"Mr. Isaac, 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. Isaac, this starts from x^0. 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)
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. Isaac, do you know anything about derivatives?"
Mavericks continued to nod his head and said two words concisely:
"learn."
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 the instantaneous velocity.
Speed = distance x 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 when t=2, what is the instantaneous speed v?
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 this 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 △t be used as the denominator when calculating speed? Few people... ahem, 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 was questioning whether lim△t→0 is equivalent to △t=0.
The essence of this question is actually a torture of nascent calculus. Is it really appropriate to use vague words 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, and there are still portraits of them in Austria, which a certain street fisherman was lucky enough to visit 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 Maverick's era, 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 has not yet summarized a systematic theory.
Seeing this, Xu Yun wrote again:
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 we have:
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 the derivation process should be left to him himself.
After a few minutes like this, the calf 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 feel better in front of William Aisku.
swish swish——
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, reincarnated and left the house.
I casually found a seat in the corner and looked up at Yunjuan Yunshu.
Just like that, two hours passed by.
Just when Xu Yun was thinking about his next move, the door of the wooden house was suddenly pushed open by someone, and Mavericks jumped out from inside 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 Maverick, 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 to overcome a series of doubts and difficulties.
The letter from John Sripoti mentioned exactly 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. Isaac."
Looking at the oriental-faced Xu Yun 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, and 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 ago.
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 physics still has a long way to go...