Expression of the day

I tried comb(2n,n) expressions. Many expressions on pi, square root of numbers and golden ratio were obtained. I used Wolfram Alpha to check(special thanks to Wolfram Alpha).

I tried Rational number series. Many expressions were obtained.

Today I decided to use both of them together. And what did I get ? Here we go . . . 

And here is the Wolfram Alpha screenshot.

I am deeply indebted to Wolfram Alpha. I mainly try mathematics only. Occasionally I also fire some Indian history queries too. It is amazing !!

Have a great day, dear blog readers !!

Back to back expressions

These are back to back expressions.

These are easy to verify. It is possibly 7th-8th class(India) material. Maybe some adults with not much of mathematics background may find them challenging. But I urge you to try them. You may have free time also due to COVID. Try these and verify.

But I feel the concept is BSc level. The concept of (mn+m-1)/(mn+m) - (mn-1)/(mn). Especially after so many (uncountable) results. Unfortunately they are not doctoral level. Otherwise I could have got a break. The gamma function result and the binomial results are advanced. But possibly only MSc level. But I am good in mathematics of electrical engineering. My basic qualification.

The expressions are 

Have a nice day, dear blog readers.

Twelve things that inspire my mathematics

I am writing an autobiography of sorts. 

There are twelve things that inspire my mathematics. These are

1. e^t. There are three different models of thinking I have on e^t.

2. 1/t!. I have done practical exercises to demonstrate the importance of 1/t!.

3. n/(n+1). I obtained an infinite series on it. This was my first major infinite expression.

4. (n+1)/n. Measuring behavior thought came up in 2010. (n+1)/n was an outcome of Measuring behaviour and also infinite series.

5. Comb(2n,n). Outcome of measuring behaviour. Two things. Fairness and unfairness.

6. Continued fractions. Vilvam (bael) tree was close to my thinking. I tried to represent Vilvam tree in continued fractions.

7.  (mn+m-1)/(mn+m)-(mn-1)/(mn). This expresses 3 consecutive numbers and 4 consecutive numbers concept. And more.

8. Geometric constructions. I was able to construct a pentagon and heptadecagon with a compass and straight edge.

9. Change produces area. My dafli playing teaches me this. Unfortunately my rhythm in calculus is not as strong as my rhythms in beats.

10. y to 1/y. I have a strong conviction here. God made integers and Pythagoras law becomes evident to me in this thread of thought.

11. Infinite series. Measuring behavior for fairness was made in 2010. And the concept of infinite series got laid.

12. #PythaShastri mantras. These are numeric utterances which makes harmony. Many of my #PythaShastri mantras have fetched results.

I thought of sharing my deep thoughts. Have a nice day dear readers !!

And now Gauss

Gauss made a brilliant equation. Integral of e to the power minus x squared for minus infinity to plus infinity equals square root of pi.

In place of x squared I put three consecutive numbers equation and four consecutive numbers equation. The result obtained is obedient. It is beautiful.

They are

I have used online Latex mathematical editor to type these.

Catalan's constant

I was working further on the complex function of mine.

I developed an expression. I tried it on Wolfram Alpha and the result show a term with Catalan's constant. It is indeed a privilege to get Catalan's constant in my expression.

Catalan's constant was first observed by Euler. However Charles Catalan did a lot of expressions and the constant is named after him. 

Below is the expression I made

Hooray !! A complex function !!

Made a complex function too. Here it is :

Have a nice day, dear blog readers !!

Hooray !! Integral and Sigma equivalence

 I have managed an integral and sigma equivalence. The equation is given below. The only other instance I know is from Parseval who did integral and sigma equivalence.

I am feeling ecstatic !!

Favorite four

 These are my favorite four expressions

You may try these for some random values and let me know if anything is wrong. I make errors inadvertently. So do let me know.

Have a nice day !!

Equations and COVID times

You take a look at the following.

If you have mathematical inclination then do the following.

1. Check for random value of n the validity of the four equations. These are approximately correct. You may put n as 17, 104 etc. 

2. Study the written text.

3. Ponder over the points written.

4. Kindly pass your comments.

5. Even if you check the four approximate expressions for values like 1, -1, 0, root 2 or i etc.; do write your comments.

My good wishes to you.

Two way reciprocals

 I am presenting below two expressions.

They are what I call "Two way reciprocals". The beauty of these two are unparalleled. They have a magnificent style and stands up with majesty like no other. 

 

Consider this equation

The other day I posted a blog post on Taylor series and Rational number series .

It is a great honor for me to have partly re-discovered this most amazing equation. As much as 70% of the world's system follow the Taylor series. It is that powerful.

Below I am pasting another expression. I feel it is damn important. And has unparalleled beauty as well. When I write damn important or unparalleled beauty I am not bragging. But I feel I am telling the truth.

Anyway forgive me if I hurt feelings due to the words in above paragraph.

The equation is

Sigma and Integral

The sigma operation came naturally to me. I made many expressions on sigma operations.

However, the integral operations did not happen. I worked on calculus yesterday night. I knew the basics. Then I managed to frame equations of similarity between sigma and integration. I took the help of Wolfram Alpha. Then I made expressions.

Here are the expressions.

Have a nice day, dear blog readers.

An expression on factorial

 Here I present an equation

I call it the walking man with arms and legs swaying in opposite ways. And its product gives factorial.

I hope you like it.

Have a nice day, dear blog readers.

Expressions on factorials

 Below I present expressions on factorials.

I hope you like them.

Have a nice day, dear blog readers.

Yet another finite expression

 Below I present three finite expressions on 1/(n+1). 

You may be aware of the following two infinite expressions on 1/(n+1)

But the one in top are finite expressions. Hope you like it.

Have a nice day, dear readers.

Equations at Ranchi

 

These equations were developed in Ranchi. They have been built from faith in God. I am just delivering his expressions. Just as no one realises that infinity is near to them; no one realises that God is still closer. 

I shall explain the significance:-

1. A finite sigma expression is damn tough. I do not know. Maybe infinite sigma expression is tougher. But I feel a finite sigma expression is tougher.

2. This combines the factorials with normal number in a mirror fashion.

3. This is an approximate expression that has been developed by sheer hard work. It took time. Understanding. But then it got developed.

4. I am taking the liberty to us comb(n,k). May I say it combines comb(n,k) with comb(k,1) in a beautiful way !!

This one is correct

In case there are errors in some of the equations, do let me know. Only this way can I correct myself.

Another expression

 I am presenting below another expression.

Approximate and Exact expressions

 

Expressions to stay cool during COVID-19

 COVID-19 is not going away. We must fight this pandemic by social distancing and maintaining and ensuring hygiene. It is better to be occupied with something.

Below are two expressions. You may verify them.

Have a nice day dear blog readers.

An Expression

 Can one write like this ?

This is another expression. 1/3 expressed in terms of 0.3+0.03+0.003+ . . . . . upto infinity. Even here the reciprocals have been used.

Two expressions

 Can one write like this ?

The above shows two expressions on reciprocals. I hope you like them.

Of Gauss's Constant G

I have belief in and fully in support of Gauss's constant G (0.8346 . . .)

There are three reasons for it.

First. 0.999 + (1/(root2 +1/(root3 +1/(root4 + . .  . is 1.8346 into 0.8346.

Second. root3/(root2+root5(root4+. . . .  is (9/8) into 0.8346 into 0.8346.

Third. My #PythaShastri mantra and my interpretation of Taylor series (for both general and special theory of relativity) together supports 0.8346. . .

Now look at this.

Wolfram results do not say it is Gauss's constant (G) in both the above cases.

I am indebted to Wolfram Alpha for making me a mathematician of sorts. But why don't they write it as G for above two cases ?

I firmly believe in Gauss and have full faith in Gauss's constant G. Gauss's constant is an undeniable truth of nature and life. 

What about you, dear readers ?

(My views - 0.8346 . . . is most certainly important.

 I am in India. I do not want to comment on this. 

Well when we talk of personality of mathematicians I have the deepest regards for Leonhard Euler. He is my god of sorts. 

And I find Niels Henrik Abel as most brilliant. A Diego Maradona of mathematics.)

My future Plan

The four consecutive numbers and the three consecutive numbers theory is over.

There is scope in solutions to quintic equations formed by four consecutive numbers and three consecutive numbers. Higher authorities need to think about it.

There is another topic. I wrote sigma of (number of destinations travelled by an office goer × significance of the destination) should equal the sigma of (number of pages written by an office goer × significance of the writing). There is scope in making complex functions with this idea. Legs and hands. The relationship. Higher authorities need to give it a thought.

If I look with "Insaaf ka tarazu"; "Haq mil raha hai". "Halka sa zyaada"."100 gram ya 200 gram zyaada". "Lekin unyay ?"

The woman(w), the man(m), the kid(k) and the god(z)

Here is the equation of Niels Abel.

I call it the woman(w), the man(m), the kid(k) and the god(z) equation.

I also feel it is the philosophy of adding.

My world of mathematics

Eminent Indian mathematician Shri C S Seshadri passed away recently. He founded the Chennai Mathematical Institute. He is known for his work in algebraic geometry. He is also known for Seshadri constant and Seshadri- Narasimhan theory.

My own research on Rational Number Series is going on. There are some very good results. I am not making them public. For fear of plagiarism. Also I am improving my prowess on mathematical equations manipulations. I lacked this. But now I am improving.

I hope to go on my own. I shall learn from internet whatever possible and I am inspired for my intuitions.

Have a good day dear blog readers.

An ambitious attempt

The rational number series was an ambitious attempt by me to make a mathematical symbol for India. Many of my blog readers may think I am being mad. But deep inside I could see a lot of connections in the series and the type of operation that I was trying.
Unfortunately, the initial paper itself did not take off. Most of the reputed Indian mathematical journals refused to publish my paper.
You can see that the rational number series gives polynomials too. Apart from approximate geometric progression. And also it can be used in prime numbers.

The best outside food I have ever had

The best Chole Bhature - It has to be the chole bhature of Nagpal in Lajpat Nagar, New Delhi. It is at a walkable distance from Moolchand Hospital.
The best Aloo Tikki - I feel the aloo tikki served in Bengali Sweets, South Extension, New Delhi is the best.
The best Tandoori item - I feel the chur chur naan served in Dwarka Sector 6, New Delhi by Raju Chur chur naan is the best.
The best Dahi Vada - The dahi vada served in Sagar Ratna (many branches), New Delhi is the best. The dahi vada is sweet.
The best dessert - I like the rabri falooda served by Gianis in Chandni Chowk, New Delhi is the best.

Equation of Life

More punch to Rational Series

It is possible to make the Rational number series more accurate and more precise.

One needs to add or subtract geometric progression of some other number terms.

I tried it on my rational number series and have met with success.

I like representing numbers this way. As Rational number series. I find it more meaningful.

But mathematicians know about it. Gauss has done work. But he and others talk of it in some other way. Moreover, Gauss was too great and was able to "smoothen" expressions. I manage to get a rough expressions of Gauss. While he was "smooth" and "so graceful".

My mind gets clean bowled by the terrific expressions of Taylor, Gauss and Maclaurin in particular. I have very deep admiration for Abel. And am awed by Euler.

I have desire to make one beautiful expression. But that is not to be. But a few practical expressions I have attempted. These are okayish, I guess.

Yet another set of two expressions

The two expressions are :-
1. ((n+1)^3/((n+2)^3)) + ((n-1)^3/(n^3)) is approximately 2 - (6/(n+2) + 2/((n+2)^2))
2. (n^3/((n+1)^3)) + ((n-1)^3/(n^3)) is approximately 2 - (6/(n+1) - 3/((n+1)^2))

Two approximate expressions

The two approximate expressions are :-
1.((n+1)^3/(n+2)^3)-((n-1)^3/(n^3)) is approximately 6/((n+1)^2)-12/((n+1)^3)
2. ((n^3)/((n+1)^3)-((n-1)^3/(n^3)) is approximately 3/((n+1)^2)-3/((n+1)^3)

These expressions are extensions of the Rational Number Series. A paper that I wrote.

Four approximate expressions

The four approximate expressions are :-
1. ((n+1)^2/(n+2)^2)+((n-1)^2)/(n^2)) is approximately 2 - 8/(2n+3)
2. (n^2/(n+1)^2)+((n-1)^2/n^2) is approximately 2 - 8/(2n+2)
3. ((n+1)^(1/2)/(n+2)^(1/2))+((n-1)^(1/2)/(n^(1/2))) is approximately 2-(1/(n+1) + 1/(4((n+1)^2)))
4. (n^(1/2)/((n+1)^(1/2))+((n-1)^(1/2)/(n^(1/2))) is approximately 2-(1/(n+1) + 3/(4((n+1)^2)))

If you have any suggestions or questions please feel free to write in comments.

Have a fine day !!

My case is strong

Relativity

God and Pythagoras theorem

The invention of i (of complex numbers)

First of all I would like to write that the invention of i is the second most beautiful thing in mathematics.
The first being trigonometric functions.
The man stood up. Defying and balancing gravity. And thereby creating y axis. And the cause for a symbol i (of complex numbers). It is not easy for a common man to "see" i. Just as he doesn't realise he can walk or stand on two legs.
The invention of i is of the 17th, 18th century.
It was Euler who used square root of (-1) in his expressions. He did not use i though. But square root of (-1).
It was Carl Gauss who advocated the use of i.
Well there are so many advantages.
To the power (x + iy) is no problem. As it is to the power x into to the power iy.
Addition is no problem. Just simple commutative law.
Subttraction no problem. Commutative.
Division is a little tough. But one can multiply with complex conjugate both the numerator and the denominator. In fact I believe conjugates are a reality and a person sort of experiences conjugates. Eating at first is a happy experience. But eating changes to unhappiness. Exercising at first is a happy experience. But exercises changes to unhappiness. So I believe. So I think.
My learning shall continue. I am interested in putting practical analogy to functions.
I read Prof. Hardy saying that Mathematics is more of an art like painting.
I believe in it.
(Please also refer to complex numbers.)

Honestly speaking

I feel like I am a Gauss in mathematics.
Carl Gauss was a very big name in mathematics. He made enormous contributions to the field of mathematics.
But one needs to introspect and try to do a right justice to oneself. And also my blog readers. Because my blog readers may be confused as to what I think of myself.

Bhool Chook Maaf . . .

As I understand

Being humorous of tilting the hour glass is quite okay to me.
Being critical of tilting the hour glass is quite okay to me.
Well my writing ( which I believe is 1,2,3,4,5,9,10(kirtivasan)) is humorous and/or critical.

Ramanujan had many LHS = RHS expressions.In fact, the continued fractions I thought critically was a great LHS = RHS expression.

I managed to make a few LHS = RHS statements. I am actually blessed with Wolphram Alpha online tool. Modern days. But during the time of Ramanujan even calculators were not there.

Edho mudinzhadhu ( Whatever I could)

Extended Poochi System

I wrote about Poochi system earlier.

And the beginning. Theta. Theta functions.

We all go 1,2,3,4,5 etc. carelessly habituated with primary school education. However, we must remember 1/1,1/2,1/3,1/4,1/5 etc. Rational numbers. 1 into 1/1 or 2 into 1/2 or 3 into 1/3 etc. are theta functions sorts. Ramanujan theta functions.

We must also remember the almost impossibility of finding things 100 percent similar. 1 to 2 is a difficult journey. Ramanujan tau functions.

1 second, 2 seconds, 3 seconds etc. are true. And so is 1 metre, 2 metres, 3 metres etc. Or 1 kg, 2 kgs, 3 kgs etc. These are of science and technology. And governed by scientific laws and functions and axioms and theorems.

1 rupee, 2 rupees are true. However 3 rupees is not. 3 rupees can be made with 1+2 or 5 -2. We must remember that. 

Co-operation or opposition is essential in money. So to speak.

My intention is not to create problems. Definitely not. God promise. But to make awareness in these difficult times.

In the image pasted above I have talked on

Extended Poochi System.

Hypotenuse.

Slanted hour glass.

Gravity aligned hour glass.

Hyperbola, ellipse, parabola and circle.

The times are difficult. I feel like stating whom I feel as the best genius. Well

1. Abdus Sattar

2. Chandrasekhar Subramanian

Among Indian mathematicians

1. Madhava of Sangamagrama

2. Srinivasa Ramanujan

Trimming your moustache

You want to trim your moustache.
You stand in front of a mirror.
You have scissors in your hand.
You want to size uniformly.
You look at the mirror and say to yourself go slow.
In case you had another mirror showing the reflection of the first mirror; then you would have said the trimming went well.
Yourself. Sizing. f(x).
First mirror. Slowly. f'(x).
Second mirror. Effect. Force. f"(x).
To me, this is the crux of Taylor series.

Gauss's constant G

Saurabh Shukla and Sanjay Mishra

I do not watch many movies these days. I have stopped a lot of internet videos too. Sometimes I watch Youtube.
Mathematics is what I desire. Well, Gauss is ahead. Certainly with Gauss constant, he is really high. I want to reach near him and learn what he said.
Anyways, there are certain actors in Bollywood who can carry the whole movie on their shoulders. They have that calibre; but are not heroes. Saurabh Shukla and Sanjay Mishra are two of them. Take for instance, the performance of Saurabh Shukla in Jolly LLB 2. Terrific. Outstanding. Or consider the performance of Sanjay Mishra in Newton. Simply great. Brilliant.
I wish to show my appreciation of these two in my blog.
Saurabh Shukla

Sanjay Mishra

Ramanujan and Continued Fractions

Better-half

Better-half is an England English word. The concept of the perpendicular bisector is due to this. The perpendicular bisector can help divide a line into any number of parts, with set-squares which can be manufactured. Possibly parallel lines can be drawn by masters as the online software GeoGebra is able to. I got it. One tangent to two semi-circles which sit on the same floor.
Perpendicular bisector is one of the top concepts of the world.
I was able to trisect an angle. GeoGebra has a tool for polygons. Hence it is possible to trisect an angle too.

Gauss error function

Constructibility of a polygon

Some Rational Ideas

1. n and 2n - n should be treated as π and 2n should be treated as π²  when one has to travel from finitely very big to infinity.
2. Simplest and hence perfect right triangle - The area of the simplest and hence perfect right triangle should be treated as zero, when one needs to prove theorems. This is my statement.
3. How many papers should an office-goer write? - This formula was developed to better rational thinking. One must choose the values of variables based on census of general populace without bias to say the least. Please refer to Number of pages to be written by office goers
4. Batthi logic -Batthi is a common jargon in SAIL. It's closest meaning in English is "outburst in anger with a sense of responsibility". Developing a logic for it is close to my thinking and I wish to make a formula for Batthi.(>, = , <; ceiling > floor for both positives and negatives). You have to trust your boss. And hope your absolute value is taken into consideration when issue is negative. Found a solution to Batthi. If the issue is negative and your absolute value is higher than your boss. You need not worry. You need to talk of accuracy. Something like "Where are you 100 percent correct sir?" or "The world is going much better and faster sir." etc. I hope you have got it.
5. Success of a committee - Two statements from the committee.
6. Wisdom - Wisdom is a bad word. Especially for young in heart people. It is like a young man shouting "I shall buy experience from a shop . . . ". Or an old man saying, " Nothing will change". A youngster is doomed !! I put forward this equation for Wisdom calculation.And that is Presence = Absence. Presence of me in Ranchi is absence of security for three young ladies at home in Delhi. (Or) Presence of an officer in office is absence of biscuits in canteen.(Or) My presence here in SAIL is absence of implementing my business plan.
7. Decimal point - Decimal point can be an irritating issue. "You are paying me 1 rupee while I have 10 million cells each wanting 1 rupee". Decimal point is a circle. 1,n squared, n cubed, n to the power of four etc. are of pi. But for n. However exercising max reps 10 times gives pi approximately. Belief in this point will give relief to irritation. If you believe that fingers are what you count with, then fine. Greater than 10? How? If you work hard and do 10 sets of max reps, then it is pi. But not in the case of n. So one can say work hard equivalent to 10 sets of max reps. then 1,10, n squared, n cubed, n to the power of four etc. all supports pi. This finishes suitably. And in a friendly manner. Databases and computer systems have ceiling and floor functions with scientific theories, axioms and lemmas backing them. So no problem in computer programs. One can argue this way too. Ceiling (LHS)= Ceiling( RHS). If asked to distribute 10 rupees among 3; then question how he obtained 10. This finishes.

My Concept of Past, Present and Future

My search is over

For the last few months I was not getting an idea as to what is it that I was looking out for in mathematics of my own type.
I somehow felt that most of the thoughts must be related and there has to be a simpler explanation for all. The constant π was important and most of the explanations of my world was explained by it. But what about thoughts? I used to think.
Then gradually I began to work with Wikipedia myself and started doing my own research. Gradually my learning and knowledge increased. I learnt of periodicity. And I began to understand the importance of elliptic functions.
Then yesterday I came across Gauss constant G. I knew there is something called as Gauss constant from the book on Mathematical constants by Steven Finch. But I never read about it.
Gauss constant is what I was looking out for. It is amazing that Gauss constant is observed in so many functions. Beta, gamma, lemniscate, Jacobi theta functions, sin(x), cos(x) and more.
It is an amazingly beautiful and a prolific number and is as useful as and as great as π.
Hats off to Carl Gauss for showing us the way with his constant, Gauss Constant G.

Tears to Joy

When my paper on "Rational Number Series" was refused by most of the important mathematical journals in India, I was in tears. I prayed to God that at least this journal should accept. Or that should accept. But none of the reputed journals found my paper suitable. I cried. 

Then, JETIR accepted my papers. Rational Number Series and Expressions for 1 got published in their journal. 

Now I tried IJMTT. A mathematical journal based in Trichy, Tamil Nadu. They accepted my paper on "Expressions of Combinations". So, a journal based on mathematics have finally accepted my paper. I am in joy. I shall write the details of the paper and provide a link later.

Good wishes to all my blog readers.

How much land does a man need?

How much land does a man need?
How many papers should a man write in a day in office? - My question
When you walk an area is covered by you. Writing or Drawing or music playing etc. which involves hands and also covers some area by you drives the feet either consciously, unconsciously or sub-consciously. - Theory
Example
1.
You go to the toilet in the morning - 1 minor activity
You go to bathroom in the morning - 1 minor activity
You water plants in the morning - 1 minor activity
You go to prayer room in the morning - 1 minor activity

2.
You go to office from home - 1 major activity
You go for lunch - 1 major activity
You go home from office - 1 major activity

3.
You go to kids' room for teaching - 1 minor activity
You go to drawing room to watch TV - 1 minor activity
You go to bedroom for sleeping - 1 minor activity

How many papers should a man write in a day in office?
3 major paper/s which has a lot of effect(Sum of category 2). And 7 minor paper/s which has less effect(Sum of category 1 and 3).

So, an equation like

∑ number_of_papers_written_in office_in_a_day x significance_of_the writing  = 

∑ number_of_destinations_traveled_in_a_day x significance_of_the_destination

should be right.

This logic seems quite okay to me. Not serious, though. But logic seems okay. What about you dear readers? Does this logic seem resonant to you? Please comment.

Finalised caricature style