A better log inspired sheet

A better log inspired sheet has been made. 

Now the results are more accurate.

Hope you like it.

Have a nice day !!

They are same

In the following, the numerator and the denominator are nearly same. Therefore they give about 1 as an answer.

You may observe that as the number in the top of binomial becomes more, more is the tendency towards equality.

And My Lord, my shepherd.

Have a nice weekend, guys !!

Approximate gamma values

 

And Wolfram screenshots to confirm the correctness

I wanted to build Indian gamma function. But my KAAL number gave only approximate results. But here in I am able to show approximate gamma function. This is truly a very happy day for me. Mathematics is not my pride. Not my ego. But my lifestyle. I rely on my mathematics a lot. To guide myself. This is the way I lead my life. And today I made approximate gamma function. In integral form.

I am finished with KAAL number discussions. And blog posts.

Approximate division by three consecutive numbers

 Here is an approximate division formula. The below divides n by m.

And the Wolfram Alpha screenshot for 1475 by 14.75

Hope you like the expression !!

My body pi and my body phi

I always wanted to express myself in terms of constants. I wanted to find what expressions give me peace of mind.

The following three give me peace of mind. Kindly note that it gives 0.61803 and 1.61803 and not 0.61803...... or 1.61803...... Only 5 places to decimals.

They give me peace. They give me serene mind. I feel as calm as a lonely man standing in a vacant beach listening to the roar of waves.

Today, Sunday, 18th January 2026, I found my peace expressions. 

It is not easy to get 5 places of approximations as in all the above. 

Have a nice day !!

Recent 2n factorial nearabout

 Given below is a recent near-about 2n factorial formula. 

And the Wolfram Alpha screenshots to confirm that the formula is correct. Numbers considered are 46 for n and 92 for 2n.

Hope you like the above expression. Have a nice day !!

All the approximate properties of 4.666

 RHS of 4.666 function for positive number

LHS of 4.666 function for positive number

RHS of 4.666 function for negative number

LHS for 4.666 function for negative number

2n factorial with 4.666 integral

Actual 2n factorial value

The log equivalent(so to speak) for 4.666 to the power 1 by x

e to the power x put in log function (so to speak)

Log x put in e to the power x (so to speak)

cos(x) part in 4.666 function (so to speak). Named One(x)

sin(x) part in 4.666 function (so to speak). Named Zero(x)

One(x) squared plus Zero(x) squared. Sin(x) squared plus Cos(x) squared (so to speak)

Argument in favor of 4.666 expressions

 Are complex number factor small i a necessity ?

Not really. The hands sway, the legs sway, the eyes sway but our ears do not sway. Our nose do not sway. Our trunk do not sway. The sway that is implied in i is not essential to understand. 

So I am here propagating the importance of KAAL number 4.666. It is a competitor for e which is 2.718. . . . 

First of all the complete 4.666 expressions. All the three involving 4.666 are approximate expressions.

The first and the third are fine but I hear a lot of rejections for the fourth one, which are approximate integrals. The criticism is mainly on the lower value of the integral.

I would like to defend it by saying that saturation is a nature principle. Many things saturate and stay stable.

Below is a saturation curve.

The lower value of integral is

and its curve is

You may observe that the lower value of integral has a nature approved principle of saturation. Because its curve is similar to saturation curve.

Therefore, I am sticking very deeply to KAAL number or 4.666. I believe this to be a truth of nature and to me the expressions of 4.666 are of nature.

2n factorial formula made accurate

You may be aware that I made 2n! formula of the type ((((n+1)(n+2))/2)^(n)). I have made it very approximate. 

The formula for (2n)! is

The Wolfram Alpha results to confirm that the formula is now very approximate. 

Have a nice weekend!!

(Today on September 8th, 2024)

If you are doing a programming in computer, then you may use the following.  The results will be more accurate.

I have been working on ((((n+1)(n+2))/2)^n) since April. 5 months. I derived the formula from Measuring Behavior expression or what I term it to be equivalent to Wallis pi. I was trying to get 0.5 factorial.  While doing that I discovered the above pattern for (2n)!. So a journey of 5 months of intense application ended yesterday night. 

Of course, James Stirling formula for n! is far more superior and has far more fidelity. 

But the expression (((n+1)(n+2))/2) is the sum of first (n+1) numbers. And that is the reason why I persisted with this formula.  Because sum of (n+1) numbers to the power n is related to 2n!. I found that interesting!!

my final statement 4

This is my final statement to the crude 2n! blog post that I made a few days back.

Hope you like this expression.

Have a nice day !!

approximate n!

 

I came very near. But then this formula escaped my grasp. I was mad with the formulation of n!. I am now relieved that this formula was done. In history by James Stirling.

Have a nice day !!

This approximation expression

 I present the following approximate expression.

Hope you like it.

Have a nice day !!

An approximate expression

 For very large values of k, the following is approximately true

Hope you like it.

Have a nice day !!

New Number

 I discovered the following.

I have checked up to n = 20 in my calculator. If you have an advanced calculator you can check and see

Maybe there is something significant here.

Have a nice day !!

An approximate expression

 I am presenting hereby an approximate expression.

The above is the sixth approximate expression of mine, including one on Pentagon. I have checked them in Wolfram Alpha for 4 or 5 cases. You can try and tell me in comments if there are discrepancies.

Have a nice day !!

Shagun

Shagun is the money given during auspicious and good occasions like marriage, birth or religious purposes. 

Many people give 101, 1001 or 10001. 

We give 108, 1008 and 10008.

I have used the function below and tested them for n at 786, 7860, 78600 and also at 100, 1000 and 10000.

And this is what I got

And people are making business with 1 rupee coined envelopes.  They are playing with 786 number of Islam.

Nehruji,  Indiraji and Rajivji. You owe an explanation to people of India.

Jai Hind !!

My binomials have recursiveness

My binomials have recursive property.

Here is a Wolfram screenshot to confirm this. The binomial terms on the LHS are a by b whole to the power of x type in both numerator and denominator. The RHS also is a by b whole to the power of k.

You may recall that my binomials if divided for sigma till n by sigma till n-k gives a by b whole to the power of k approximately. Below is what I mean to say 

What I mean is below

And here are screenshots to confirm that my just above statement is true

Have a nice day !!