Han Li, The Mathematical Genius

Chapter 25: Han - Mathematical Prodigy - Li (Seeking Reads!!!)

In the room, Xu Yun was speaking eloquently:

"Mr. Newton, Sir Han Li has calculated 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, this starts from x^0. It's more convenient to discuss starting from 0, can you understand?"

Little Ox nodded, signaling that he understood.

Then Xu Yun continued to write:

Assume the conclusion holds when n=k, 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

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)

Next, Xu Yun drew a circle around f(k+1) and asked:

"Mr. Newton, do you know about Derivatives?"

Little Ox continued to nod, and concisely uttered two words:

"I do."

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 calculus.

It is now the end of 1665, and Little Ox's understanding of Derivatives has actually reached a rather profound level.

In terms of differentiation, Little Ox's starting point was instantaneous velocity.

Velocity = Distance / Time, this is a formula that elementary school students know, but what about instantaneous velocity?

For example, if we know the distance s = t^2, what is the instantaneous velocity v when t=2?

The thinking of a mathematician is to transform problems they haven't learned into problems they have learned.

So Newton thought of a very clever method:

Take a "very short" time interval Δt, and first calculate the average velocity during the time interval from t=2 to t=2+Δt.

v = s/t = (4Δt + Δt^2)/Δt = 4 + Δt.

As Δt becomes smaller and smaller, 2+Δt becomes closer and closer to 2, and the time interval becomes narrower and narrower.

As Δt approaches 0, the average velocity approaches the instantaneous velocity.

If Δt becomes 0, the average velocity 4+Δt becomes the instantaneous velocity 4.

Of course.

Later, Berkeley discovered some logical issues with this method, namely whether Δt is actually 0.

If it is 0, how can Δt be used as the denominator when calculating velocity? It is well known, cough cough, even elementary school students know that 0 cannot be a divisor.

If it is not 0, then 4+Δt will never become 4, and the average velocity will never become the instantaneous velocity.

According to the modern concept of Calculus, Berkeley was questioning whether lim Δt→0 is equivalent to Δt=0.

The essence of this problem is actually a questioning of nascent Calculus, is it appropriate to define precise mathematics using "infinitely subdividing" and vague terms like "motion"?

The series of discussions triggered by Berkeley was the famous Second Crisis in Mathematics.

Some pessimists even claimed that the edifice of mathematics and science was about to collapse, that our world was all false--and then these people actually jumped to their deaths. Their portraits remain in Austria. A certain failed fisherman once had the honor of visiting them; they looked like the seven dwarfs, and it's unclear if they were meant for admiration or posthumous condemnation.

This matter would not be fully explained and concluded until the appearance of Cauchy and Weierstrass, and it was they who truly defined the tree that many students later failed.

But that was later. In Little Ox's era, the practicality of nascent mathematics was prioritized, so rigor was relatively neglected.

Many people at this time were using mathematical tools for research while simultaneously improving and optimizing those tools based on the results they obtained.

Occasionally, some unlucky individuals, while calculating, would suddenly realize that their research for their entire life was actually wrong.

In conclusion.

At this point in time, Little Ox was quite familiar with differentiation, but had not yet summarized a systematic theory.

Xu Yun saw this and 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!

From the assumption, we know that 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.

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!

Finally, Xu Yun wrote:

In summary, for any n:

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

With his discourse complete, Xu Yun put down his pen and looked at Little Ox.

At this very moment.

The founding father of physics in later generations was staring intently at the draft paper before him with his large ox-like eyes.

Indeed.

At Little Ox's current research progress, he couldn't quite grasp the true intrinsic meaning of tangents and areas.

But anyone familiar with mathematics knows that the generalized Binomial Theorem is actually a special case of the Taylor series for complex functions.

This series is compatible with the Binomial Theorem, and its coefficients are also compatible with binomial coefficients.

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

However, Xu Yun held something back here, not telling Little Ox that when n is a negative number, it becomes an infinite series.

Because according to the normal historical timeline, infinitesimals originated from Little Ox, so it would be best to let him derive the process himself.

After a few minutes, Little Ox finally snapped back to reality.

He completely ignored Xu Yun beside him, leaped back to his seat in one stride, and began calculating furiously.

Watching Little Ox completely immersed in his calculations, Xu Yun didn't get angry. After all, this founding father had such a temper; perhaps he was only a little more amiable in front of William Askew.

Rustle, rustle, rustle--

Soon.

The sound of the pen tip meeting the paper echoed, and formulas were rapidly laid out.

Seeing this, Xu Yun pondered for a moment and turned to leave the room.

He casually found a spot in the corner, looked up at the drifting clouds.

Just like that, two hours passed.

As Xu Yun was contemplating his next move, the wooden door was suddenly pushed open from the inside, and Little Ox burst out, his face flushed with excitement.

His eyes were bloodshot, and he forcefully waved the draft paper in his hand at Xu Yun:

"Fat Fish, negative numbers, I've derived negative numbers! Everything is clear now!

The binomial exponent doesn't matter whether it's positive or negative, an integer or a fraction; the binomial coefficients hold true for all conditions!

Yang Hui's Triangle, yes, the next step is to study Yang Hui's Triangle!"

Perhaps due to his excitement, Little Ox didn't even notice his wig falling to the floor.

Looking at the flushed Little Ox, Xu Yun couldn't help but feel a surge of exhilaration at changing history.

According to the normal trajectory.

Little Ox wouldn't have his epiphany and overcome a series of doubts and difficulties until January of the following year, after receiving a letter from John Tissleporty.

And that letter from Johnslyporty mentioned Pascal's publicly revealed triangular diagram.

This means...

This crucial point in the history of mathematics in this timeline has been altered for the first time!

With the preliminary results from the binomial expansion, Little Ox would inevitably, and not long from now, construct a preliminary model of Fluxions with the assistance of Yang Hui's Triangle.

As a result.

The name Yang Hui's Triangle will be engraved on the foundation of the mathematical throne, the position it rightfully deserves!

Even after hundreds of years of change, of seas turning into mulberry fields, no one will ever be able to shake it!

The light of Huaxia's ancient sages will never be tarnished in this timeline!

With this thought, Xu Yun took a deep breath and strode forward:

"Congratulations, Mr. Newton."

Looking at Xu Yun, who had an Eastern face, Little Ox's face also showed a hint of emotion.

Sir Han Li, whom he had never met, had, with just a few jotted-down notes, brought him clarity. And through his disciple Fat Fish, separated by countless generations, he had opened a door for him.

Then, to what heights must Sir Han Li's own knowledge reach?

A genius who could devise such an expansion could undoubtedly be called a mathematical prodigy!

He had originally thought Mr. Descartes was invincible, but he never expected there to be someone even more formidable!

It seems his path in mathematics and physics is still long and arduous.

Note:

Why is the out-of-circle index negative?