Just as Gregorian pressed forward step by step, Ella's calculations were initially obtained.

"Master... I can't draw a graph as you ask. I want to double the area, that is, the product of the side length of the new square is two. Because the side length of the square is equal, the product of the number itself and itself is two. I originally wanted to calculate what kind of number this is... but I can't calculate it."

Gottfried was attacking Gregory's continuous problems, and Ella's words gave him a chance to change the topic. He hurriedly said: "How do you calculate?"

"I referred to the figure you drew at the door. You used the method of calculating the area of ​​the circle by using two polygons. I also used the same method. First, I got the number between four-thirds and three-halfs, and then continued to look for the fraction between the two... But no matter how I searched, I couldn't find out what this number was."

Ella's words also attracted Gregory's attention. He left behind his investigation into the ancient church of Abraham and said, "Is it just that you didn't calculate deeply enough?"

"No, I also made some special evidence for this, and then found that... this number is impossible to exist."

A light flashed in Gottfried's eyes: "Oh? Tell me about your proof process."

"First of all, the first axiom, multiplying any integer by two will become an even number, right?"

Gregory nodded aside: "Yes, this is an unspoken axiom."

"Secondly, the second axiom is that the square of an even number is an even number, and the square of an odd number is an odd number, isn't it right?"

"it goes without saying."

"Then, I assume that the simplest fraction of this number is manifested in a/b, and its square is 2, that is, (axa)/(bxb)=2, in other words, 2(bxb)=(axa). According to the first axiom, (axa) will be an even number, and according to the second axiom, a is also an even number."

“Verified.”

"Since a is an even number, then a must be divided by 2 to get another integer, right?"

"certainly."

"We represent this integer with s. Then a is equal to 2s. Substitute the formula before, and it becomes 2(bxb)=(2sx2s)=4(sxs), and after simplification, it is (bxb)=2(sxs). According to the first axiom, (bxb) will be an even number, and according to the second axiom, b is an even number."

"Oh, a and b are even numbers, which is a magical discovery. But what does this mean?"

"Don't forget, we set a/b at the beginning to be the simplest fractional representation of this number! If a and b are both even numbers, then they will be able to divide the same number two, which is no longer the simplest! But even if we set the new numbers c and d, let them be one-half of a and b respectively, and then represent this number as c/d, we can prove that c and d are both even numbers through the above method! If we divide this way, this number will never have the simplest fractional representation!"

Ella's words were like throwing a huge rock into a calm lake, causing every muscle on Gregori's face to twitch. He tried to repeat Ella's proof process without finding any problems. But this conclusion made him unacceptable: "You mean that the numerator and denominator of this number can be divided infinitely by two, and keep itself an integer? Is this infinite number... a projection of the god?"

"So I can't draw this figure...a square with two areas, its side length...is strange."

"Don't try to draw anymore!" Gregory suddenly shouted irritably, "It's normal to wonder, because we can't understand the infinite gods! Let it exist there, and never measure it!"

Gottfried laughed as he listened to the argument between the two.

"Do you know the Pythagoras theorem?" he asked suddenly.

Ella and Gregory turned their attention to Gottfried together: "You mean that the square of the oblique side of a right triangle is equal to the sum of the squares of two right angle sides, right? This is Pythagoras' most famous theorem. Why should we mention this?"

"Girl...you draw a diagonal on that square with one side. What is the length of this diagonal?"

Ella continued to draw without thinking, but she stopped halfway through the line and said in a trembling voice: "This line has two squares?"

"Okay, now, using this line as the side length of the new square, has the problem been solved?"

"Wait a minute! Stop!" Ella interrupted Gottfried, "...This should be an infinite number, but why has it now become a limited, traceable line segment?"

Before Gottfried could speak, Ella used her trembling hands to finish the unfinished line. Then, the line lay quietly on the ground, starting from one point to another, without any magic at all.

"You... have measured the infinite?" Gregory first looked at Gottfried in disbelief, then shook his head desperately, "No, this is impossible! This line must have been made by the demon's hand, a prank from the demon!"

"Hey, are you talking nonsense?" Ella couldn't help but say, "Although it is indeed incredible, this is the line I drew with my own hands. Why did it become a devil's hand!"

"He is right, this number comes from the demon." Gottfried said, "Don't you want to know about the Pythagoras school? They think that everything is counted, but they have spent all their lives learning and have not been able to use any number to represent this line segment! It is obvious that this line segment is right in front of you, but it is impossible to express it in the form of numbers, which makes the concept of 'everything is counted' extremely ridiculous."

"Then what……?"

"They couldn't solve this number, so they solved the person who discovered it. His name was Sibersus, who was the incarnation of the demon by his classmates from Pythagoras, and was lifted up and threw it into the sea on the spot! But they could not put this number into the sea. Since then, the magic of the Pythagoras school has declined. Although I am indeed a member of the Pythagoras school, now the Pythagoras school only studies mathematics and no longer has any connection with magic."

"…So, how did the Pythagoras school deal with this number in the end?"

"They abandoned the idea that everything is a number, and they separated the geometric figures from the number. Numbers are numbers, and geometry is geometry. In this way, the length problem of this line segment is avoided."

Ella sat down on the ground. Gottfried's words just meant that she could not learn the magic of the Pythagoras school.

Gregory breathed a sigh of relief.
Chapter completed!