Chapter 25 Han·Mathematical Genius·Li (please read it!!!!)
In the room, Xu Yun was talking:
"Mr. Newton, Sir Han Li calculated that when the exponent in the binomial theorem is a fraction, we can use e^x=1+x+x^2/2!+x^3/3!+...+x^n /n!+… to calculate.”
As he spoke, Xu Yun picked up the pen and wrote a line of words on the paper:
When n=0, e^x>1.
"Mr. Newton, we start from x^0 here. It is more convenient to use 0 as the starting point for discussion. Can you understand?"
Maverick nodded, indicating that he understood.
Xu Yun then continued to write:
Assume that the conclusion holds when n=k, that is, e^x>1+x/1!+x^2/2!+x^3/3!+……+x^k/k! (x>0)< br>
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>0）
Then Xu Yun drew a circle on f(k+1) and asked:
"Mr. Newton, do you know anything about derivatives?"
Mavericks continued to nod, and said two words concisely:
"Understood."
Friends who have studied mathematics should all know this.
Derivatives and integrals are the most important components of calculus, and derivatives are the basis of differential and integral calculus.
It is now the end of 1665, and Mavericks' understanding of derivatives has actually reached a relatively profound level.
In terms of derivation, Maverick's intervention point is instantaneous velocity.
Speed = distance/time, this is a formula that all primary school students know, but what about instantaneous speed?
For example, if we know the distance s=t^2, then what is the instantaneous speed v when t=2?
The thinking of a mathematician is to transform unstudied problems into learned problems.
So Newton thought of a very clever way:
Take a "very short" time period △t, and 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 becomes smaller and smaller, 2+△t becomes closer and closer to 2, and the time period becomes narrower and narrower.
As △t gets closer and closer to 0, the average speed gets closer and closer to the instantaneous speed.
If △t is as small as 0, the average speed 4+△t becomes the instantaneous speed 4.
Of course.
Later, Berkeley discovered some logical problems with this method, that is, whether △t is 0.
If it is 0, then how can we use △t as the denominator when calculating speed? Few people know, even elementary school students 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 was 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 a moving and fuzzy word like "infinite subdivision" to define precise mathematics?
The series of discussions that Berkeley triggered became the famous second mathematical crisis.
Some pessimistic parties even claimed that the building of mathematics and physics was about to collapse, and that our world was all false - and then these guys really jumped off the building. There are still portraits of them in Austria. A certain street fisherman was lucky enough to visit them once. Like the seven dwarfs, I don't know if it is used to be admired or to whip corpses.
It was not until the appearance of Cauchy and Weierstrass that this incident was fully explained and concluded, and it truly defined the tree that many classmates hung in later generations.
But that was something later. In this era of Mavericks, the practicality of freshmen mathematics was given top priority, so rigor was relatively ignored.
Many people in this era use mathematical tools to conduct research and use the results to improve and optimize the tools.
Occasionally, there will be some unlucky people who are calculating and suddenly realize that their research in this life is actually wrong.
all in all.
At this point in time, Mavericks is still relatively familiar with derivation, but he has not yet summarized a systematic theory.
Seeing this, Xu Yun wrote again:
Taking the derivative of 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, f(k+1)'>0
Then when x=0.
f(k+1)=e^0-1-0/1!-0/2!-0/k+1!=1-1=0
So when x>0.
Because the derivative is greater than 0, f(x)>f(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>0) is established!
Finally Xu Yun wrote:
To sum up, for any n:
e^x＞1+x/1!+x^2/2!+x^3/3!+……+x^n/n!（x＞0）
After finishing his discussion, Xu Yun put down his pen and looked at Mavericks.
I only see this moment.
The founder of later physics was staring at the draft paper in front of him with his bull's eyes wide open.
True.
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 a complex variable function.
This series is compatible with the binomial theorem, and the coefficient symbols are compatible with the combinatorial symbols.
Therefore, the binomial theorem can be extended from natural numbers to complex powers, and the combination definition can also be extended from natural numbers to complex numbers.
It's just that Xu Yun left a trick here and didn't tell Mavericks that when n is a negative number, it is an infinite series.
Because according to the normal historical line, infinitesimal quantity came from Mavericks, so it would be better to leave the derivation process to him.
After a few minutes like this, Mavericks finally came back to his senses.
He simply ignored Xu Yun beside him, rushed back to his seat, and quickly began to calculate.
Looking at Mavericks who was immersed in calculations, Xu Yun was not angry. After all, the founder had this kind of temper, and he might be better off in front of William Aisku.
Shashasha——
soon.
The sound of the pen tip making contact with the manuscript paper was heard, and formulas were quickly listed one after another.
Seeing this, Xu Yun thought for a moment, then turned and left the house.
He casually found a seat in the corner and looked up at Yunjuan Yunshu.
In this way, two hours passed by.
Just when Xu Yun was thinking about his next move, the door of the wooden house was suddenly pushed open, and Mavericks rushed out with an excited look on his face.
His eyes were filled with bloodshot eyes, and he waved the manuscript paper in his hand vigorously towards Xu Yun:
"Fat fish, negative numbers, I introduced negative numbers! Everything is clear!
It doesn’t matter whether the binomial exponent is a positive or negative number, an integer or a fraction, the combined 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 he was too excited, but Maverick didn’t notice at all, and his wig was knocked to the ground.
Looking at the red-faced Mavericks, Xu Yun could not help but feel a sense of excitement about changing history.
Follow the normal trajectory.
The Mavericks will have to wait until they receive a letter from John Tisripotti in January next year before they will be able to overcome a series of doubts and difficulties.
The letter from John Sripoti refers to the triangular figure disclosed by Pascal.
That is to say
This node in the history of space-time mathematics has been changed for the first time!
With the preliminary results of binomial development, it will not take long for Mavericks to build a preliminary calculus model with the help of Yang Hui's triangle.
From this.
The name Yang Hui's Triangle will also be engraved on the foundation of the throne of mathematics, where it belongs!
Even if the world changes in the next hundreds of years, no one can shake it!
The light of China's 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 the oriental face in front of him, Maverick's face also felt a surge of emotion.
Sir Han Li, whom he had never met before, could clear up the clouds for him by just leaving a few essays. He could open a door for himself just by relying on the hands of Fei Yu, a disciple who had not known how many generations before.
So what height can Sir Han Li's knowledge reach?
A genius who can come up with this kind of expansion is not an exaggeration to be called a mathematical genius, right? I originally thought that Mr. Descartes was invincible, but I didn't expect that there was someone more brave than him!
It seems that my road to mathematics and science still has a long way to go.
Note:
Why is the out-of-circle index negative.
(End of chapter)