?
Seems x, x squared, x cubed . . . etc.
This is important.
Finally
I wrote fingers and gaps as
1
10
100
1000
10000 etc.
Now, putting Concepts of numbers 1 to 4 and the above model of fingers and gaps will give us understanding of pi, e,sin and cos. Also the understanding of the most famous Euler equation e^i.pi + 1 = 0.
Moving ahead
Are there equations which gives complex numbers and complex conjugates in nature?
Are there any more understanding required with respect to complex numbers?
I am keen.
Concepts of numbers 5
tan theta is important and can be observed by a man. Just two limbs of a right triangle.
Now I had given equations on growth of rational numbers, their squares and their square roots. The growth on all the above can be expressed in terms of natural numbers. This is important from a calculus point of view as calculus is about slope or tan functions.
(Above . . .) And I claim that the limbs of a right triangle can be expressed as rational numbers.
Learning elliptic functions may be easier for me because of the two beats that I play on dafli.
This concludes.
Tan x expression of Colin McLaurin is a genius of highest quality. And also other trigonometry series.
That's the difference between Kasparov and others.
Expecting training from chess grandmasters is not feasible. Just 1900 rating and age is 50.
Expecting maths education is also not feasible. I am a salary earner.
But what is feasible is steel crystals math. Why this does not happen?
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