The law of the Pythagorean theorem is fast, what is the Pythagorean law

This is like this, it is a property of the Pythagorean number of consecutive numbers, and it can be proved as follows, the Pythagorean number of consecutive numbers is set as.

A square + B square = (b + 1) square.

a 2+b 2=(b+1) 2,a 2+b 2=b 2+2b1, so there is, a 2=

The formula 2b+1 is what your classmate said, after the square is added and subtracted by 1 and divided by 2 is the other two numbers, as long as you use the number a that conforms to the Pythagorean theorem to operate, you will be able to find the other two consecutive numbers.

Not if you do it with other numbers.

The left side of these equations is in the form of a sum of two numbers, while the right side is the product of the same two numbers, and the sum on the left side is equal to the product on the right. There are infinitely many equations that satisfy this relation, and the key to writing such equations is to find the relationship between the two numbers involved in the equation.

Considering that the equations listed in the question all contain an integer and a finite decimal (2+2=2*2, which is a special case, it is advisable to think of one of the 2s as a finite decimal and does not affect the analysis results) - this is a limit 1

Let these two numbers be a and b respectively, where a is an integer and b is a finite decimal, then, according to the above law, there should be a+b=a*b, so, b=a (a-1).

Obviously, when a>2, a is coprime with a-1, and to make b a finite decimal, then the denominator (a-1) can only contain prime factors 2 and 5

Therefore, the value range of a is a=(2 m)*(5 n)+1, where m,n are any natural numbers, and the symbol 2 m represents the m power of 2.

According to this rule, the following equation can be written.

5+ 17+ 21+ 26+ etc., you can verify them one by one.

For example, 4+4 3=4*4 3 7+7 6=7*7 6 8+8 7=8*8 7, etc.

It should be easier to find patterns from these formulas with fractions

One angle is 90 degrees, the other two are 45 degrees, the ratio of the side is 1 to 1, the ratio is 2, the angle is 90, the other is 60 degrees, 30 degrees, the ratio of the side is 1, the root number is 2, the ratio is 3, this is what the teacher said before, I don't know if it helps.

Pythagorean law refers to 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.

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.

In China, Shang Gao during the Zhou Dynasty proposed a special case of the Pythagorean theorem of "Pythagorean three, four strings, five". In the West, the Pythagoreans of ancient Greece in the 6th century BC were the first to propose and prove this theorem, and they 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.

In Euclid's Geometry Primitives, the following proof of the Pythagorean theorem is given. Let abc be a triangle with a right angle, where a is a right angle. Draw a straight line from point A to the opposite edge so that it is perpendicular to the side of the opposing group.

Extending this line divides the square on the opposite side in two, with an area equal to the other two squares.

The idea of the proof is to draw a straight line from point A to the opposite edge, so that it is perpendicular to the opposite edge. Extend this line to divide the square on the side of the answer hand into two, and convert the two squares above into two rectangles of the same area below by means of a triangle of equal height and bottom.

What the Pythagorean law is is introduced below:

The Pythagorean theorem is a fundamental theorem in elementary geometry. The so-called Pythagorean theorem means that in a right triangle, the sum of the squares of two right-angled sides is equal to the square of the hypotenuse. This theorem has a very long history and has been studied by almost all ancient civilizations (Greece, China, Egypt, Babylon, India, etc.).

The Pythagorean theorem is known as the Pythagorean theorem in the West, and is said to have been the name of the ancient Greek mathematician and philosopher Pythagoras (572 BC).497 BC?(Image at right) was first discovered in 550 BC.

But the Pythagorean method of proving the Pythagorean theorem has been lost. The famous Greek mathematician Euclid (330 BC, 275 BC) gave a good proof in his magnum opus, Geometry (Vol. 4, Proposition 47).

The discovery and application of this mathematical theorem in ancient China was much earlier than that of Pythagoras. At the beginning of one of China's earliest mathematical works, "Zhou Ji Suanjing", there is a conversation between Zhou Gong asking 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 in the sky, and the earth cannot be measured by a ruler in a section, so how can you get data about heaven and earth? "

When the right triangle' moment'The result is a right-angled edge' hook'Equal to 3, the other right-angled side' strand'When equal to 4, then it's hypotenuse'strings'It must be 5. This principle was summed up by Dayu when he was controlling the water. "

If it is said that Dayu's water control cannot be accurately verified because the age is as long as Hu Sen, then the dialogue between Zhou Gong and Shang Gao can be determined in the Western Zhou period around 1100 BC, more than 500 years earlier than Pythagoras. The Pythagorean 3 strands and 4 strings 5 mentioned in it are a special application of the Pythagorean theorem. So now it is very appropriate in mathematics to call it the Pythagorean theorem.

The magical Pythagorean law model, guess what principle is the formula derived?

"The Pythagorean theorem: In any right triangle, the sum of the squares of the two right-angled sides must be equal to the squares of the hypotenuses. This theorem is also known as the "Shang Gao Theorem" in China and the "Pythagorean Theorem" in foreign countries.

The Pythagorean theorem (also known as Shang Gao's theorem, Pythagorean theorem) is a basic geometric theorem that was discovered by Shang Gao as early as the Shang Dynasty in China. It is said that after Pythagoras discovered this theorem, he beheaded a hundred cows in celebration, hence the "Hundred Oxen Theorem".

The Pythagorean theorem states:

The sum of squares of the two right-angled sides of a right-angled triangle (i.e., the "hook" and "strand" for the short one is the hook and the long one is the strand) is equal to the square of the side length of the hypotenuse (i.e., the "string").

That is, if the two right-angled sides of a right-angled triangle are a and b, and the hypotenuse is c, then.

a^2+b^2=c^2

The Pythagorean theorem has now found about 400 ways to prove it, making it one of the most provable theorems among mathematical theorems.

The Pythagorean theorem is actually a special form of the cosine theorem.

Shang Gao, a famous mathematician in ancient China, said: "If the hook is three, the strand is four, then the string is five." "It is recorded in the Nine Chapters of Arithmetic. ”

Origin of the Pythagorean theorem.

There is a very important theorem in trigonometry, which is called the Pythagorean theorem in China, also known as the Shanggao theorem. Because of the mention of "Zhou Ji Sutra", Shang Gao said"Hook three strands, four strings, five"Words.

In fact, it was discovered by the ancient working people of our country through long-term measurement experience. They found that when the short right-angled side (hook) of a right-angled triangle is 3 and the long right-angled side (strand) is 4, the opposite side of the right-angled triangle (chord) is exactly 5. Whereas.

This is a special case of the Pythagorean theorem. Later, through long-term measurement practice, it was found that as long as it is a right triangle, its three sides have such a relationship. i.e. there are many groups of positive integers that are equivalent to them.

The Zhou Ji Sutra also said that Xia Yu had initially applied this theorem in actual measurement. The book also records that a mathematician named Chen Zi applied this theorem to measure the height of the sun, the diameter of the sun, and the length and width of the heavens and the earth. The Egyptians 5,000 years ago also knew a special case of this theorem, that is, hook 3, strand 4, chord 5, and used it to determine right angles.

Later, it gradually spread to the general situation.

The base of the pyramid is square, facing east, west, north and south, and the visible direction is measured accurately, and the four corners are strictly right angles. To measure a right angle, of course, you can use the method of making a vertical line, but if you reverse the Pythagorean theorem, that is, as long as the three sides of the triangle are , or conform to the formula, then the angle opposite the chord must be a right angle.

By 540 B.C., the Greek mathematician Pythagoras noticed that the three sides of a right triangle were , or , had such a relationship: ,.

He wondered: Do all the three sides of a right triangle conform to this law? Conversely, if the three sides conform to this law, is it a right triangle?

He collected many examples, and the results were positive on both issues. He was so happy that he killed a hundred cows to congratulate him.

Since then, Westerners have called this theorem the Pythagorean theorem.

In China, the description of this theorem was first seen in the "Zhou Ji Sutra".

The book was written about the first century BC in the Western Han Dynasty), and there is a passage in the book in which Shang Gao (c. 1120 BC) answered the question of Zhou Gong, "Gou Guangsan.

Stock repair four, Jingyu five", which means that the two right-angled sides of the right-angled triangle are 3 and 4, and the hypotenuse is 5 The book also records Chen Zi (

716 BC) replied to Rong Fang's question: "If you seek evil solstice, take the sun as the hook, the high of the day as the stock, and the Pythagorean multiplication, and open the square to remove it, and get the evil to the sun", the evil in ancient Chinese is obliquely solved, so this sentence clearly states the content of the Pythagorean theorem Zhao Shuang of the Three Kingdoms (about the 3rd century), in his mathematical document "Pythagorean Circle Diagram" (as a note to the "Zhou Ji Sutra", which is retained in the book) used the string diagram to skillfully prove the Pythagorean theorem, as shown in Figure 2 He painted the triangle in red, and its area is called "Zhu Shi" , the middle square is painted yellow called "middle yellow solid", also called "poor solid" He wrote: "According to the chord diagram, it can be multiplied by Pythagorean as Zhu Shi 2, and times as Zhu Shi 4, and multiplied by the difference between Pythagorean to become Zhong Huang Shi, plus the difference between the real and the strings" If you use the current symbols, use a, b, and c respectively to mark the length of the hook, strand, and string, which Zhao Shuang said, that is, 2AB+(a-b)2=C2, and simplify it to obtain A2+B2=C2