Who delineates the forbidden zone for science
Whoever becomes the enemy of science
As usual, the intern Xiaotian was going to the printing room to print a bunch of documents...
When Xiaotian watched one piece after another of A4 paper come out, a question suddenly flashed in his mind: Why is the A4 paper this size? Is there any "mystery" in it?
Therefore, Xiao Tian took out an A4 paper and asked the almighty supermodel: I have a question, the size of the A4 paper should not be arbitrarily determined, right?
Supermodel Jun: Well, it's a clever little day, of course, it's not set casually.
Xiao Tian: Then tell me the origin of it.
Supermodel Jun: Actually, A4 paper is obtained after the A0 paper is folded in half 4 times.
There are 3 categories of international standard paper, namely: A paper, B paper, and C paper. Among them, A paper is the most widely used.
So, as the most widely used type of paper, what is special about A paper.
We assume that a rectangle is a long and b wide, and that a small rectangle obtained after folding it in half once is b long and a/2 wide.
According to the characteristics of the whole set of A paper being folded in half by A0 paper, a: b = b: a / 2 can be obtained, simplified to obtain a² = 2b², and then transformed to get a: b = √2.
It can be seen that the size of A4 paper is not arbitrarily determined, and the length-to-width ratio of the entire set of A paper is √2.
Sky: Oh, that's right.
Supermodel Jun: By the way, don't underestimate this root number 2, it is the number that triggered the first mathematical crisis.
Xiao Tian was surprised: No, so powerful?!
Supermodel: That's when I tell the story of Pythagoras.
In 500 BC, there was a cattle man named Pythagoras. If you are a little stranger to this cattle man, then the Pythagorean theorem should know, that is, in a right triangle, the sum of the squares of the two right angles is equal to the square of the hypotenuse.
In China, this is known as the "Pythagorean theorem".
He founded a school of mathematics called the Pythagorean school, which held that integers, like atoms, make up everything in the universe and can describe everything in the universe. The relationships between the universes can be expressed in terms of integers or ratios of integers, and apart from that, there is nothing else...
Xiao Tian: This kind of thinking is a bit childish...
Supermodel Jun: I haven't finished talking about it yet. If you look at the description below, you may think that Pythagoras is very reasonable.
Let's look at the question first: Can integers, and fractions divided by two integers, occupy the entire axis of numbers?
Let's start with integers (i.e., fractions with a denominator of 1) and put them on the number axis:
Next, insert all numbers with denominator 2 (half of the number above) in the gap.
Then insert the number with denominator 4:
As the denominator continues to grow, we insert more and more numbers, and the points inserted into the number axis will become more and more dense.
According to this line of thinking, no matter how small the number between two fractions, we can insert numbers with a denominator that are larger.
Xiao Tian: Well, it seems that this is really the case...
Supermodel Jun: However, this view is completely wrong!
Pythagoras had a student named Hibsos. He is studious, observant and analytical.
One day, he ran to Pythagoras and asked him: What is the length of the diagonal of a square with a side length of 1?
Pythagoras was stunned to hear this question, according to his proof of the theorem, the square of the diagonal length of a square with a side length of 1 should be equal to 2, so what number is squared equal to 2?
Pythagoras searched for a long time without finding it, hoping to find two very large numbers divided by them, and the result was equal to this number. But no matter how large the numerator and denominator of the found fraction, this ratio can only be very close, but it cannot be accurately equal to 2 square (at that time there was no expression of √2, only considered to be an irreducible quantity).
This was contrary to the Pythagorean school's theory of "all things are numbered (i.e., rational numbers)", a discovery that made the leaders of the school panic and anger, believing that it would shake their dominance in academia...
Thus, the Pythagorean school established a new discipline: no one was allowed to divulge the secret of the existence of root number 2 (i.e., irrational numbers).
However, the naïve Hippathos once inadvertently spoke to someone about his discovery, and the result was that he was considered a "traitor" of the school, imprisoned, tortured in every way, and finally drowned in the Aegean Sea...
Xiao Tian: Oh, what a pity...
Supermodel Jun: However, the truth cannot be drowned after all, and the Bi school's erasure of truth is "unreasonable".
In honor of Hebsos, a venerable scholar who dedicated himself to truth, the irreducible quantity was named the irrational number, and the number of all the universe (the ratio of integers to two integers) previously considered by Pythagoras was called rational numbers.
The discovery of irrational numbers and the later "Zeno Paradox" (indirect factors) led to a revolution in mathematical thought, which the history of science refers to as the "first mathematical crisis".
Zeno believes that a person who goes from point A to point B must first walk 1/2 of the distance, then walk 1/2 of the total distance, and then walk the remaining 1/2... And so on, and you can never reach the end. Assuming that the person's speed is unchanged, the time for a period of time is divided by 2 each time, and the time is 1/2+1/4+1/8+, the time is limited to the actual time required, that is, the distance between the person and the destination can be any small, but it cannot be reached. In fact, the paradox itself limits the time, of course it can not reach...
Later, it was proved that not only did irrational numbers exist, but there were far more irrational numbers than rational numbers.
In the above method of inserting the score with the increasing denominator, if we think about it carefully, we will find that no matter how much is carried out, there will be countless gaps in the number axis, and these gaps are filled by irrational numbers...
If we put a random needle between 0 and 1, you can say that you have almost a 100% chance of getting an irrational number!
Small Day: This probability...
Supermodel Jun: So, from The dedication of Hibsos to knowledge, people can learn this lesson, "Science is endless, whoever draws a forbidden area for science becomes the enemy of science and is eventually buried by science."
"Super mathematical modeling" (WeChat supermodeling), learn a little knowledge every day, easily understand various thinking, and be a fun rationalist. 600,000 mathematical elites are paying attention!