In the room, Xu Yun was talking enthusiastically:
"Mr. Isaac, Sir Han Li discovered that when the exponent in the binomial theorem is a fraction, it can be calculated using e^x=1+x+x^2/2!+x^3/3!+……+x^n/n!+……."
As he spoke, Xu Yun picked up a pen and wrote a line on the paper:
When n=0, e^x＞1.
"Mr. Isaac, we start from x^0 here, using 0 as the starting point is more convenient for discussion, can you understand?"
Little Newton nodded, indicating that he understood.
Then Xu Yun continued to write:
Assuming that the conclusion holds 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
So 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 understand derivatives?"
Little Newton continued to nod, succinctly blurting out two words:
"Understand."
Friends who have studied mathematics should know.
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Derivatives and integrals are the most important components of calculus, and derivatives are the foundation of differential and integral calculus.
It is now the end of 1665, and Little Newton's understanding of derivatives has actually reached a quite profound level.
In terms of differentiation, Little Newton's entry point is instantaneous velocity.
Velocity = distance x time, this is a formula that elementary school students know, but what about instantaneous velocity?
For example, if we know that the distance is s=t^2, then when t=2, what is the instantaneous velocity v?
The thinking of mathematicians is to transform unfamiliar problems into familiar ones.
So Newton came up with a clever solution:
Take a "very short" time interval △t, first calculate the average velocity within the time interval from t=2 to t=2+△t.
v=s/t=(4△t+△t^2)/△t=4+△t.
As △t becomes smaller and smaller, 2+△t gets closer and closer to 2, and the time interval becomes narrower.
As △t approaches 0, the average velocity gets closer and closer to the instantaneous velocity.
If △t becomes small enough to be 0, the average velocity 4+△t becomes the instantaneous velocity 4.
Of course.
Later, Berkeley discovered some logical problems with this method, that is, whether △t is really 0.
If it is 0, then how can △t be used as the denominator when calculating velocity? Even elementary school students know that 0 cannot be a divisor.
If it is not 0, 4+△t can never become 4, and the average velocity can never become the instantaneous velocity.
According to the concept of modern calculus, Berkeley was questioning whether lim△t→0 is equivalent to △t=0.
The essence of this problem is actually a questioning of the birth of calculus, whether it is appropriate to define precise mathematics with such vague terms as "infinite division".
The series of discussions sparked by Berkeley led to the famous Second Mathematical Crisis.
Some pessimists even claimed that the mathematical edifice was about to collapse, and our world was all false—then these people really jumped off the building, leaving their portraits in Austria, whether they were meant to be admired or whipped, no one knows.
This matter did not have a complete explanation and conclusion until the appearance of Cauchy and Weierstrass, and it truly defined the tree that many students hang on in later generations.
But that was a later matter. In Little Newton's era, the practicality of new mathematics was the priority, so rigor was relatively neglected.
Many people in this era use mathematical tools to conduct research while using the results obtained to improve and optimize the tools.
Occasionally, there are also unlucky ones who suddenly realize that their lifelong research is actually wrong.
In short.
At this point in time, Little Newton is quite familiar with differentiation, but he has not yet deduced a systematic theory.
Seeing this, Xu Yun wrote again:
Differentiating f(k+1), we get f(k+1)'=e^x-1+x/1!+x^2/2!+x^3/3!+……+x^k/k!
According to the assumption, f(k+1)'＞0
So 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) holds true!
Finally, Xu Yun wrote:
In conclusion, for any n:
e^x＞1+x/1!+x^2/2!+x^3/3!+……+x^n/n! (x＞0)
Having finished his discourse, Xu Yun put down his pen and looked at Little Newton.
At this moment.
This ancestor of future physics was staring intently at the draft paper in front of him with his cow-like eyes wide open.
Indeed.
At Little Newton's current research progress, he still doesn't quite understand the true inherent meanings of tangents and areas.
But anyone who understands mathematics knows that the generalized binomial theorem is actually a special case of the Taylor series of complex functions.
This series is compatible with the binomial theorem, and the coefficients are also compatible with the combination symbols.
So the binomial theorem can be extended from natural number powers to complex number powers, and the definition of combinations can also be extended from natural numbers to complex numbers.However, Xu Yun held back a bit here, not informing Little Newton that when n is a negative number, it would result in an infinite series.
Because according to the normal historical timeline, infinitesimal quantities were derived by Little Newton himself, so it's better to leave the derivation process to him.
Minutes passed like this, and only then did Little Newton come back to his senses.
Ignoring Xu Yun beside him, he darted back to his seat and began to calculate rapidly.
Seeing Little Newton fully engrossed in his calculations, Xu Yun wasn't upset. After all, this was just the kind of temperament the old master had. He might be a bit better in front of William Escu.
Soon, the sound of the pen tip touching the draft paper rang out, and formulas were quickly listed.
Seeing this, Xu Yun thought for a moment and left the room.
He found a spot in the corner, looked up, and watched the clouds roll by.
In this way, two hours passed quickly.
Just as Xu Yun was considering his next move, the wooden door was suddenly pushed open, and Little Newton rushed out, his face full of excitement.
His eyes were bloodshot, and he waved the draft paper in his hand at Xu Yun:
"Fat Fish, negative numbers, I've figured out negative numbers! Everything is clear now!
The binomial exponent doesn't care if it's positive or negative, an integer or a fraction, the combination number holds for all conditions!
Pascal's triangle, yes, the next step is to study Pascal's triangle!"
Perhaps due to his excitement, Little Newton didn't notice that his wig had fallen to the ground.
Seeing Little Newton's flushed face, Xu Yun couldn't help but feel a sense of excitement about changing history.
According to the normal trajectory.
Little Newton would have to wait until next January to receive a letter from John Tisliboti before he could crack a series of difficult questions.
And in John Tisliboti's letter, he mentioned Pascal's publicly disclosed triangular figure.
That is to say......
The node of mathematical history in this timeline has been changed for the first time!
With the initial results of the binomial expansion, it won't take Little Newton long to construct a preliminary model of calculus with the help of Pascal's triangle.
As a result.
The name of Pascal's triangle will be engraved on the base of the throne of mathematics, the place that rightfully belongs to it!
Even if hundreds of years pass and the world changes, no one will be able to shake it!
The glory of the Chinese sages will never be obscured in this timeline!
Thinking of this, Xu Yun took a deep breath and quickly stepped forward:
"Congratulations, Mr. Isaac."
Looking at Xu Yun's oriental face, Little Newton also felt a sense of emotion.
The never-met Sir Han Li, whose few notes alone could clear the clouds for him, and through the hand of Fat Fish, a disciple from who knows how many generations later, could open a door for him.
So, how high could Sir Han Li's knowledge be?
The genius who could come up with such an expansion, wouldn't it be an understatement to call him a mathematical genius?
He originally thought that Mr. Descartes was invincible, but he didn't expect that there was someone even more courageous than him!
It seems that his mathematical journey is still a long way to go......
......
Note:
Why is the exponent out of the circle negative.....