Pythagorean theorem is also called quotient height theorem, Pythagorean theorem or Pythagorean theorem.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of two right angles. If the two right angles of a right triangle are A and B and the hypotenuse is C, then A? +b？ =c？ That is, α * α+b * b = c * C

Summary: When the exponent becomes n, the equal sign becomes less than the sign.

According to textual research, human beings have known this theorem for at least 4000 years!

The first chapter of China's earliest mathematical work "Weekly Parallel Calculations" contains the relevant contents of this theorem: Duke Zhou asked: "I heard that doctors are good at counting, so I want to ask the ancients to set up a calendar of weeks and days." The sky cannot rise step by step, and the earth cannot be measured. How many times can I go out? "Shang Gao replied:" The number method comes from the circle, the circle comes from the square, the square comes from the moment, and the moment comes from 998 1. Therefore, the moment is considered as three, the stock is four and the diameter is five. Outside is the square, half an hour, a circle is * * *. If you get that the moments of three, four, five and two are twenty and twenty-five, this is called product moments. Therefore, Yu rules the world because this number is born. "That is to say, a rectangle folded diagonally is called a right triangle. If the hook (short right side) is 3 and the rope (long right side) is 4, then the chord (hypotenuse) must be 5. From the above conversation, we can clearly see that people in ancient China discovered and applied Pythagorean theorem, an important mathematical principle, thousands of years ago.

The earliest documents in the west proved to be given by Pythagoras. It is said that when he proved Pythagorean theorem, he was ecstatic and killed a hundred cows to celebrate. Therefore, western countries also call Pythagorean Theorem "Hundred Cows Theorem". Unfortunately, Pythagoras' proof method has long been lost, and we have no way of knowing his proof method.

In fact, in earlier human activities, people have realized some special cases of this theorem. In addition to the above two examples, it is said that the ancient Egyptians also used the law of "hooking three strands, four strings and five" to determine the right angle. However, this legend has aroused the suspicion of many mathematical historians. For example, Professor M. Klein, an American mathematical historian, once pointed out: "We don't know whether the Egyptians realized the Pythagorean theorem. We know that they have rope puller (surveyor), but they tied a knot on the rope, divided the whole length into 3, 4 and 5 sections, and then used them to form a right triangle, which has never been confirmed in any literature. " However, archaeologists discovered several pieces of ancient Babylonian clay tablets, which were completed around 2000 BC. According to expert research, one of them is engraved with the following question: "A stick with a length of 30 units stands upright on the wall. How far is its lower end from the corner when its upper end slides down by 6 units? " This is a special case of a triangle with a ratio of three sides of 3:4:5. Experts also found that there was a strange number table carved on another clay tablet, in which * * * was engraved with four columns and fifteen rows of numbers, which was a Pythagorean number table: the rightmost column was the serial number from 1 to 15, while the left three columns were the values of strands, hooks and strings respectively, and a * * recorded/kloc. This shows that Pythagorean theorem has actually entered the treasure house of human knowledge.

Pythagorean theorem is a pearl in geometry, which is full of charm. For thousands of years, people have been eager to prove it, including famous mathematicians, painters, amateur mathematicians, ordinary people, distinguished dignitaries and even the president of the country Perhaps it is precisely because Pythagorean theorem is important, simple, practical and more attractive that it has been repeatedly demonstrated for hundreds of times. 1940 published a proof album of Pythagorean theorem, which collected 367 different proof methods. In fact, that's not all. Some data show that there are more than 500 ways to prove Pythagorean theorem, and only the mathematician Hua in the late Qing Dynasty provided more than 20 wonderful ways to prove it. This is unmatched by any theorem. (The detailed proof of Pythagorean theorem is not included because the proof process is complicated. ※.)

People are interested in Pythagorean theorem because it can be generalized.

Euclid gave a generalization theorem of Pythagorean theorem in Elements of Geometry: "A straight side on the hypotenuse of a right triangle has an area equal to the sum of the areas of two similar straight sides on two right angles".

From the above theorem, the following theorem can be deduced: "If a circle is made with three sides of a right-angled triangle as its diameter, the area of the circle with the hypotenuse as its diameter is equal to the sum of the areas of two circles with two right-angled sides as its diameter".

Pythagorean theorem can also be extended to space: if three sides of a right triangle are used as corresponding sides to make a similar polyhedron, then the surface area of a polyhedron on the hypotenuse is equal to the sum of the surface areas of two polyhedrons on the right side.

If three sides of a right-angled triangle are used as balls, the surface area of the ball on the hypotenuse is equal to the sum of the surface areas of two balls made on two right-angled sides.

2) Example 2

On Pythagorean Theorem

Pythagorean theorem is a pearl in geometry, so it is full of charm. For thousands of years, people have been eager to prove it, including famous mathematicians, amateur mathematicians, ordinary people, distinguished dignitaries and even national presidents. Perhaps it is precisely because of the importance, simplicity and attractiveness of Pythagorean theorem that it has been repeatedly hyped and demonstrated for hundreds of times. 1940 published a proof album of Pythagorean theorem, which collected 367 different proof methods. In fact, that's not all. Some data show that there are more than 500 ways to prove Pythagorean theorem, and only the mathematician Hua in the late Qing Dynasty provided more than 20 wonderful ways to prove it. This is unmatched by any theorem.

Among these hundreds of methods of proof, some are very wonderful, some are very concise, and some are very famous because of the special identity of witnesses.

In foreign countries, especially in the west, Pythagorean theorem is usually called Pythagorean theorem. This is because they think that the right triangle has the property of "hook 2+ chord 2= chord 2", and Pythagoras, an ancient Greek mathematician, was the first to give a strict proof.

In fact, in earlier human activities, people have realized some special cases of this theorem. In addition to the Pythagorean theorem discovered in China more than 0/000 years ago, it is said that the ancient Egyptians also used the law of "hooking three strands, four chords and five" to determine the right angle. However, this legend has aroused the suspicion of many mathematical historians. For example, Professor M. Klein, an American mathematical historian, once pointed out: "We don't know whether the Egyptians realized the Pythagorean theorem. We know that they have people who pull the rope (surveyors), but the theory that they tied a knot on the rope, divided the whole length into three sections, 3, 4 and 5, and then used it to form a right triangle has never been confirmed in any literature. " However, archaeologists discovered several pieces of ancient Babylonian clay tablets, which were completed around 2000 BC. According to expert research, one of them is engraved with the following question: "A stick with a length of 30 units stands upright on the wall. How far is its lower end from the corner when its upper end slides down by 6 units? " This is a special case of a triangle with a side length of 3:4:5; Experts also found that there is a strange number table engraved on another board, in which * * * is engraved with four columns and fifteen rows of numbers, which is a Pythagorean number table: the rightmost column is the serial number from 1 to 15, while the left three columns are the values of stocks and hook chords respectively, and a * * * records/kloc-0.

Proof method:

Take four identical right triangles first. Put an (a+b) square, and the area of the beige square in the middle is c2. Figure (1) If you change the position of the triangle, you will see two beige squares with an area of (a2, b2). The areas of the four triangles in Figure (2) are unchanged, so the conclusion is: a2+b2 = c2.

History of Pythagorean Theorem;

Shang Gao was a native of China in 1 1 century BC. At that time, China's dynasty was the Western Zhou Dynasty, which was a slave society. In ancient China, it was about the Warring States period.

Zhou Shu Yun Jing, a mathematical work in the Western Han Dynasty, recorded a dialogue between Shang Gao and Duke Zhou. Shang Gao said, "... so the moment is folded, and the shares are changed four times."

The phrase "Gao Gao" means that when the two right-angled sides of a right-angled triangle are 3 (short side) and 4 (long side) respectively, the diameter.

The angle (i.e. chord) is 5. In the future, people will simply call this fact "three strands, four strings and five". This is the famous Pythagorean theorem.

Regarding the discovery of Pythagorean theorem, Zhou said: "So, Yu ruled the world because of the birth of this number." "This number" means "hook"

"Three strands, four strings and five" means that the relationship between three strands, four strings and five was discovered when Dayu was in charge of water control.

Zhao Shuang:

Wu people from the end of the Eastern Han Dynasty to the Three Kingdoms period.

He took notes for Zhou Pian and wrote the Pythagorean Square.

Zhao Shuang's proof is ingenious and innovative. He proved the consistency between algebraic expressions by cutting, cutting, spelling and supplementing geometric figures.

Equivalence is not only rigorous but also intuitive, which is inseparable from the unity of form and number and the close combination of algebra and geometry in ancient China.

The unique style sets a good example. Later mathematicians mostly inherited this style and developed it. Liu Hui, for example, proved it later.

Pythagorean theorem is also a method to prove numbers in form, but the division, combination, displacement and complement of specific numbers are slightly different.

The discovery and proof of Pythagorean theorem by ancient mathematicians in China has a unique contribution and position in the history of mathematics in the world, especially among them.

The thinking method of "unity of form and number" is of great significance to scientific innovation. In fact, the thinking method of "unity of form and number" is correct.

It is an extremely important condition for the development of mathematics. As Wu Wenjun, a contemporary mathematician in China, said, "In China's traditional mathematics, the relationship between quantity and quantity.

Descartes invented analytic geometry in17th century, which is the traditional thought of China.

After hundreds of years of pause, ideas and methods reappear and continue. "

At the beginning of China's earliest mathematical work "Parallel Calculation of Classics in Weeks", there was a dialogue in which the Duke of Zhou asked Shang Gao for mathematical knowledge:

Duke Zhou asked, "I heard that you are very proficient in mathematics. Excuse me: there is no ladder to go up in the sky, and you can't go with a ruler. "

A measurement, so how can we get the data about heaven and earth? "

Shang Gao replied: "The number comes from the understanding of the other party and the circle. There is a principle: when a right triangle is a moment.

When one right-angled side "hook" equals 3 and the other right-angled side "strand" equals 4, then its hypotenuse "chord" must be 5. This truth was summed up when Dayu was in charge of water conservancy.