Han Li, the Mathematical Prodigy (Please follow the story!)

Han Li, the Mathematical Prodigy (Please follow the story!)

Inside the room, Xu Yun was speaking with great enthusiasm:

"Mr. Newton, Sir Han Li worked out 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 with x^0. It's more convenient to use zero as the starting point for the discussion. You understand, right?"

Little Ox nodded to show he understood.

Then Xu Yun continued writing:

Suppose 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

Now, 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)

Xu Yun then circled f(k+1) and asked:

"Mr. Newton, are you familiar with derivatives?"

Little Ox nodded again and succinctly offered two words:

"I am."

Friends who have studied mathematics should know this.

Derivatives and integrals are the most important parts of calculus, and derivatives are the foundation of differential and integral calculus.

It was already the end of 1665, and Little Ox's understanding of derivatives had actually reached a fairly advanced level.

His path into differentiation had begun with instantaneous velocity.

Velocity = distance/time. Even elementary school students knew that formula, but what about instantaneous velocity?

For example, if you knew that the distance was s=t^2, what would the instantaneous velocity v be when t=2?

A mathematician's way of thinking was to turn an unsolved problem into one they already knew how to solve.

So Newton came up with a clever method:

Take a "very short" time interval △t, and first calculate the average velocity between t=2 and t=2+△t.

v=s/t=(4△t+△t^2)/△t=4+△t.

As △t got smaller and smaller, 2+△t got closer and closer to 2, and the time interval grew narrower and narrower.

As △t got closer and closer to 0, the average velocity got closer and closer to the instantaneous velocity.

If △t became 0, then the average velocity 4+△t became the instantaneous velocity 4.

Of course.

Later, Berkeley discovered some logical problems with this method: namely, whether △t was actually 0.

If it was 0, how could you use △t as the denominator when calculating velocity? Even ahem, even elementary school students knew that you couldn't divide by 0.

But if it wasn't 0, 4+△t could never become 4, and the average velocity could never become the instantaneous velocity.

In terms of modern calculus, Berkeley was questioning whether lim△t→0 was equivalent to △t=0.

The heart of the problem was that it challenged the foundations of calculus in its infancy. Was it really appropriate to define precise mathematics using a vague notion of "infinite subdivision"?

The series of discussions sparked by Berkeley's challenge became the famous Second Mathematical Crisis.

Some pessimists even declared that the edifice of mathematics was about to collapse and that our world was all a lie—and then those idiots really did jump off buildings. Their portraits were still kept in Austria. A certain washed-up fisherman had once had the good fortune to see them; they looked like the seven dwarfs. It was hard to tell whether they were there to be admired or to be desecrated.

It wasn't until Cauchy and Weierstrass came along that the issue was finally explained and settled, and the concept that would make so many future students want to hang themselves was properly defined.

But that would happen later. In Little Ox's time, the practical applications of this nascent mathematics came first, so rigor was relatively neglected.

Many people of the era used mathematical tools in their research while improving and refining those tools based on the results they obtained.

Every now and then, some poor soul would be working through a problem only to suddenly discover that the research he had spent his whole life on was wrong.

In short.

At this point in time, Little Ox was fairly familiar with differentiation, but he hadn't yet gathered it into a systematic theory.

Xu Yun saw this and wrote:

Taking the derivative of f(k+1), we get f(k+1)'=e^x-1+x/1!+x^2/2!+x^3/3!+...+x^k/k!

By 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

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!

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 explanation, Xu Yun set down his fountain pen and looked at Little Ox.

At that very moment.

The future patriarch of physics was staring wide-eyed at the sheet of scratch paper in front of him.

To be sure.

With Little Ox's research at its current stage, he couldn't yet fully understand the true meaning of tangents and areas.

But anyone who knew mathematics knew that the generalized Binomial Theorem was actually a special case of the Taylor series for complex functions.

This series was compatible with the Binomial Theorem, and its coefficients were compatible with the notation for combinations.

So the Binomial Theorem could be extended from natural-number powers to complex-number powers, and the definition of combinations could likewise be extended from natural numbers to complex numbers.

Xu Yun had held something back, though. He hadn't told Little Ox that when n was negative, it became an infinite series.

In the normal course of history, infinitesimals had come from Little Ox himself, so it was best to leave the derivation to him.

A few minutes passed before Little Ox finally came back to himself.

He completely ignored Xu Yun, darted back to his seat, and began working out the calculations at top speed.

Watching Little Ox immerse himself completely in the calculations, Xu Yun didn't get upset. That was simply how the patriarch was. Perhaps he was a little more pleasant around William Askew.

Scritch, scritch—

Soon.

The sound of pen nib against paper rang out, and formula after formula appeared in rapid succession.

Xu Yun thought for a moment, then turned and left the room.

He found a spot in the corner, looked up, and watched the clouds drift by.

Just like that, two hours passed.

Just as Xu Yun was considering his next move, the wooden door was suddenly pushed open from within, and Little Ox rushed out, his face alight with excitement.

His eyes were bloodshot. He waved the paper in his hand at Xu Yun:

"Fat Fish, negative numbers—I've figured out negative numbers! It all makes sense now!

"We don't need to care whether the exponent in the Binomial Theorem is positive or negative, or whether it's an integer or a fraction. The combinations work for every case!

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

Perhaps because he was too excited, Little Ox didn't even notice that his wig had fallen to the ground.

Looking at Little Ox's glowing face, Xu Yun couldn't help feeling a thrill at having changed history.

Under the normal course of events.

Little Ox wouldn't have had his breakthrough and solved a series of difficult questions until he received a letter from John Tisriporty the following January.

The letter from Johnslyporty had mentioned Pascal's published triangular diagram.

In other words—

The course of mathematical history had just changed for the first time!

With his initial results on the Binomial Theorem, it wouldn't be long before Little Ox constructed an early model of the Fluxions model with the help of Yang Hui's Triangle.

And so.

The name Yang Hui's Triangle would be engraved on the foundation of the throne of mathematics, in the place where it had always belonged!

Even after the changes of centuries, through the vicissitudes of time, no one would ever be able to dislodge it!

The light of Huaxia's sages would never be obscured on this timeline!

Thinking of this, Xu Yun took a deep breath and hurried forward:

"Congratulations, Mr. Newton."

Looking at Xu Yun's Oriental face, Little Ox's own face filled with emotion.

Sir Han Li, whom he had never met, had left only a few notes, yet they had cleared the clouds from his path. By borrowing the hand of Fat Fish, a student separated from him by who knew how many generations, he had opened a door for him.

Just how learned could Sir Han Li himself be?

A genius who could come up with an expansion like this surely deserved to be called a mathematical prodigy, didn't he?

He had once thought Mr. Descartes was invincible, but who would have guessed there was someone even more formidable?

It seemed his journey through mathematics still had a long way to go.

Note:

Why is the out-of-circle exponent negative?