"Quick! Count how many radishes are crushed!" the tattered man and the dirty man shouted in panic. Each of them was holding a generation of radishes in their hands.
But the girl suddenly laughed on the ground: "Ha...haha...hahahaha! I figured it out!"
There is no doubt that the man with tattered clothes is Gregory, the man with dirty clothes is Gottfried, and the girl is Ella.
The three of them were on their way to Ditmarschen, passing through the city of Hamburg. Along the way, Ella kept pestering Gottfried to learn the mathematics of the Pythagorean school. Gottfried found it troublesome, so he
He tried to persuade her to give up a few geometry problems, but Ella quickly solved them. Gottfried had to keep making the geometry problems more difficult. Along the way, he found the auxiliary lines needed to solve the problems.
It has gone from one or two to a dozen, but Ella still learns it easily.
Hamburg is very close to Stade, but Ella was so obsessed with doing geometry problems that she never thought about going back.
"Gottfried! Look what you did! You drove this innocent girl crazy!"
Gregory was complaining on the side. Ella was studying mathematics, and it was sad for him. He also had some basic knowledge in mathematics. Gottfried set up the questions and Ella did it. He had nothing to do, so he just stood aside quietly and silently.
Thinking about it, it wasn't like he couldn't keep up, but as the difficulty increased, his brain was in a state of confusion most of the time. Every time, he had to wait until Ella was halfway through solving the problem before he woke up as if from a dream.
"I just taught her a few geometry problems! It must be just because she broke her brain in that fall just now!"
Gottfried hurriedly put aside the relationship, but his confidence was obviously insufficient. Even in his eyes, Ella had learned a bit of a nagging state in the past few days. He was worried that Ella's soul would interact with magic if she continued to learn like this.
There is another unrelated sense of "ascension".
"I didn't break my brain!" Ella protested, "I just suddenly figured out something!"
As she spoke, she picked up a radish and said "1", picked up a second radish and said "2", then picked up a third radish and said "3". She put
Put the three radishes back on the ground, raised his head and said, "Do you understand?"
"Only three radishes intact?"
"No! I'm talking about the origin of numbers! 1, 2, 3...the way these numbers are written, the Qiqiu Empire and the Tianfang Empire are completely different, but they all mean the same meaning. And no matter which country's language, numbers
It will not be one or two less than other countries. It will not be said that the Qiqiu Empire only has 1, 3 and no 2. It will not say that the Tianfang Empire has only 1, 2 and no 3. Why is this? Because these numbers all come from
The need for measurement of real things! This is how integers were born, and they are the same no matter which country you are in!"
After speaking, Ella divided the three radishes into three parts, gave one to Gregory, and another to Gottfried, leaving one for herself. Then she looked at the two people and asked: "Understood."
Is it done?”
Gregory glanced at Carrot and asked, "Do you want to say that this is how fractions were born?"
"That's right! Whether it is integers or fractions, they were all created because of people's application needs! For a long time, these two kinds of numbers can meet all people's needs, so that everyone thinks that there are only these two kinds of numbers.
—There is really only one, since any integer can be expressed as a fraction."
Ella could not hide the excitement of being young in her heart.
"But now, we have discovered a special number in geometry problems - rather than admitting that the numerator and denominator of this number are even numbers no matter how many twos it divides, it is better to think that this number cannot be expressed as a fraction at all, which is more in line with people's thinking.
Rational, right? Since fractions are not enough, why can’t we create a new number? Just like people create fractions?”
After saying that, Ella looked at Gottfried expectantly. She thought she had solved the problem that troubled the Pythagoreans. Unexpectedly, Gottfried smiled as if he had expected it.
Got up.
"Congratulations on taking the first step to master infinity. You are indeed talented because you spend much less time than me."
Ella was startled: "You already knew this answer?"
"This is a matter of course. Since this number can be represented by a finite line segment, it cannot be infinite. Although it can be expressed as a decimal and continue indefinitely, it is a finite and measurable number. You
You mistakenly think it is infinite, just because the tools you use are limited to fractions. If you change a tool, everything will become vaster."
As he spoke, Gottfried knelt down and wrote a root symbol on the ground.
"I use such a symbol to represent this type of number. Because the square of the number we are discussing is two, then just write two under this symbol."
"It's meaningless." Gregory snorted, "It's just a symbolic symbol. It's not a number at all."
"Of course this is a number, because it obeys certain mathematical rules. And I can also mark it on the number line. You see, make a same right triangle on the number line, and then draw it with a circular orbit, you can find
Its specific position on the number line..."
"It's a little trick of playing with nouns." Gregory said, "Look at your own symbol, it's almost exactly the same as the division method used by businessmen. You just define the process as the result, but you have no idea how the process should be carried out.
No answer."
But what Ella cares about is completely different from Gregory.
"Since this number can be expressed numerically and can even find its position on the number line, what are the Pythagoreans still wondering about?"
"First of all, the way this number is expressed, its position on the number axis, and its algorithm were all discovered by me. Most members of the Pythagoreans did not know it."
After a pause, Gottfried continued:
"Secondly, even so, the Pythagorean school's magical principle of 'everything is number' is problematic. Because the expression of this number on the number line must rely on geometry, and the problem of geometry can be completely separated from the number...
In the study of geometry, members of the Pythagoreans became more and more aware that most geometric problems could only be solved with geometric techniques, but not with numbers and formulas. The Magic of the Pythagoreans
The strength is based on the user's understanding of mathematics, but the deeper the mathematical foundation, the more aware of the insurmountable gap between geometric identities. So... if you want to learn the magic of the Pythagoreans, you still have to count.
Alright."
"Let me interject a word." Gregory lifted the sack in his hand, "Why on earth did we... buy so many radishes?"
