In fact, in earlier human activities, people have realized some special cases of this theorem. In addition to the Pythagorean theorem discovered by 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 right angles. 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.
Whether it is the ancient Egyptians, Babylonians or China who first discovered Pythagorean theorem, the same property discovered by our ancestors in different periods and places is obviously not just the private property of any nation, but the common wealth of all mankind. It is worth mentioning that the harvest after discovering the same nature of this * * * is not exactly the same. Let's briefly introduce the Pythagorean Theorem and Pythagorean Theorem as examples:
1. Pythagorean theorem
Pythagoras was a famous ancient Greek. Pythagoras, born in the 6th century BC, traveled to Egypt, Babylon (another way to put it, India) and other places in his early years, and then moved to Crotone in the south of the Italian peninsula, where he organized a secret group integrating politics, religion and mathematics-Pythagoras School, which attached great importance to mathematics and tried to explain everything with numbers. They claim that number is the origin of all things in the universe, and the purpose of studying mathematics is not to be practical, but to explore the mysteries of nature. One of their great contributions to mathematics is conscious recognition and emphasis; Mathematical things such as numbers and figures are abstractions of thinking, which are completely different from actual things or actual images. Some people in primitive civilized society (such as Egyptians and Babylonians) also know how to think about numbers from physical objects, but their awareness of the abstract nature of this thinking is completely different from that of Pythagoras school. Moreover, before the Greeks, geometric thought was inseparable from physical objects. For example, the Egyptians thought that a straight line was the edge of a tight rope or field; A rectangle is the boundary of a field. Another feature of this school is the close connection between arithmetic and geometry.
Because of this, the Pythagorean school found a formula that belongs to both arithmetic and geometry to represent the side length of a right-angled triangle composed of three integers: if 2n+ 1 are two right-angled sides, then the hypotenuse is (but this rule cannot represent all integer Pythagorean arrays). It is precisely because of the above reasons that this school found the so-called "incommensurable measure" by searching and studying the integral pythagorean number-for example, the ratio of the hypotenuse to the right side of an isosceles right triangle, that is, the ratio of the diagonal of a square to its side, cannot be expressed by the ratio of integers. For this reason, they call those ratios that can be expressed by the ratio of integers "commensurability ratio", that is, two quantities can be measured in commensurability units, but those that cannot be expressed in this way are called "commensurability ratio". As we wrote today, the ratio of L is the ratio of incommensurability. As for the proof that 1 is incommensurable, it is also given by Pythagoras school. This proof points out that if the hypotenuse of an isosceles right triangle can be congruent with the right side, then the same number is both odd and even. The proof process is as follows: Let the ratio of hypotenuse to right angle of isosceles right triangle be α: β, and let this ratio be expressed as the minimum integer ratio. According to Pythagoras theorem, there is. Since even number is even number, α must be even number, because the square of any odd number must be odd number (any odd number can be expressed as 2n+ 1, so it is still odd number. But α: β is irreducible, so β is not even but odd. Since α is an even number, α=2γ can be set. So ... So, here, it is even, so β is even, but β is odd, which creates a contradiction. Regarding the proof of Pythagoras theorem, the earliest written material preserved by human beings is proposition 47 in the first volume of Geometry written by Euclid (around 300 BC): "The square on the hypotenuse of a right triangle is equal to the sum of the squares on two right angles". The proof is carried out by region.