Outside the house.
Watching little Newton rush back into the house, Xu Yun vaguely sensed something and quickly followed.
"Bang——"
As soon as he entered the house, Xu Yun heard a sound of a heavy object hitting something.
He looked in the direction of the sound, only to see little Newton standing by the desk with a frustrated look on his face, his left hand clenched into a fist, the knuckles pressing heavily on the desk.
Clearly, little Newton had just thrown a punch at the desk.
Seeing this, Xu Yun walked over and asked:
"Mr. Isaac, what is this....."
"You wouldn't understand."
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Little Newton waved his hand irritably, but after a few seconds, he seemed to think of something:
"Fat Fish, you——or Sir Han Li, do you understand mathematical tools?"
Xu Yun gave him a blank look and asked:
"Mathematical tools? You mean a ruler? Or a compass?"
Hearing this, little Newton's heart sank, but he couldn't just stop halfway through his sentence, so 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 some particles that are invisible to the naked eye.
To calculate this rate of change, we need to use another tool that can continuously accumulate, to calculate the product of the refraction angle.
For example, the product of n times a+b is taken from a or b in a+b, such as (a+b)^2=a^2+2ab+b^2... Forget it, I guess you wouldn't understand."
Xu Yun gave him a half-smiling look and said:
"I understand, it's Pascal's triangle."
"Right, so let's prepare to go to Uncle William's..... Wait, what did you say?"
Little Newton was originally speaking according to his own thoughts, but when he heard Xu Yun's words, he was taken aback, then suddenly looked up and stared at him:
"Pascal's triangle? What is that?"
Xu Yun thought for a moment, then reached out to little Newton:
"Could you pass me the pen, Mr. Isaac?"
If this had been a day earlier, when little Newton had just met Xu Yun, Xu Yun's request would have been rejected outright.
He might even have been met with another 'You're not worthy'.
But after the recent derivation of the dispersion phenomenon, little Newton had developed a slight interest and recognition for Xu Yun——or the Sir Han Li behind him.
Otherwise, he wouldn't have bothered to explain so much to Xu Yun just now.
So faced with Xu Yun's request, little Newton unusually handed over the pen.
Xu Yun took the pen and quickly drew a diagram on the paper:
.............1
.......1......1
....1......2......1
1.....3.......3.........1 (Please ignore the ellipsis, without it the starting point will automatically indent, dizzy)
.......
Xu Yun drew eight lines in total, with the two outermost numbers on each line being 1, forming an equilateral triangle.
Those familiar with this image should know that this is the famous Pascal's triangle, also known as Yang Hui's triangle——in the international mathematical world, the latter is more widely accepted.
But in fact, Yang Hui discovered this triangle more than four hundred years earlier than Pascal:
Yang Hui was born in the Southern Song Dynasty, and in 1261, he preserved a precious diagram in his "Detailed Explanation of the Nine Chapters of Mathematics"——the "Root Method Origin" diagram, which is the oldest existing triangular diagram.
However, due to some well-known reasons, Pascal's triangle is much more widely spread, and some people don't even recognize the name of Yang Hui's triangle.
Therefore, despite Yang Hui's original records, this mathematical triangle is still called Pascal's triangle.
But it's worth mentioning that......
Pascal studied this triangle in 1654 and officially announced it in late November 1665, which is.....
Still a whole month away!
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, whether it's the deflection angle or its own "seven-in-one" appearance, it directly points to the calculus tool.
The concept of 1/7 is directly linked to the fractional statement of the exponent.
If little Newton, who encountered the dispersion phenomenon, didn't think of his own 'fluxion method' that he was struggling with, then he might as well go to bed.
Little Newton sees the dispersion phenomenon——little Newton becomes curious——little Newton calculates data——little Newton thinks of the fluxion method——Xu Yun brings up Yang Hui's triangle.
This is a perfect logical progression trap, a situation from physics to mathematics.
As for the reason why Xu Yun drew this diagram, it's simple:
Yang Hui's triangle is a thorn in the heart of every mathematician!
Yang Hui's triangle was originally a mathematical tool invented by our ancestors and has solid evidence, so why should it be named after someone else because of the frustration of modern times?
He couldn't and didn't have the power to manage the original timeline, but at this point in time, Xu Yun wouldn't let Yang Hui's triangle share its name with Pascal!
With Newton as a guarantee, Yang Hui's triangle is Yang Hui's triangle.
A term that belongs only to China!
Then Xu Yun let out a sigh of relief and continued to draw a few lines on it:
"Mr. Isaac, look, the two sides of this triangle are made up of the number 1, and the rest of the numbers are equal to the sum of the two numbers on its shoulders.
From the diagram, any number C(n,r) is equal to the sum of the two numbers C(n-1,r-1) and C(n-1,r) on its shoulders."
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
As Xu Yun wrote down the cubic column, Little Newton's expression gradually became serious.
By the time Xu Yun wrote down the sixth power, Little Newton could no longer sit still.
He simply stood up, snatched Xu Yun's pen, and began to write himself:
(a+b)^6=a^6+6a^5b+15a^4b^2+20a^3b^3+15a^2b^4+6ab^5+a^6!
It's clear.
The numbers in the nth row of Pascal's triangle have n terms, the sum of the numbers is 2 to the power of n-1, and the coefficients in the expansion of (a+b) to the nth power correspond to each term in the (n+1)th row of Pascal's triangle!
Although this expansion is not difficult for Little Newton, it can even be considered as the basic operation of binomial expansion.
However, this is the first time someone has so intuitively expressed the square root number in a graphical form!
More importantly, the m numbers in the nth row of Pascal's triangle can be represented as C(n-1, m-1), which is the combination number of selecting m-1 elements from n-1 different elements.
This is undoubtedly a huge help for the binomial derivation that Little Newton is currently working on!
But......
Little Newton's brow gradually furrowed again:
The appearance of Pascal's triangle can be said to have opened up a new line of thought for him, but it didn't help much with the problem he was currently stuck on, which was the expansion of (P+PQ) m/n.
Because Pascal's triangle involves the issue of coefficients, and what Little Newton is troubled by is the issue of exponents.
Little Newton now is like an experienced cyclist.
When he turned a mountain road and suddenly found a flat plain a hundred meters ahead with a magnificent view, but there was a huge pile of fallen rocks blocking the road just over ten meters ahead.
Just as Little Newton was in a dilemma, Xu Yun slowly said another sentence:
"By the way, Mr. Isaac, Sir Han Li also did some research on Pascal's triangle.
He later found that the exponent of the binomial does not necessarily need to be an integer, fractions and even negative numbers seem to be feasible."
"He didn't explain the proof method for negative numbers, but he did leave the proof method for fractions."
"He called it....."
"Han Li Expansion!"
.....
Note:
Some readers have been asking these days, let me reiterate, this is a science fiction novel, there will be real-life scenarios later......
A book of several million words, we're just getting started, and some people are saying the protagonist hasn't done anything....
The pace at which I write has always been slow, and the setup will be a bit longer, in the last book of 1.4 million words, only one person has reached the Foundation Establishment stage.....
I said when I started the book, if you want to read the kind of book where the protagonist starts off killing all around and becomes a billionaire in twenty chapters, you can look elsewhere, I can't write that kind of book.
Meeting Newton in the first chapter, throwing out the law of universal gravitation in the third chapter, returning to reality in the fifth chapter, does that make sense?
Moreover, although the protagonist's pace is slow, both I and the feedback from most readers indicate that the plot so far is readable, and that's enough.
Qidian has always been a tolerant platform, when did not writing fast-paced books deserve criticism?
Scratching my head, puzzled.