In the room, Xu Yun was speaking eloquently:
"Mr. Newton, Sir Han Li calculated and 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. Newton, it starts from x^0 here, using 0 as the starting point is more convenient for discussion, you understand this, right?"
Little Niu nodded, indicating his understanding.
Then Xu Yun continued writing:
Assume that when n=k, the conclusion holds, i.e., 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 circled f(k+1) and asked:
"Mr. Newton, are you familiar with derivatives?"
Little Niu kept nodding, succinctly uttering two words:
"Familiar."
Friends who have studied mathematics should all know.
Derivatives and integrals are the most important components of calculus, and derivatives are the foundation of differential and integral calculus.
It was already late 1665, and Little Niu's understanding of derivatives was actually quite profound by this point.
In the field of deriving, Little Niu's entry point was instantaneous velocity.
Speed = distance/time, this is a formula even elementary school students know, but what about instantaneous velocity?
For example, if the distance s=t^2 is known, what is the instantaneous velocity v when t=2?
Mathematical thinking is about transforming problems not yet learned into ones that have been learned.
So Newton came up with a very clever way:
Take a "very short" time period △t, first calculate the average speed within the period from t= 2 to t=2+△t.
v=s/t=(4△t+△t^2)/△t=4+△t.
As △t becomes smaller, 2+△t gets closer to 2, and the time period becomes narrower.
When △t approaches 0, the average speed gets closer to the instantaneous speed.
If △t reduces to 0, the average speed of 4+△t becomes the instantaneous speed of 4.
Of course.
Later, Berkeley discovered some logical problems with this method, namely, whether △t is really 0.
If it is 0, then how can △t be used as a denominator when calculating speed? A rare.... ahem, even elementary school students know 0 cannot be a divisor.
If it's not 0, 4+△t will never really become 4, and the average speed will never become the instantaneous speed.
According to modern calculus concepts, Berkeley was questioning whether lim△t→0 is equivalent to △t=0.
The essence of this question was actually a challenge to early calculus, questioning if using terms like "infinitely small" to define precise mathematics is truly appropriate.
The series of discussions triggered by Berkeley led to the famously known second mathematical crisis.
Some pessimists even announced that the edifice of mathematical physics was about to collapse, and our world was illusionary—then these people really did jump off buildings, leaving their pictures in Austria, as if they were set up for public admiration or posthumous contempt, resembling seven dwarves, as some unfortunate angler once had the opportunity to witness.
This issue was not fully resolved until the emergence of Cauchy and Weierstrass, who provided thorough explanations and conclusions, and truly defined the tree that many students failed afterward.
But that was a later matter. In Little Niu's era, the practicality of new mathematics was placed in the forefront, thus strictness was relatively ignored.
Many people in this era were using mathematical tools to conduct research while simultaneously refining and optimizing the tools with the results obtained.
Occasionally, some unlucky individuals would suddenly realize that their life's research was wrong while calculating.
All in all.
At this point in time, Little Niu was still relatively familiar with derivation, but had not yet summarized a systematic theory.
Seeing this, Xu Yun wrote again:
Derivating f(k+1) gives 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
Therefore, when x > 0.
Since the derivative is greater than 0, f(x) > f(0)=0
Thus, 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:
To sum up, for any given n:
e^x ＞ 1 + x/1! + x^2/2! + x^3/3! + ... + x^n/n! (x ＞ 0)
Having finished the discussion, Xu Yun put down his pen and looked at Little Niu.
At this very moment.
The patriarch of future physics was staring intently at the draft paper before him, his eyes wide open, as if scorched into his memory.
Indeed.
At Little Niu's current stage of research, it's still not easy to fully understand the true inner meaning 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 combinatorial symbols.
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.
However, Xu Yun kept a card up his sleeve by not telling Little Niu that n being a negative number actually refers to an infinite series.
Because, according to normal historical trajectory, infinitesimals are supposed to come from Little Niu's hand, so it's better to leave the derivation process to him.
And so, after a few minutes, Little Niu came back to his senses.
Ignoring Xu Yun beside him, he quickly darted back to his seat and began calculating with great speed.
Watching Little Niu engrossed in his calculations, Xu Yun didn't get upset. After all, this patriarch has a temperament like this, perhaps only in front of William Asku would he be relatively milder.
Scratch, scratch, scratch—
Soon enough.
The sound of pen on paper started, one equation after another was quickly written out.
Xu Yun pondered for a moment upon seeing this, then turned and left the room.
Finding a spot in the corner at random, he looked up at the drifting clouds.
And just like that, two hours passed by.
As Xu Yun was contemplating his next move, the wooden house door suddenly swung open from the inside, and Little Niu came rushing out, full of excitement.
His eyes were bloodshot as he vigorously waved the draft paper at Xu Yun:
"Fat Fish, negative numbers, I figured out negative numbers! Everything is clear now!
For the binomial exponent, it doesn't matter if it's positive or negative, integer or fraction, the combination number holds true for all conditions!
Yang Hui's Triangle, yes, the next step is to study Yang Hui's Triangle!"
Perhaps due to being overly excited, Little Niu didn't even realize his wig had fallen to the ground.
Looking at the flushed Little Niu, Xu Yun couldn't help but feel a sense of triumph at the historical change.
Following the normal trajectory.
Little Niu was supposed to receive a letter from John Tisliboti next January, only then would he suddenly overcome a series of riddles and difficulties.
And in John Siriboti's letter, it mentioned Pascal's publicly known triangular figure.
Which means...
The node in this timeline's mathematical history has been changed for the first time!
With the initial results of the binomial expansion, it won't be long before Little Niu, with the help of Yang Hui's Triangle, constructs the preliminary model of the Fluxion Technique.
As a result.
The name Yang Hui's Triangle will also be engraved on the mathematical throne's foundation, in a place it rightfully belongs!
Regardless of the world's changes, regardless of the seas rising and falling over centuries, none can shake it!
The light of Huaxia's ancient wisdom will never dim in this timeline!
Thinking of this, Xu Yun couldn't help but take a deep breath and walked forward quickly:
"Congratulations, Mr. Newton."
Faced with Xu Yun's oriental features, Little Niu also felt a strong wave of emotion.
That Sir Han Li, whom he never met, could part the clouds with just a few jotted notes, opening a door for him through the hand of this Fat Fish, a disciple separated by untold generations.
Then to what heights could Sir Han Li's own knowledge reach?
It wouldn't be too much to call a genius who conceived such an expansion a mathematical prodigy?
Originally, I thought Mr. Descartes was already invincible in the world, but I didn't expect there to be someone even braver!
It seems the road of my mathematical and physical journey remains long and arduous...
...
Note:
Why are the binomial exponents negative.....