This Spacetime, The Only Name!

Outside the room.

Seeing Little Ox rush back inside, Xu Yun vaguely realized something and quickly followed him.

"Bang—"

As soon as he entered the room, Xu Yun heard the sound of something heavy hitting.

He looked over and saw Little Ox standing by the desk with a look of annoyance, his left fist clenched, knuckles pressing heavily on the table.

It was obvious that Little Ox had just thrown a deliberate punch at the desk.

Seeing this, Xu Yun walked over and asked:

"Mr. Newton, what is this?"

"You wouldn't understand."

Little Ox waved his hand impatiently, but after a few seconds, he seemed to remember something:

"Fat Fish, do you—or that Sir Han Li—know anything about mathematical tools?"

Xu Yun feigned ignorance again, looking at him and asking:

"Mathematical tools? Do you mean rulers? Or compasses?"

Hearing this, Little Ox's heart sank halfway, but since he had started speaking, he couldn't stop. He continued:

"Not physical tools, but a set of theories that can calculate rates of change.

For example, the dispersion phenomenon just now, that's an instantaneous rate of change, and it might even involve certain particles invisible to the naked eye.

To calculate this rate of change, we need to use another tool that can be continuously accumulated to calculate the product of the angles of refraction.

For example, multiplying n instances of a+b together, is the product of choosing either 'a' or 'b' from a+b. For example, (a+b)^2 = a^2 + 2ab + b^2. Forget it, I don't think you'd understand."

Xu Yun looked at him with a half-smile and said:

"I understand, it's Yang Hui's Triangle."

"Hmm, so prepare yourself and we'll go see Uncle William later. Wait, what did you say?"

Little Ox was speaking his mind when he heard Xu Yun's words clearly. He froze, then suddenly looked up, staring intently at him:

"Yang Fei San Jiao? What's that?"

Xu Yun thought for a moment and extended his hand to Little Ox:

"Could you hand me a pen, Mr. Newton?"

If this were a day ago, when Little Ox had just met Xu Yun, Xu Yun's request would have been rejected 100%.

He might have even been told, "You think you're worthy?"

But after deriving the dispersion phenomenon not long ago, Little Ox now felt a hint of interest and recognition towards Xu Yun—or rather, towards the Sir Han Li behind him.

Otherwise, he wouldn't have explained so much to Xu Yun just now.

Therefore, facing Xu Yun's request, Little Ox, uncharacteristically, handed him the pen.

Xu Yun took the pen and quickly drew a diagram on the paper:

1

11

121

1331 (Please ignore the ellipses; if omitted, the starting point will auto-indent, it's annoying)

Xu Yun drew a total of eight rows. The two outermost numbers in each row were 1, forming an equilateral triangle.

Friends familiar with this diagram should know that this is the famous Yang Hui's Triangle, also known as Pascal's Triangle—in the international mathematics community, the latter is more widely accepted.

However, the truth is that Yang Hui discovered this triangle more than four hundred years before Pascal:

Yang Hui was born in the Southern Song Dynasty. In his 1261 work "Detailed Explanation of the Nine Chapters' Algorithms," he preserved a precious diagram—the "Diagram of the Method of Extracting Roots"—which is the oldest traceable triangular diagram in existence.

But due to certain well-known reasons, Pascal's Triangle has spread much wider, and some people don't even recognize the name Yang Hui's Triangle.

Therefore, despite Yang Hui's original written record, this mathematical triangle is still called Pascal's Triangle.

However, it is worth mentioning.

Pascal studied this triangular diagram in 1654 and officially published it in late November 1665, which is still a full month from now!

This is why Xu Yun started with the dispersion phenomenon:

The dispersion phenomenon is a very typical differential model, even more classic than Universal Gravitation. Both the deflection angle and its inherent "seven-in-one" appearance directly point to the tools of calculus.

The concept of 1/7 is directly linked to the fractional representation of exponents.

If Little Ox, after encountering the dispersion phenomenon, didn't think of his own intractable 'Fluxions', then he could truly go to sleep.

Little Ox sees the dispersion phenomenon → Little Ox becomes curious → Little Ox measures data → Little Ox thinks of Fluxions → Xu Yun introduces Yang Hui's Triangle.

This was a perfect logical progression trap, a setup from physics to mathematics.

The reason Xu Yun drew this diagram was simple:

Yang Hui's Triangle was a thorn in the side of every mathematics practitioner!

Yang Hui's Triangle was a mathematical tool originally invented by our ancestors, with conclusive evidence to prove it. Why should it be forced to bear another's name due to recent unfortunate circumstances?

He couldn't control or influence the original timeline, but at this point in time, Xu Yun would not allow Yang Hui's Triangle to share its name with Pascal!

With Old Master Newton as a guarantor, Yang Hui's Triangle would remain Yang Hui's Triangle.

A term belonging solely to Huaxia!

Xu Yun then let out a deep breath and continued drawing lines on the paper:

"Mr. Newton, you see, the two slanted sides of this triangle are composed of the number 1, and all other numbers are the sum of the two numbers directly above them.

This diagram illustrates that any number C(n,r) is equal to the sum of the two numbers above it, C(n-1,r-1) and C(n-1,r)."

As he spoke, Xu Yun wrote a formula on the paper:

C(n,r)=C(n-1,r-1)+C(n-1,r)(n=1,2,3,···n)

And

(a + b)^2= a^2 + 2ab + b^2

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 6ab^3 + b^4

(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5

When Xu Yun reached the cubic expansion, Little Ox's expression gradually became serious.

By the time Xu Yun wrote out the expansion for the sixth power, Little Ox could no longer sit still.

He abruptly stood up, snatched the pen from Xu Yun, and began writing himself:

(a + b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + a^6!

It was clear.

The numbers in the nth row of Yang Hui's Triangle had n terms, and their sum was 2 to the power of n-1. The coefficients in the expansion of (a+b) to the nth power corresponded sequentially to each term in the (n+1)th row of Yang Hui's Triangle!

Although this expansion presented no difficulty for Little Ox, and could even be considered basic operation for binomial expansion.

This was, however, the first time anyone had visually represented the roots of an equation so intuitively with a diagram!

More importantly, the mth number in the nth row of Yang Hui's Triangle could be represented as C(n-1,m-1), which is the number of combinations of choosing m-1 elements from n-1 distinct elements.

This would undoubtedly be a huge help for Little Ox's ongoing subsequent derivations of binomial expansions!

However...

Little Ox's brow furrowed again:

The appearance of Yang Hui's Triangle had opened up a new line of thought for him, but it didn't offer much help for the problem he was currently stuck on, namely the expansion of (P+PQ)m/n.

This was because Yang Hui's Triangle dealt with coefficients, while Little Ox was struggling with exponents.

Little Ox was now like an experienced driver.

As he rounded a mountain pass, he suddenly saw a clear, beautiful vista stretching out a hundred meters ahead. However, a massive pile of fallen rocks blocked his path about ten meters away.

While Little Ox was contemplating this dilemma, Xu Yun slowly said another sentence:

"Oh, right, Mr. Newton, Sir Han Li has also done some research on Yang Hui's Triangle.

He later discovered that the exponents in binomial expansions don't necessarily have to be integers; fractions and even negative numbers seem to be feasible."

"He didn't explain his proof for negative numbers, but he left behind a method for proving it with fractions."

"He called it..."

"Han Li Expansion!"