Outside the house.
Watching Little Niu hurriedly run back inside, Xu Yun vaguely realized something and quickly followed him.
"Bang—"
As soon as he entered, Xu Yun heard the sound of a heavy object hitting.
He looked over and saw Little Niu standing by the desk with a face full of frustration, his left hand clenched into a fist pressing heavily on the table.
Obviously, Little Niu had just given the desk a deliberate punch.
Seeing this, Xu Yun stepped forward and asked:
"Mr. Newton, what are you....."
"You don't understand."
Little Niu waved his hand irritably, but quickly thought of something:
"Fat Fish, you — or that Sir Han Li, are you familiar with mathematical tools?"
Xu Yun feigned ignorance and looked at him, asking:
"Mathematical tools? Are you talking about rulers? Or compasses?"
Upon hearing this, Little Niu's heart sank halfway, but he couldn't stop halfway and continued:
"Not physical tools, but a theory capable of calculating rates of change.
For example, the dispersion phenomenon just now, which is an instantaneous rate of change, might even involve some particles invisible to the naked eye.
To calculate this rate of change, we need another tool capable of continuous accumulation to calculate the product of the refraction angle.
For example, multiplying n instances of a+b results in a product from selecting either letter a or b from a+b, like (a+b)^2=a^2+2ab+b^2... Forget it, I doubt you understand."
Xu Yun looked at him with a half-smile and said:
"I understand, it's Yang Hui's Triangle."
"Hmm, so let's prepare to go to William's uncle later... Wait, what did you say?"
Little Niu was initially following his train of thought, but upon hearing Xu Yun's words, he froze and then suddenly looked up, staring at him:
"Yang Fei San Jiao? What's that?"
Xu Yun thought for a moment and reached his hand out to Little Niu:
"Can you pass me the pen, Mr. Newton?"
If this had been a day ago when Little Niu first saw Xu Yun, this request would have been outright rejected by Little Niu.
Might even receive a retort of 'Do you even deserve it?'.
But with the recent derivation of the dispersion phenomenon, Little Niu at this point had developed a vague interest and recognition towards Xu Yun—or rather, the Sir Han Li behind him.
Otherwise, he wouldn't have bothered explaining those words to Xu Yun earlier.
Therefore, facing Xu Yun's request, Little Niu rarely did pass the pen over.
Xu Yun took the pen and quickly drew a diagram on paper:
.......1
.... 1...1
....1...2...1
1.....3....3......1 (Please ignore ellipses, without them the starting point shifts, dizzying)
....
Xu Yun drew a total of eight rows, with each row's outermost two numbers being 1, forming an equilateral triangle.
Friends familiar with this image should know, it's the famous Yang Hui's Triangle, also called Pascal's Triangle—in international math, the latter is better accepted.
But actually, Yang Hui discovered this triangle over four hundred years prior to Pascal:
Yang Hui was born in the Southern Song period, and in his 1261 "Detailed Explanations of the Nine Chapters on Mathematical Art", preserved a precious image—the "Origin of Square Root Calculation" diagram, the oldest existing traceable triangular image.
However, due to some well-known reasons, the spread of Pascal's Triangle is much broader, and some people don't even recognize the name Yang Hui's Triangle.
Thus, despite Yang Hui's original notes, this mathematical triangle is still called Pascal's Triangle.
But it's worth mentioning...
Pascal researched this triangular diagram in 1654, officially publishing it in late November 1665, from now.....
Exactly one month to go!
This is why Xu Yun started with the dispersion phenomenon:
Dispersion is a very typical differential model, even more classic than universal gravitation, regardless if it's the angle of deflection or the "seven-in-one" representation itself, it directly points to calculus tools.
The 1/7 concept directly connects to fractional exponent representation.
Seeing dispersion, Little Niu not thinking about his stuck 'Fluxion Technique' would be rather foolish.
Little Niu observes the dispersion—Little Niu becomes curious—Little Niu measures data—Little Niu thinks about the Fluxion Technique—Xu Yun introduces Yang Hui's Triangle.
It's a perfect logical progression trap, a transition from physics to mathematics.
The reason Xu Yun draws this diagram is simple:
Yang Hui's Triangle is a thorn in the heart of every math professional!
Yang Hui's Triangle is originally a tool our ancestors invented with solid evidence, why should it be forced under someone else's name due to modern grievances?
He can't bother with the original timeline nor has the ability to change it, but at this point in time, Xu Yun won't let Yang Hui's Triangle be shared with Pascal!
With Mr. Niu's endorsement, Yang Hui's Triangle is Yang Hui's Triangle.
A term belonging solely to Huaxia!
Then Xu Yun let out a deep breath and continued drawing lines on top:
"Mr. Newton, see here, both slanted edges of this triangle are composed of the number 1, and all other numbers equal the sum of the two numbers on its shoulders.
Any number C(n, r) illustrated in this graph equals the sum of the two numbers C(n-1, r-1) and C(n-1, r) on its shoulders."
Speaking, Xu Yun wrote an equation on 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 wrote the cubic section, Little Niu's expression gradually became serious.
By the time Xu Yun reached the sixth power, Little Niu couldn't sit still anymore.
Directly stood up, snatched Xu Yun's pen, and wrote for himself:
(a + b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + a^6!
Clearly.
Yang Hui's Triangle has n numbers in the nth row, the sum of the numbers is 2 raised to the power of n-1, and the coefficients in the expansion of (a+b) raised to the nth power correspond sequentially to each term in the (n+1) row of Yang Hui's Triangle!
Although for Little Niu this expansion was trivial, almost basic binomial expansion operation.
Yet it's the first time someone has visually expressed square roots with a diagram!
Moreover, the m numbers in the nth row of Yang Hui's Triangle can be expressed as C(n-1, m-1), representing the combination number of selecting m-1 elements from n-1 different elements.
This is a tremendous support for Little Niu's ongoing binomial subsequent derivation!
However...
Little Niu's brows gradually furrowed:
The emergence of Yang Hui's Triangle opened up a new line of thinking for him, but didn't help much with his current stuck issue, which is the expansion of (P+PQ)m/n.
Because Yang Hui's Triangle deals with coefficients, Little Niu's headache is about exponents.
Little Niu at present feels like an experienced cyclist.
Who, upon turning a mountain road, suddenly sees a wide open plateau ahead, the scenery magnificent, yet sees a huge rock pile blocking ten meters ahead.
While Little Niu was struggling, Xu Yun slowly spoke another sentence:
"By the way, Mr. Newton, Sir Han Li also researched Yang Hui's Triangle.
Later he discovered that the exponent of a binomial doesn't necessarily need to be an integer, fractions even negative numbers seem feasible."
"He didn't detail the method for negatives, but did leave the method for fractions."
"He called it....."
"Han Li Expansion!"
.....