Into Unscientific

Chapter 25 Han · Mathematics Wizard · Li (seeking to follow up!!!!!!)

In the room, Xu Yun was talking eloquently:

"Mr. Newton, Sir Han Li calculated that when the exponent in the binomial theorem is a fraction, you can use e^x = 1+x+x^2/2!+x^3/3!+...+x^n /n!+... to calculate."

As he spoke, Xu Yun picked up a pen and wrote a line on the paper:

When n=0, e^x\u003e1.

"Mr. Newton, here starts from x^0. It is more convenient to use 0 as the starting point for discussion. Do you understand?"

The Maverick nodded, indicating that he understood.

Then Xu Yun continued to write:

Assuming that the conclusion is true when n=k, that is, e^x＞1+x/1!+x^2/2!+x^3/3!+...+x^k/k!（x＞0）

Then e^x-[1+x/1!+x^2/2!+x^3/3!+…+x^k/k!]＞0

Then when n=k+1, let the function f(k+1)=e^x-[1+x/1!+x^2/2!+x^3/3!+…+x^( k+1)/(k+1)]! (x\u003e0)

Then Xu Yun drew a circle on f(k+1) and asked:

"Mr. Newton, do you know anything about derivatives?"

Maverick continued to nod his head, and said two words concisely:

"learn."

Friends who have studied mathematics should know it.

Derivatives and integrals are the most important components of calculus, and derivatives are the basis of differential integration.

It is now the end of 1665, and Mavericks' understanding of derivatives has actually reached a relatively profound level.

In terms of derivation, Mavericks' intervention point is the instantaneous velocity.

Speed ​​= distance/time, this is a formula that elementary school students know, but what about the instantaneous speed?

For example, knowing the distance s=t^2, then when t=2, what is the instantaneous speed v?

The thinking of a mathematician is to transform unlearned problems into learned problems.

So Newton thought of a very clever way:

Take a "very short" time period △t, first calculate the average speed during the time period from t= 2 to t=2+△t.

v=s/t=(4△t+△t^2)/△t=4+△t.

When △t is getting smaller and smaller, 2+△t is getting closer to 2, and the time period is getting narrower and narrower.

When △t is getting closer to 0, then the average speed is getting closer to the instantaneous speed.

If △t is small to 0, the average speed 4+△t becomes the instantaneous speed 4.

Of course.

Later, Berkeley discovered some logical problems of this method, that is, whether △t is 0 or not.

If it is 0, how can △t be used as the denominator when calculating the speed? Few people cough, elementary school students also know that 0 cannot be used as a divisor.

If it is not 0, 4+△t will never become 4, and the average speed will never become the instantaneous speed.

According to the concept of modern calculus, Berkeley is questioning whether lim△t→0 is equivalent to △t=0.

The essence of this question is actually a torture of the nascent calculus. Is it really appropriate to use the moving and fuzzy words like "infinite subdivision" to define precise mathematics?

The series of discussions triggered by Berkeley is the famous second crisis of mathematics.

There are even some pessimistic parties claiming that the Mathematics Building is about to collapse, and that our world is all false—then these goods really jumped off the building, and there are still their portraits in Austria. A street fishing guy was lucky enough to visit once, Like the seven dwarfs, I don't know whether it is used to be admired or to flog corpses.

This matter did not have a complete explanation and conclusion until the appearance of Cauchy and Weierstrass, and it really defined the tree that many students in later generations hung.

But that was later. In the age of Mavericks, the practicality of freshman mathematics was given top priority, so strictness was relatively ignored.

Many people in this era use mathematical tools to do research, and use the results to improve and optimize the tools.

Occasionally, there will be some unlucky people who are calculating and suddenly find that their research in this life is actually wrong.

all in all.

At this point in time, Mavericks is quite familiar with derivation, but he has not yet summarized a systematic theory.

Seeing this, Xu Yun wrote:

Deriving f(k+1), we can get f(k+1)'=e^x-1+x/1!+x^2/2!+x^3/3!+…+x^ k/k!

From the assumption that f(k+1)'\u003e0

Then when x=0.

f(k+1)=e^0-1-0/1!-0/2!-0/k+1!=1-1=0

So when x\u003e0.

Because the derivative is greater than 0, so f(x)\u003ef(0)=0

So when n=k+1 f(k+1)=e^x-[1+x/1!+x^2/2!+x^3/3!+…+x^(k+1 )/(k+1)]! (x\u003e0) established!

Finally Xu Yun wrote:

To sum up, for any n we have:

e^x＞1+x/1!+x^2/2!+x^3/3!+…+x^n/n!（x＞0）

After the discussion, Xu Yun put down the pen and looked at the Mavericks.

Only at this moment.

The patriarch of later generations of physics is staring at the draft paper in front of him with his bull's eyes wide open.

Indeed.

With the current research progress of Mavericks, it is not easy to understand the true inner meaning of tangent and area.

But anyone who knows mathematics knows that the generalized binomial theorem is actually a special case of the Taylor series of complex variable functions.

This series is compatible with the binomial theorem, and the coefficient signs are also compatible with the composite signs.

Therefore, the binomial theorem can be extended from natural number powers to complex number powers, and the combination definition can also be extended from natural numbers to complex numbers.

It's just that Xu Yun kept a hand here, and didn't tell Maverick that when n is negative, it is an infinite series.

Because according to the normal historical line, the infinitesimal amount was created by Mavericks, and it is better to leave the derivation process to him.

After a few minutes like this, the Mavericks just came back to their senses.

I saw him directly ignoring Xu Yun who was beside him, rushed back to his seat, and quickly started to calculate.

Looking at Mavericks who was devoting himself to calculations, Xu Yun was not angry either. After all, this patriarch had such a temper, and he might be relatively better in front of William Escue.

rustle——

soon.

The sound of the tip of the pen touching the manuscript paper sounded, and formulas were quickly listed.

Seeing this, Xu Yun thought for a moment, then turned and left the room.

Randomly found a place in the corner, looked up at Yunjuan Yunshu.

Just like that, two hours passed by.

Just when Xu Yun was thinking about what to do next, the door of the wooden house was suddenly pushed open, and Xiao Niu rushed out from inside with an excited expression on his face.

His eyes were bloodshot, and he vigorously waved the manuscript in his hand to Xu Yun:

"Fat fish, negative numbers, I introduced negative numbers! Everything is clear!

The binomial index does not care whether it is positive or negative, integer or fraction, the combination number is true for all conditions!

Yang Hui triangle, yes, the next step is to study Yang Hui triangle! "

I don't know if it was because of being too excited, Maverick didn't even notice that his wig was shaken to the ground.

Seeing the red-faced calf, Xu Yun couldn't help but feel a sense of excitement about changing history.

Follow the normal trajectory.

The Mavericks will not be able to overcome a series of doubts and difficulties until they receive a letter from John Tisripodi in January next year.

And in the letter of John Sripodi, it is Pascal's public triangular figure that is mentioned.

That is to say

This node in the history of space-time mathematics has been changed for the first time!

With the initial results of the binomial development, Mavericks will certainly not be long before building a preliminary flow model with the help of Yang Hui's triangle.

From this.

The name Yang Hui Triangle will also be engraved on the base of the Mathematics Throne, where it should have been!

Even if the world changes in the next few hundred years and the vicissitudes of life, no one can shake it!

The light of Chinese sages will never be dusted in this timeline!

Thinking of this, Xu Yun couldn't help but take a deep breath, and walked forward quickly:

"Congratulations, Mr. Newton."

Looking at Xu Yun with an oriental face in front of him, Maverick's face also showed emotion.

Sir Han Li, who had never met, could see the light of day for himself with just a few essays left behind, and he could open a door for himself with the help of Fei Yu, a disciple who didn't know how many generations apart.

So what kind of height can Sir Han Li's own knowledge reach?

A genius who can come up with this kind of expansion is not an exaggeration to be called a mathematical genius, right?

Originally, I thought that Mr. Descartes was invincible, but I didn't expect that there would be someone more brave than him!

It seems that my road of mathematics and science still has a long way to go.

Note:

Why is the out-of-circle index negative.