How Einstein proved the Pythagorean theorem

Einstein and the Pythagorean theorem[1] Wang Bonian, Song Limin, Shi Zhaoshen (University of Shanghai for Science and Technology, Shanghai 200093, China) [Abstract] Based on the analysis of reliable and original biographical materials of Einstein, Einstein's "Autobiography" and Euclid's "Geometric Primitives", it can be confirmed that Einstein independently came up with a proof of the Pythagorean theorem when he was 12 years old, and this is the simplest and best of the many proofs. However, this is not innovative, as it is present in the Geometry Originals. Einstein's natural curiosity, keen rational thinking, diligent inquiry, and the education of the enlighteners were necessary for this miracle to happen.

Keywords] Albert Einstein; Pythagorean theorem; Euclid; In June 2004, the 58th session of the United Nations decided that 2005 would be the World Year of Physics. For the first time in the history of the United Nations, the World Year has been named after a science, to commemorate the 100th anniversary of Albert Einstein's miraculous publication of five epoch-making academic papers in 1905, as well as the 50th anniversary of the death of the greatest physicist of the 20th century.

Einstein was not a mathematician, but a theoretical physicist. He applied tensor analysis, which was in the creation stage at that time, to general relativity, which not only found an effective mathematical tool for this theory, but also played an important role in promoting and improving the development of tensor analysis in mathematics. In addition, Einstein also made direct contributions to mathematics in terms of Einstein's summation convention [1] and Einstein tensors [2].

This article does not examine the relationship between Einstein and tensor analysis.

Einstein and the Pythagorean theorem.

The Pythagorean theorem formula is that the square of a plus the square of b is equal to the square of c. If the two right-angled sides of a right-angled triangle are a, b, and the hypotenuse is c, then the formula is: a2+b 2=c 2.

The Pythagorean theorem now has about 500 ways to prove it, making it one of the most provable theorems in mathematics. The Pythagorean theorem is one of the important mathematical theorems discovered and proven by mankind in the early days, one of the most important tools for solving geometric problems with algebraic ideas, and one of the links between numbers and shapes.

Introduction to the Pythagorean Theorem:

The Pythagorean theorem is a basic geometric theorem that states that the sum of the squares of the two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse. In ancient China, the right triangle was called the Pythagorean shape, and the smaller of the right-angled sides was the hook, the other long right-angled side was the strand, and the hypotenuse was the chord, so this theorem was called the Pythagorean theorem, and some people called the Shanggao theorem.

There are about 500 ways to prove the Pythagorean theorem, which is one of the most provable theorems in mathematics. The Pythagorean theorem is one of the important mathematical theorems discovered and proven by mankind in the early days, one of the most important tools for solving geometric problems with algebraic ideas, and one of the links between numbers and shapes.

In China, Shang Gao during the Shang Dynasty proposed a special case of the Pythagorean theorem of "Pythagorean three strands and four". In the West, the first to propose and prove this theorem was the Pythagoreans of ancient Greece in the 6th century BC, and he used the deductive method to prove that the square of the hypotenuse of a right triangle is equal to the sum of the squares of two right angles.

I think it's a mistake on the part of the editors and the proofreaders, and if they don't see the problem on the part of the editors before they are printed, they don't take it seriously.

This is the lowest level of error, and it is easy to mislead students, and many students have learned from textbooks and should be corrected even if they are corrected.

I don't think this is a mistake, it's just a simplification of the original complex theory.

Draw a square, and then take the side of the square as the hypotenuse, make four congruent right-angled triangles into the square, and let the right-angled sides of the four triangles intersect in pairs, so that a small square will appear inside the square, then the area of the four right-angled triangles is equal to 4 2*abAdd the square of the area (a-b) of the small square in the middle, this is the area of the big square, and finally calculate the square of a + the square of b. It is equal to the square of the side of the square (or the hypotenuse of the right triangle) c.

This leads to the Pythagorean theorem.

This theorem is also known in China"Quotient high theorem", as it is called in foreign countries"Pythagorean theorem; "。Why does a theorem have so many names? Shang Gao was a Chinese in the 11th century BC.

At that time, China's dynasty was the Western Zhou Dynasty, which was a period of slave society. In ancient China, around the Warring States Period, the Western Han Dynasty's mathematical work "Zhou Ji Suanjing" records a conversation between Shang Gao and Zhou Gong. Shang Gao said:

Therefore, the folding moment, the hook wide three, the stock repair four, the meridian five. "What is"Hooks, strands"This? In ancient China, people called the upper part of an arm that was bent at right angles"Tick", the lower part is called"shares"。

The meaning of Shang Gao's passage is that when the two right-angled sides of a right triangle are 3 (short side) and 4 (long side) respectively, the corner of the radius (that is, the string) is 5. Later on, people simply said this fact"Hook three strands, four strings, five"。

Since the content of the Pythagorean theorem was first seen in Shang Gao's words, people called this theorem"Quotient high theorem"。At the beginning of one of China's earliest mathematical works, "Zhou Ji Suanjing", there is a conversation in which Zhou Gong asked Shang Gao for mathematical knowledge: Zhou Gong asked:

I heard that you are very proficient in mathematics, and I would like to ask you: there is no ladder to go up to the sky, and the earth cannot be measured by a ruler, so how can we get data about the heavens and the earth? Shang Gao replied:

The number of ** is born in the other party and the circle of these forms hungry to know. There is a principle: when the right triangle 'moment' gives a right-angled side 'hook' equal to 3 and another right-angled side 'strand' equal to 4, then its hypotenuse 'string' must be 5.

This principle was summed up by Dayu when he was controlling the water. From the above quotation of this dialogue, we can clearly see that the ancient people of our country have discovered and applied the Pythagorean theorem, an important mathematical principle, thousands of years ago. Readers who know a little about plane geometry know that the so-called Pythagorean theorem refers to the fact that in a right triangle, the sum of the squares of two right-angled sides is equal to the square of the hypotenuse.

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