What is the Pythagorean number?
The Pythagorean school believes that "1" is the first principle of numbers, the mother of all things and the mother of wisdom; "2" is the principle of opposition and negation, and it is an opinion; "3" is the shape and shape of everything; "4" is justice and a symbol of the creator of the universe; "5" is odd and even, and the combination of men and women is also marriage; "6" is the life of God and the soul; "7" is an opportunity; "8" is harmony, fraternity and friendship; "9" is rational and powerful; "10" contains all the numbers, perfect and beautiful.
The golden section of Pythagoras: (a: b =
Pythagoras school believes that the ratio of the orbits of the sun, the moon and the stars to the distance from the earth is equal to three harmonious intervals, namely octave, fifth degree and fourth degree.
The Pythagorean school thinks that summer is dominated by heat, winter by cold, spring by dryness and autumn by wetness, and the best season is the harmonious and balanced distribution of cold, heat, dryness and wetness in quantity.
The Pythagorean school lists ten pairs of opposites from the mathematical point of view, namely, quantitative contradictions, namely, finite and infinite, one and many, odd and even, square and rectangle, good and evil, light and dark, straight and curved, left and right, yang and yin, dynamic and static, among which finite and infinite, one and many opposites are the most basic opposites, and everything in the world is attributed to these ten pairs of opposites.
Everything counts.
Pythagoras school is the first school to put the concept of number in a prominent position. They attach great importance to mathematics and try to explain everything with numbers. It is claimed that number is the origin of all things in the universe, and the purpose of studying mathematics is not to use it but to explore the mysteries of nature. They abstract the number five from five apples, five fingers and so on. This is a common thing today, but it was a great progress in philosophy and practical mathematics at that time. In practical mathematics, it makes arithmetic possible. Philosophically, this discovery makes people believe that numbers are the foundation of the material world.
At the same time, he arbitrarily exaggerates that immaterial and abstract numbers are the origin of the universe, and thinks that "everything is a number", "number is the essence of everything" and "the principle of existence", and the whole universe is a harmonious system of numbers and their relationships. Pythagoras mystified numbers, believing that numbers are the mother of gods, the origin of the universe, and the principle of opposition and negation in nature.
After the Pythagorean theorem was put forward, hippasus, a member of his school, considered a question: What is the diagonal length of a square with a side length of 1? He found that this length can not be expressed by integer or fraction, but only by a new number. Hippasus's discovery led to the birth of the first irrational number √2 in the history of mathematics. The appearance of small √2 set off a huge storm in the mathematics field at that time. It directly shook the Pythagorean school's mathematical belief and made the Pythagorean school panic. In fact, this great discovery is not only a fatal blow to Pythagoras school. This was a great shock to the thoughts of all the ancient Greeks at that time. The paradox of this conclusion lies in its conflict with common sense: any quantity can be expressed as a rational number within any precision range. This is a widely accepted belief not only in Greece at that time, but also in today's highly developed measurement technology. However, the conclusion that is convinced by our experience and completely in line with common sense is overturned by the existence of a small √2! How contrary to common sense and ridiculous this should be! It just subverts the previous understanding. To make matters worse, people are powerless in the face of this absurdity. This directly led to the crisis of people's understanding at that time, which led to a big storm in the history of western mathematics, known as the "first mathematical crisis."
pythagorean theorem
Pythagoras himself is famous for discovering Pythagoras theorem (called Pythagoras theorem in the west). The Babylonians knew this theorem for a long time (in ancient China, it was a dialogue between Shang Yang and Zhou Gong in the mathematical book Zhou You Shu Jing, which was written in the 2nd century BC to 1 century BC. Shang Gao said: "... so fold the moment, tick three, fix four, and cross the corner five." Quotient height means that when two right-angled sides of a right-angled triangle are 3 (short side) and 4 (long side) respectively, the radius angle (chord) is 5. In the future, people will simply describe this fact as "hooking three strands, four strings and five". This is China's famous Pythagorean theorem. ), but the earliest proof can probably be attributed to Pythagoras. He proved by deduction that the square of the hypotenuse of a right triangle is equal to the sum of the squares of two right angles, that is, Pythagorean Theorem.
Pythagorean theorem is an elementary geometric theorem, one of the important mathematical theorems discovered and proved by human beings in the early days, one of the most important tools to solve geometric problems with algebraic ideas, and one of the ties between number and shape. "Hooking three strands, four chords and five" is one of the most famous examples of Pythagorean theorem. When integers a, b and c satisfy a? +b？ =c？ In this case, (a, b, c) is called a pythagorean array. That is to say, if the two right angles of a right triangle are A and B and the hypotenuse is C, then A? +b？ =c？ . It also has a long history in the history of Chinese mathematics, and it is the most important thing in the middle calculation. There is a description of "hooking three strands, four chords and five" in the Parallel Computation Classic of Zhou Dynasty, and Zhao Shuang's Notes on Parallel Computation describes the Pythagorean Theorem as "Pythagorean Theorem is a chord reality that multiplies each other. In addition to prescriptions, it is a string. " .
number theory
Pythagoras did a lot of research on number theory, and divided natural numbers into odd numbers, even numbers, prime numbers, perfect numbers, square numbers, triangular numbers and pentagonal numbers. Pythagoras believes that number provides a conceptual model for the universe. Quantity and shape determine the form of all natural objects. Numbers have not only quantity, but also geometry. In this sense, they understand number as the form and image of natural objects, and it is the total root of all things. Because there are geometric points when there are numbers, there are surfaces when there are wires, and there are three-dimensional points when there are points. When there are solids, there are four elements, namely fire, air, water and earth, which make up everything, so numbers come before things. All phenomena and laws in nature are determined by numbers, and they all obey the "harmony of numbers", that is, obey the relationship of numbers.
Pythagoras further proved his theory by explaining the relationship between numbers and physical phenomena. He once proved that when three strings are used to produce a musical tone and its fifth and eighth tones, the ratio of the lengths of the three strings is 6:4:3. Starting from the viewpoint that the sphere is the most perfect geometric shape, he thinks that the earth is spherical and puts forward the viewpoint that the sun, the moon and the planets move in a uniform circle. He also thinks that ten is the perfect number, so there must be ten luminous bodies moving in the sky.
theory
He also has a theory that the earth revolves around a "central fire" along a fixed point of a sphere in space, and there is an "anti-earth star" on the other side to balance it. This "central fire" is the altar of the universe and people will never see it. The distance between these ten celestial bodies and the central fire has the same proportional relationship with the interval between syllables, thus ensuring the harmony of the planets and playing the music of celestial bodies.
integer
Pythagoras and his school have many creations in mathematics, especially interested in the changing law of integers. For example, a number whose sum of all factors (except itself) is equal to itself is called a perfect number (such as 6,28,496, etc.). ), the sum of a number is less than its factor called abundance; A number greater than the sum of its factors is called a deficit.
Other contributions
In geometry, the Pythagorean school proved that the sum of the interior angles of a triangle is equal to two right angles. Studied the golden section; The method of regular pentagon and similar polygon was found. It is also proved that there are only five regular polyhedrons-regular tetrahedron, regular hexahedron, regular octahedron, regular dodecahedron and regular icosahedron.
In terms of music, Pythagoras corresponds the harmony of intervals with the harmonious order of the universe and stars, and brings music into his abstract interpretation theory of the world centered on numbers. His discussion on the relationship between chord length ratio and music harmony has had a scientific bud. Great contribution to the law of five degrees.