2021-12-01 11:25 (수)
Pythagorean theorem causes a catastrophe
Pythagorean theorem causes a catastrophe
• 조송현
• 승인 2021.03.22 17:41
• 업데이트 2021.03.22 17:50
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The re-reading of the 'The Universe-View Odyssey' in English (8) The Pythagoras statue of the Greek island of Samos. It stands with a right triangle-shaped sculpture that embodies the immortal Pythagorean theorem he proved. Source: samos-beaches.com

Even if you don't like math, you'll have a rough idea of what Pythagorean theorem is about. The Pythagorean theorem proves that the square of the length of the sides in a right triangle is equal to the square of the other two sides.

a² + b² = c²

At first glance, this is a very simple arrangement. At that time, Egypt and Babylonia also knew about this and actually applied it. But they didn't 'prove' that this theorem applies to all right triangle. The Pythagorean school proved this.

Pythagoras's proof is beyond dispute. This demonstration perfectly demonstrated that every right triangle in the universe satisfies the Pythagorean theorem. It was such an important discovery at the time that the Pythagorean school reportedly thanked God by offering 100 cows to the altar.

This discovery was a landmark in the history of mathematics and a major turning point in the history of human civilization. There are two reasons why the proof is considered so important. The first is that it developed the concept of proof. The conclusions made through mathematical proofs have deeper credibility than any conclusion in the world, because they are the product of perfect logic that progresses step by step. Thales, an ancient Greek philosopher, developed a rudimentary proof of geometry, but not comparable to Pythagoras. Pythagoras demonstrated mathematical propositions in a more rigorous and elegant way than Thales by taking the concept of proof to a few dimensions.

The second significance of the Pythagorean theorem is that abstract mathematics has been successfully linked to real objects. Pythagoras was the first to discover the surprising fact that 'mathematics applies to the scientific world, so mathematics provides the logical basis for dominating the world.' Mathematics helps science start from the foundation of strict truth.

faced a disaster through the Pythagorean theorem

However, it is ironic that the Pythagorean school faced disaster through its famous theorem. They could not hide their embarrassment by translating the existence of an irrational number through the theorem. Draw a right triangle with two sides of a right angle that are one length each. The square of the length of this right triangle's sides is that by Pythagorean theorem, 1²' + 1² = x² = 2. That is, the length of the sides is 'the number of squares to 2'. What is this number like?

Today, in mathematics, the number of squares to 2 is written √2 and read root 2. But this expression didn't exist at that time. The Pythagorean school expected all the geometry they developed to be represented by the ratio of integers or integers, rational numbers. No, it had to be. rational number is a very simple number, and no matter how long or short distance is expressed asrational number. If all geometry is represented by rational numbers alone, understanding geometry will be as easy as that. It's not just geometry. As the Pythagorean school said, 'All things are numbers,' numbers were believed to be the building blocks of all things and the principles of all things.

At this time, the number they thought of was the number that appeared as the ratio of an integer to an integer. These numbers are called rational numbers, which have integers and fractional numbers, and are represented by a decimal number, which is either a finite decimal or circulating decimal. The number was also an object of religious worship. If the existence of a strange number other than the rational number is revealed, wouldn't the rational number religion collapse? Wouldn't the monotheists be in a state of religious panic if they witnessed the descent of a pagan god?

The Pythagorean school discovered some numbers that could not be represented by integers or ratios through the immortal mathematical theorem they discovered by them. That is, the square number of two had to be assumed to be some strange number, which cannot be expressed by the integer ratio. It's not just the number of squares that make it two. If the lengths of two sides of a right triangle are 1 and 2, respectively, the hypotenuse of a right-angled triangle are squared according to the Pythagorean theorem (X² = 1² +2²) and are 'number 5'. You can't find this with the ratio of an integer or an integer. Pythagorean theorem also gave birth to a number of odd numbers that cannot be represented by rational numbers, such as number of squares to be 8 and number of squares to be 10(the 'rational' in rational numbers comes from ratio).

The Pythagorean school tried to somehow express this value in conventional rational number, but it all ended in vain. The Pythagorean school had discovered a catastrophic but grim fact that there were numbers that could not represent the proportion of the two integers. The reason why the question of 'Prove that √2 is not rational number' has often appeared in previous college entrance exams is probably because the birth of irrational number, including √2, has this historical meaning in mathematics.

The Pythagorean school fell into a psychological panic due to the discovery of an irrational number of people. Because the number they worshiped at the religious level only meant integers and the ratio of rational number. They had atomic views on quantity and number, so it was only natural to fall into a dilemma. The discovery of an indescribable number in the proportion of two integers, the creation of God, shook all the belief systems of their sect.

By the time such a dangerous discovery was made, the Pythagorean school was forming a systematic denomination dedicated to the study of the power and mystery of numbers. Hippasus, a member of these denominations, is known to have been guilty of leaking the secrets of the existence of the irrational number to the outside world. There have been countless legends in the aftermath of this incident. They include the legend that Hippasus was expelled from the denomination, the legend that Pythagoras hanged or drowned the traitor himself, and the legend that the cult members sank the ship after putting him on board and sending him far away. The Pythagorean school's belief that the integer is sacred ended with the death of Hippasus.

The Pythagorean school solved the dilemma caused by the discovery of irrational number in two mathematical and physical ways. The mathematical solution was to abandon most of the arithmetic and focus on geometry. Geometry does not represent √2 as a constant number, but it can be represented as a constant length of line segment. Ancient Greek geometry was born after the secrets of irrational number were known to the world. Geometry deals with lines, faces, and angles. It's all continuous. It is said that research on the concept of continuum began only after the discovery of irrational numbers. A rational number is stated in a finite number of terms. On the other hand, the expression of an irrational number, such as a pi(π), is inherently infinite. This means that in order to completely define the number of irrational number digits, it must be expressed in myriad digits.

In terms of physics, They started to think about the physical properties of the unit points, excluding the atomic number of characteristics. According to the Pythagorean mathematical universeview, there should be nothing between integers and rational number. Isn't this universe consistent with atomic theory that it is a vacuum between atoms and atoms? But once it was revealed that there were an infinite number of groups between the rational numbers, it would be hard to believe in atomic theory. So it was around this time that the ether hypothesis, which contrasted with atomic theory, looked up. The ether hypothesis assumes that space is filled with an infinite number of unknown materials, just as there are countless overpopulations between the rational numbers.

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