Just as Gregory was pressing closer, Ella's calculations initially got the results.
"Master... I can't draw the figure according to your request. To make the area double, that is to say, the product of the side lengths of the new square is two. Since the side lengths of the squares are equal, that is to say, the number itself and itself
The product of is two. I wanted to calculate what kind of number this was...but I couldn't."
Gottfried was struggling with Gregory's continuous questions. Ella's words gave him a chance to change the subject. He hurriedly said: "How do you calculate?"
"I referred to the figure you drew at the door. You used the method of pinching two polygons to calculate the area of a circle. I also used the same method. First, I found that the number is between four-thirds and two-thirds.
between three, and then kept looking for the fraction between those two...but no matter how hard I looked, I couldn't figure out what that number was."
Ella's words also attracted Gregory's attention. He put aside his pursuit of the ancient church of Abraham and said from the side: "Is it just that your calculations are not in-depth enough?"
"No, I went out of my way to prove it, and then I found out... that this number simply cannot exist."
A light flashed in Gottfried's eyes: "Oh? Tell me about your proof process."
"First of all, the first axiom, any integer multiplied by two will become an even number, right?"
Gregory nodded aside: "Yes, this is a self-evident axiom."
"Secondly, the second axiom, the square of an even number is an even number, and the square of an odd number is an odd number, is it true?"
"it goes without saying."
"Then, I assume that the simplest fractional representation of this number is a/b, and its square is 2, that is, (a×a)/(b×b)=2. In other words, 2(b×b
)=(a×a). According to the first axiom, (a×a) will be an even number, and according to the second axiom, a will also be an even number.”
"Exactly."
"Since a is an even number, then a must be divided by 2 to get another integer, right?"
"certainly."
"We represent this integer as s. Then a is equal to 2s. Substituting into the previous formula, it becomes 2(b×b)=(2s×2s)=4(s×s). After simplification, it is (b
×b)=2(s×s). According to the first axiom, (b×b) will be an even number, and according to the second axiom, b is an even number.”
"Oh, a and b are both even numbers. What a magical discovery. But what does this mean?"
"Don't forget, we assumed at the beginning that a/b is the simplest fractional representation of this number! If a and b are both even numbers, then they must be divided by two, which is no longer the simplest form! But even if
We set new numbers c and d, let them be one-half of a and b respectively, and then express this number as c/d. We can also prove again that c and d are even numbers through the above method! So
If we continue to divide it, this number will never be able to have the simplest fractional representation!"
Ella's words were like throwing a huge stone into a calm lake, making every muscle on Gregory's face begin to twitch. He tried to repeat Ella's proof process, but found no problems.
But this conclusion was unacceptable to him: "You are saying that the numerator and denominator of this number can be divided by two infinite times and remain an integer? This infinite number... is it the projection of a god?"
"So I can't draw this shape...a square with area two, and its side lengths...weird."
Valley
"Stop trying to draw!" Gregory suddenly shouted angrily, "Strangeness is normal, because we cannot understand the infinite gods! Let it exist there, and never measure it!"
Gottfried listened to the argument between the two and laughed.
"Do you know the Pythagoras theorem?" he asked suddenly.
Ella and Gregory both turned their attention to Gottfried: "You mean, the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two right sides, right? This is Pythagoras
The most famous theorem. Why do you need to mention this?”
"Girl... you draw a diagonal line on the square with side length one. What is the length of this diagonal line?"
Ella continued drawing without thinking, but she stopped halfway through the line and said in a trembling voice: "The square of this line is two?"
"Okay, now, use this line as the side length of the new square. Is the problem solved?"
"Wait a minute! Stop!" Ella interrupted Gottfried, "...this should be an infinite number, but why is it now a finite line segment that can be drawn?"
Before Gottfried could speak, Ella completed the unfinished line with trembling hands. Then, the line lay quietly on the ground, starting from one point and ending at another point.
, it doesn’t reflect the slightest bit of magic at all.
"You...measured infinity?" Gregory first looked at Gottfried in disbelief, then shook his head desperately, "No, this is impossible! This line segment must have come from the hands of the devil. It was
A prank from the devil!”
"Hey, are you talking nonsense!" Ella couldn't help but say, "Although it is indeed unbelievable, these are the lines I drew with my own hands, how come they were made by the hands of the devil!"
"He is right, this number comes from the devil." Gottfried said at the side, "Don't you want to know about the Pythagoreans? They think that everything is number, but they spent their whole life
From what I've learned, I can't use any number to represent this line segment! This line segment is clearly right in front of me, but there is no way to express it in the form of numbers, which makes the idea that 'everything is number' seem ridiculous."
"Then what……?"
"They couldn't solve the number, so they solved the person who discovered the number. His name was Sibersus. He was regarded as the incarnation of the devil by his Pythagorean classmates, and he was lifted up and thrown into the sea on the spot!
But they could not throw this number into the sea. Since then, the magic of the Pythagoreans has declined. Although I am indeed a member of the Pythagoreans, today the Pythagoreans only
Study mathematics and have nothing to do with magic."
"...So, how did the Pythagoreans deal with this number in the end?"
"They abandoned the idea that "everything is number" and separated geometric figures from numbers. Numbers are numbers, and geometry is geometry. In this way, the issue of the length of this line segment is avoided."
Ella sat down on the ground. What Gottfried just said meant that she could no longer learn Pythagorean magic.
Gregory breathed a sigh of relief.
"That's right. Even if it can be symbolized by graphics and symbols, it cannot be measured by specific numbers. This is infinity."
