1. In life, we often use the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Do you know who invented these numbers? These digital symbols were first invented by ancient Indians, and then spread to * * *, and then from * * * to Europe. Europeans mistakenly think that it was invented by * * * people, so it is called "* * * number". Because it has been circulating for many years, people still call them * * *. Now, the number * * * has become a universal digital symbol all over the world.

2. Nine Jiu Ge is the multiplication formula we use now. As early as the Spring and Autumn Period and the Warring States Period BC, Jiujiu songs have been widely used by people.

In many works at that time, there were records about Jiujiu songs. The original 99 songs started from "99 8 1" to "22 gets 4", with 36 sentences.

Because it started with "998 1", it was named 99 Song. The expansion of Jiujiu songs to "11" was between the 5th century and10th century.

It was in the 13 and 14 centuries that the order of Jiujiu songs changed from "one to one" to "9981". At present, there are two kinds of multiplication formulas used in China. One is a 45-sentence formula, usually called "Xiao Jiujiu"; There is also a sentence 8 1, which is usually called "Big Uncle Nine".

3. The circle is a seemingly simple but actually wonderful circle. The ancients first got the concept of circle from the sun and the moon on the fifteenth day of the lunar calendar.

Even now, the sun and the moon are used to describe some round things, such as the Moon Gate, Qin Yue, Moon Shell, Sun Coral and so on. Who drew the first circle? The stone balls made by the ancients more than 100 thousand years ago are quite round.

As mentioned earlier, Neanderthals 18000 years ago used to drill holes in animal teeth, gravel and stone beads, some of which were very round. Neanderthals drilled holes with pointed devices, but they couldn't get in on one side and then drilled from the other.

The tip of the stone tool is the center of the circle, and half of its width is the radius. Turn around and you can drill a round hole. Later, in the pottery age, many pottery were round.

Round pottery is made by putting clay on a turntable. When people start spinning, they make round stones or ceramic cocoons.

Banpo people (in Xi 'an) built round houses 6000 years ago, with an area exceeding 10 square meter. The ancients also found that rolling logs was more economical.

Later, when they were carrying heavy objects, they put some logs under big trees and stones and rolled them around, which was of course much more labor-saving than carrying them. Of course, because the log is not fixed under the weight, you have to roll the log rolled out from the back to the front and pad it under the front of the weight.

About 6000 years ago, Mesopotamia made the world's first wheel-a round board. About 4000 years ago, people fixed round boards under wooden frames, which was the original car.

Because the center of the wheel is fixed on a shaft, and the center of the wheel is always equal to the circumference, as long as the road surface is flat, the car can move forward in a balanced way. You can make a circle, but you don't necessarily know its nature.

The ancient Egyptians believed that the circle was a sacred figure given by God. It was not until more than 2,000 years ago that China's Mozi (about 468- 376 BC) defined the circle: "One China has the same length".

It means that a circle has a center and the length from the center to the circumference is equal. This definition is 100 years earlier than that of the Greek mathematician Euclid (about 330 BC-275 BC).

Pi, the ratio of circumference to diameter, is a very strange number. The Book of Weekly Calculations says that "the diameter is three times a week", and pi is considered to be 3, which is only an approximate value.

When the Mesopotamians made the first wheel, they only knew that pi was 3. In 263 AD, Liu Hui of Wei and Jin Dynasties annotated Nine Chapters of Arithmetic.

He found that "the diameter is three times that of a week" is just the ratio of the circumference to the diameter of a regular hexagon inscribed in a circle. He founded secant technology, and thought that when the number of inscribed sides of a circle increased infinitely, the circumference was closer to the circumference of a circle.

He calculated the pi = 3927/1250 of the inscribed circle of the regular 3072-sided polygon. Would you please convert it into decimal and see what it is? Liu Hui applied the concept of limit to solving practical mathematical problems, which is also a great achievement in the history of mathematics in the world. Zu Chongzhi (AD 429-500) continued to calculate on the basis of previous calculations, and found that the pi between 3. 14 15926 and 3. 14 15927 was the earliest value in the world accurate to seven decimal places. He also used two fractional values to express pi: 22/7 is called approximate ratio.

Please change these two fractions into decimals and see how many decimals are the same as the known pi today. In Europe, it was not until 1000 years later16th century that the Germans Otto (A.D. 1573) and Antoine Z got this value. Now that there is an electronic computer, pi has been calculated to more than 10 million after the decimal point.

4. Besides counting numbers, mathematics needs a set of mathematical symbols to express the relationship between numbers and shapes. The invention and use of mathematical symbols are later than numbers, but they are much more numerous.

Now there are more than 200 kinds in common use, and there are more than 20 kinds in junior high school math books. They all had an interesting experience.

For example, there used to be several kinds of plus signs, but now the "+"sign is widely used. +comes from the Latin "et" (meaning "and").

/kloc-in the 6th century, the Italian scientist Nicolo Tartaglia used the initial letter of "più" (meaning "add") to indicate adding, and the grass was "μ" and finally became "+". The number "-"evolved from the Latin word "minus" (meaning "minus"), abbreviated as m, and then omitted the letter, it became "-".

It is also said that wine merchants use "-"to indicate how much a barrel of wine costs. After the new wine is poured into the vat, a vertical line is added to the "-",which means that the original line is erased, thus becoming a "+"sign.

/kloc-In the 5th century, German mathematician Wei Demei officially determined that "+"was used as a plus sign and "-"was used as a minus sign. Multipliers have been used for more than a dozen times, and now they are commonly used in two ways.

One is "*", which was first proposed by the British mathematician Authaute at 163 1; One is "",which was first created by British mathematician heriott. German mathematician Leibniz thinks: "*".

2. What mathematical knowledge do you have?

1 There is only one straight line between two points. The shortest line segment between two points is 3. The same angle or the complementary angle of the same angle is equal. 4. The same angle or the complementary angle of the same angle is equal. 5. Only one straight line is perpendicular to the known straight line. 6. Among all the line segments connected with points on a straight line, the shortest parallel axiom of a vertical line segment passes through a point outside the straight line. There is only one straight line parallel to this straight line. If both lines are parallel to the third line, the two lines are parallel to each other. The isosceles angles are equal and the two straight lines are parallel to each other. 10, the offset angles are equal, and the two straight lines are parallel to each other. 1 1 is complementary to the inner corner of the side, and the two straight lines are parallel to each other. 13, two straight lines are parallel. The internal dislocation angle is equal to 14, and the two straight lines are parallel. Theorem The sum of two sides of a triangle is greater than the third side 15. The difference between two sides of the reasoning triangle is less than the third side 17. The sum of the interior angles of a triangle is the theorem that the sum of the three interior angles of a triangle is equal to 180 18. The two acute angles of a right triangle complement each other 19. The outer corner of a triangle. The sum of two non-adjacent internal angles is 20. Inference 3 An outer angle of a triangle is larger than the corresponding side of any inner angle that is not adjacent to it, 2 1 congruent triangles. The corresponding angles are equal. The 22-angle axiom has two triangles with equal angles. The 23-angle axiom has two angles and two equilateral triangles. It is inferred that there are two angles, and the opposite side of one angle corresponds to a triangle with two equal sides. The 25-sided axiom has three sides, corresponding to a triangle with two equal sides. 26 hypotenuse, right-angled side axiom has hypotenuse and right-angled side corresponding to two equal right-angled triangles. Theorem 1 The distances between points on the angular bisector are equal. Theorem 2 Reach the point where both sides of an angle are equidistant. On the bisector of this angle, the bisector of angle 29 is the property theorem that the distances from the isosceles triangle to all points on both sides of angle *** 30 are equal. The two base angles of an isosceles triangle are equal to 3 1. It is inferred that the bisector of the top angle of the isosceles triangle of 1 bisects the bottom and is perpendicular to the bisector of the top angle of the isosceles triangle of the bottom 32. The heights of the center line and the base coincide with each other. Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60 34 isosceles triangle. If a triangle has two equal angles, then the opposite sides of the two angles are also equal (equilateral) 35 Inference 1 A triangle with three equal angles is an equilateral triangle 36 Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle 37 in a right triangle. If an acute angle is equal to 30, then the right-angled side it faces is equal to half of the hypotenuse. The median line of the hypotenuse of a right triangle is equal to half of the hypotenuse. Theorem 39 A point on the vertical line of a line segment is equal to the distance between the two endpoints of this line segment. The inverse theorem and the point where the two endpoints of a line segment are equal. On the perpendicular bisector of this line segment, the perpendicular bisector of 4 1 line segment can be regarded as the theorem of all points with equal distance from both ends of the line segment 1. Two figures that are symmetrical about a line are conformal. Theorem 43 Theorem 2 If two figures are symmetrical about a straight line, then the symmetry axis is Theorem 44 of the perpendicular line connecting the corresponding points. Theorem 3 Two graphs are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry. 45 Inverse Theorem If the straight line connecting the corresponding points of two graphs is bisected vertically by the same straight line, then the two graphs are symmetrical about this straight line. 46 Pythagorean Theorem The sum of squares of two right-angled sides A and B of a right triangle is equal to the square of hypotenuse C, that is, A+B = the inverse theorem of Pythagorean Theorem a+b=c 47. If the lengths of three sides of a triangle are related, a+b=c, then this triangle is a right triangle, and the sum of the internal angles of the quadrilateral of Theorem 48 is equal to 360 49. Theorem 360: The sum of inner angles of polygons is equal to (n-2)* 180 5 1 Inferring that the sum of outer angles of any polygon is equal to 360 52 parallelogram property theorem 1 Parallelogram Diagonal Equality 53 Parallelogram Property Theorem 2 Parallelogram Opposite Sides Equal 54 Inference that the parallel line segments sandwiched between two parallel lines are equal 55 Parallelogram Property Theorem 3 Parallelogram Diagonal bisection 56 Parallelogram Decision Theorem 1 Two sets of parallelograms with equal diagonal lines are parallelograms 57. Parallelogram Decision Theorem 2. Two sets of equal parallelograms with opposite sides are parallelograms 58. Parallelogram Decision Theorem 3. A set of parallelograms with equal diagonal lines are parallelograms 59. Parallelogram Decision Theorem 4. A set of parallelograms with equal opposite sides are parallelograms 60. Rectangular property theorem 1 The four corners of a rectangle are right angles 665438. +0 Rectangular Property Theorem 2 Rectangular Diagonal Equivalence 62 Rectangular Decision Theorem 1 A quadrilateral with three right angles is a rectangle 63 Rectangular Decision Theorem 2 A parallelogram with equal diagonal is a rectangle 64 rhombic Property Theorem 1 All four sides of a rhombus are equal 65 rhombic Property Theorem 2 The diagonals of the rhombus are perpendicular to each other. And each diagonal bisects a set of diagonal lines. 66 rhombic area = half of diagonal product, that is, S=(a*b)÷2 67 rhombic decision theorem 1 quadrilateral with four equal sides is rhombic 68 rhombic decision theorem 2 parallelograms with orthogonal diagonals are rhombic 69 square property theorem 1 four corners of a square are right angles. All four sides are equal. Theorem of 70 Square Properties 2 Two diagonal lines of a square are equal and bisected vertically, and each diagonal line bisects a set of diagonal lines. 7 1 theorem 1 congruence of two graphs with central symmetry. Theorem 2 About two graphs with central symmetry, a straight line connecting symmetrical points passes through the center of symmetry and is split in two by the center of symmetry. Inverse Theorem If a straight line connecting the corresponding points of two graphs passes through a point and is equally divided by the point, then the two graphs are symmetric about the point. The two angles of the isosceles trapezoid on the same base are equal to 75. The two diagonals of an isosceles trapezoid are equal. A trapezoid with two equal angles on the same base is an isosceles trapezoid 77. The trapezoid with equal diagonal lines is an isosceles trapezoid 78. If a set of parallel lines cut on a straight line are equal, then the line segments cut on other straight lines are also equal. 79 Inference 1 Crossing a straight line with one waist parallel to the bottom, the other waist 80 must be equally divided. Inference 2 A straight line passing through a triangle with one side parallel to the other side must bisect the third side 8 1. The center line of a triangle is parallel to the third side and equal to half of it. The midline theorem of trapezoid.

3. Little knowledge of mathematics, for the sixth grade.

1, Yang Hui triangle is a triangle table arranged by numbers. The general form is as follows:11113314641151. 1 561721353521kloc-0/........................................................................... Yanghui triangle's most essential feature is that its two hypotenuses are all composed of the number1,and the rest are equal to the sum of its upper two numbers.

In fact, ancient mathematicians in China were far ahead in many important mathematical fields. The history of ancient mathematics in China once had its own glorious chapter, and the discovery of Yang Hui's triangle was a wonderful one.

Yang Hui was born in Hangzhou in the Northern Song Dynasty. In his book "Detailed Explanation of Algorithms in Nine Chapters" written by 126 1, he compiled a triangle table as shown above, which is called an "open root" diagram.

And such triangles are often used in our Olympic Games. The simplest thing is to ask you to find a way. Now we are required to output such a table through programming.

2. Mathematicians triggered by a story Chen Jingrun, a famous mathematician, made great contributions to overcoming Goldbach's conjecture and founded the famous "Chen Theorem", so many people affectionately called him "the prince of mathematics". But who would have thought that his achievement originated from a story?

1937, diligent Chen Jingrun was admitted to Huaying College in Fuzhou. At this time, during the period of War of Resistance against Japanese Aggression, Professor Shen Yuan, director of the Department of Aeronautical Engineering in Tsinghua University, returned to Fujian to attend the funeral, unwilling to stay in his hometown because of the war. Several universities got the news and wanted to invite Professor Shen to give lectures. He declined the invitation.

As he is an alumnus of Huaying, he came to this middle school to teach mathematics to his classmates in order to report to his alma mater. One day, Teacher Shen Yuan told us a story in math class: "A Frenchman discovered an interesting phenomenon 200 years ago: 6 = 3+3, 8 = 5+3, 10 = 5+5, 12 = 5+7, 28 = 5+23.

Every even number greater than 4 can be expressed as the sum of two odd numbers. Because this conclusion has not been proved, it is still a guess.

Euler said: Although I can't prove it, I am sure this conclusion is correct. It is like a beautiful light ring, shining with dazzling brilliance in front of us not far away.

..... "Chen Jingrun stare eyes, absorbed. From then on, Chen Jingrun became interested in this wonderful question.

In his spare time, he likes going to the library. He not only read the counseling books in middle schools, but also eagerly read the textbooks of mathematics and physics courses in these universities. Therefore, he got the nickname "bookworm".

Interest is the first teacher. It is such a mathematical story that aroused Chen Jingrun's interest and his diligence and made him a great mathematician.

3. People who are crazy about science, because of endless research, often get some logical but absurd results (called "paradoxes"), and many great mathematicians take an evasive attitude because they are afraid of falling into it. During the period of 1874- 1876, Cantor, a young German mathematician less than 30 years old, declared war on the mysterious infinity.

With hard sweat, he successfully proved that points on a straight line can correspond to points on a plane one by one, and can also correspond to points in space one by one. In this way, it seems that there are "as many" points on the 1 cm long line segment as there are points in the Pacific Ocean and the whole earth. In the following years, Cantor published a series of articles about this kind of "infinite * * *" problem, and drew many amazing conclusions through strict proof.

Cantor's creative work has formed a sharp conflict with the traditional mathematical concept, which has been opposed, attacked and even abused by some people. Some people say that Cantor's theory of * * * is a kind of "disease", Cantor's concept is "fog in fog", and even Cantor is a "madman".

Great mental pressure from the authority of mathematics finally destroyed Cantor, making him exhausted, suffering from schizophrenia and being sent to a mental hospital. True gold is not afraid of fire, and Cantor's thought finally shines.

At the first international congress of mathematicians held in 1897, his achievements were recognized, and Russell, a great philosopher and mathematician, praised Cantor's work as "probably the greatest work that can be boasted in this era." But at this time, Cantor was still in a trance, unable to get comfort and joy from people's reverence.

1918 65438+1October 6th, Cantor died in a mental hospital. Cantor (1845- 19 18) was born in a wealthy family of Danish Jewish descent in Petersburg, Russia. /kloc-moved to Germany with his family at the age of 0/0, and was interested in mathematics since childhood.

He received his doctorate at the age of 23 and has been engaged in mathematics teaching and research ever since. His theory of * * * is considered as the basis of all mathematics.

4. Mathematicians' "forgetfulness" On the 60th birthday of Professor Wu Wenjun, a mathematician in China, as usual, he got up at dawn and buried himself in calculations and formulas all day. Someone specially chose to visit at home this evening. After greeting, he explained his purpose: "I heard from your wife that today is your sixtieth birthday, and I came to congratulate you."

Wu Wenjun seemed to hear a message and suddenly said, "Oh, really? I forgot. " The bearer was secretly surprised and thought, how can a mathematician not even remember his birthday because his mind is full of numbers? In fact, Wu Wenjun has a good memory for dates.

Nearly sixty years old, he conquered a difficult problem for the first time-"machine certificate". This is to change the working mode of "a pen, a piece of paper, a head" for mathematicians, and realize mathematical proof with electronic computers, so that mathematicians have more time to do creative work. In the course of his research on this subject, he clearly remembers the date of installing the electronic computer and compiling more than 300 "instruction" programs for the computer.

Later, when a birthday visitor asked him in a chat why he couldn't even remember his birthday, he replied knowingly, "I never remember those meaningless numbers." In my opinion, what does it matter if the birthday is one day earlier and one day later? So, I don't remember my birthday, my wife's birthday, my child's birthday. He never wants to celebrate his or his family's birthday, even my wedding day.

However, some figures must be remembered, and it is easy to remember ... "5. Routine steps under the apple tree 1884 1984 In the spring of 1984, Adolf leonid hurwicz, a young mathematician, came to Koenigsberg from G? ttingen as an associate professor, when he was less than 25 years old.

4. Little knowledge of mathematics

Archimedes 1, sand calculation, is a book devoted to the study of calculation methods and theories.

Archimedes wanted to calculate the number of grains of sand in a big sphere full of the universe. He used a very strange imagination, established a new counting method of order of magnitude, determined a new unit, and put forward a model to represent any large number, which is closely related to logarithmic operation. 2. Using 96 circumscribed circles and inscribed circles to measure the circle, the pi is 3.1408 3. By skillfully using exhaustive method, it is proved that the surface area of the ball is equal to four times the area of the great circle of the ball; The volume of a ball is four times that of a cone. The base of this cone is equal to the great circle of the ball, which is higher than the radius of the ball.

Archimedes also pointed out that if there is an inscribed sphere in an equilateral cylinder, the total area of the cylinder and its volume are the surface area and volume of the sphere respectively. In this book, he also put forward the famous "Archimedes axiom".

4. "Parabolic quadrature method", which studies the quadrature problem of curves and figures, and establishes a conclusion by exhaustive method: "Any arch (i.e. parabola) surrounded by the cross section of a straight line and a right-angled cone is four-thirds of the area of a triangle with the same base height." He also verified this conclusion again by mechanical weight method, and successfully combined mathematics with mechanics.

5. On Spiral is Archimedes' outstanding contribution to mathematics. He made clear the definition of spiral and the calculation method of spiral area.

In the same book, Archimedes also derived the geometric method of summation of geometric series and arithmetic series. 6. Plane balance is the earliest treatise on mechanical science, which is about determining the center of gravity of plane graphics and three-dimensional graphics.

7. Floating Body is the first monograph on hydrostatics. Archimedes successfully applied mathematical reasoning to analyze the balance of floating body, and expressed the law of floating body balance with mathematical formula. 8. On the cone and sphere, it is about determining the volume of the cone formed by parabola and hyperbola rotation and the volume of the sphere formed by ellipse rotation around its long axis and short axis.

Pythagorean Theorem 1 Pythagorean Theorem: Anyone who has studied algebra and geometry will hear about Pythagorean Theorem. This famous theorem is widely used in many branches of mathematics, architecture and measurement. The ancient Egyptians used their knowledge of this theorem to construct right angles. They tie ropes every 3, 4 and 5 units. Then straighten the three ropes to form a triangle. They know that the diagonal of the largest side of a triangle is always a right angle (32+42=52). Pythagoras theorem: given a right triangle, the square of the hypotenuse of the right triangle is equal to the sum of the squares of the two right sides of the same right triangle. And vice versa: if the sum of squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle. Although this theorem was named after the Greek mathematician Pythagoras (about 540 BC), there is evidence that the history of this theorem can be traced back to the era of Hammurabi in ancient Babylon 1000 years ago. The name of this theorem is attributed to Pythagoras. Probably because he was the first to record the certificate he wrote at school. The conclusion of Pythagoras theorem and its proof spread all over the world in all continents, cultures and periods. In fact, this theorem has been proved more than any other discovery! 2. The Pythagorean school holds that any number can be expressed as an integer or a ratio of integers. But a student named Hibbs found that if the side length of an isosceles right triangle is 1, then according to the Pythagorean theorem (that is, Pythagorean theorem, which is just called in the west, it was actually discovered by our ancestors first! . ), the square of the length of the hypotenuse should be 1+ 1=2, and the number with the square equal to 2 cannot be expressed by integers or fractions.

He told others about this discovery, but this discovery overthrew the basic idea of the "Bi" school. So he was thrown into the river and executed.

Later, people affirmed this discovery and named it irrational number to distinguish Bipai's rational number. Memory of irrational numbers √ 2 √ 1.4 1: only meaningful √ 3 √ 1.7320: laying goose eggs together √ 5 √ 2.2360679: two geese laying six eggs (delivering babies) and six wives and uncles √ 7 √ 2.64579.

I need three math knowledge and stories (the shorter the better).

Say four, very brief: when Gauss was in primary school, the teacher asked his students to calculate1+2+3+...+98+99+100.

The teacher himself is beside, counting honestly. Gauss finished the calculation quickly and told him that the method was to add the first and last numbers and multiply them by 50, which surprised the teacher. In the 6th century AD, Herbers, a scholar of Pythagoras School, discovered this irrational number when he was studying the diagonal length of a square with a length of 1. The irrational number was not recognized by Pythagoras school and was submerged in the sea, which caused the first crisis in the history of mathematics, that is, the irrational number was not recognized and prevented its spread.

Once, Abel, a famous mathematician, wrote a letter to Homer, his teacher. The date of the letter was triple radical number 604438+0438+09, which involved prescriptions. It was written as13.5089.00000000606 (year), 365 * 0.5908275 = 2/kl.

Hua had a visit abroad. On the plane, a passenger next to him was reading a math magazine. The above question is: what is the root number 593 19 of the third degree? Hua blurted out that it was 39 after reading it, which surprised everyone. The algorithm he explained was omitted.

6. What is the little knowledge of mathematics?

Look at Yang Hui Triangle!

Yang Hui Triangle is a triangular numerical table arranged by numbers, and its general form is as follows:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 2 1 35 35 2 1 7 1

… … … … …

The most essential feature of Yang Hui Triangle is that its two hypotenuses are all composed of the number 1, and the other numbers are equal to the sum of the two numbers on its shoulders. In fact, ancient mathematicians in China were far ahead in many important mathematical fields. The history of ancient mathematics in China once had its own glorious chapter, and the discovery of Yang Hui's triangle was a wonderful one. Yang Hui was born in Hangzhou in the Northern Song Dynasty. In his book "Detailed Explanation of Algorithms in Nine Chapters" written by 126 1, he compiled a triangle table as shown above, which is called an "open root" diagram. And such triangles are often used in our Olympic Games. The simplest thing is to ask you to find a way. Now we are required to output such a table through programming.

Odd number * odd number = odd number

Odd+even = odd

Odd+odd = even

Odd even number = even number

Even number+even number = even number

Even number * even number = even number

Silence is better than sound.

There is no lack of artistic conception in mathematics that silence is better than sound. 1903, in a math lecture in new york, mathematician Le Ke walked to the podium. He didn't say a word, but wrote down the calculation results of two numbers on the blackboard with chalk, one is 2-1 to the 67th power, and the other is19370721* 7665438+. Why is this?

Because Lecco has solved the problem that has been unclear for 200 years, that is, 2 is the power of 67-is1a prime number? Since it is equal to the product of two numbers, it can be decomposed into two factors, thus proving that 2 is the power of 67-1is not a prime number, but a composite number.

Cole only gave a short silent report, but it took him three years to reach a conclusion on all Sundays. The courage, perseverance and hard work contained in this simple formula are more attractive than the voluminous report.

7. A little knowledge about mathematics

The history of ancient mathematics in China once had its own glorious chapter.

In foreign countries, this is also called Pascal Triangle. And such triangles are often used in our Olympic Games. The simplest thing is to ask you to find a way.

Now we are required to output such a table through programming. At the same time, this is also the law of the quadratic coefficient of polynomial (a+b) n after bracket opening, that is, 0 (a+b) 0 (0ncr0)1(a+b)1ncr0) (1ncr65438). (2 NCR 1)(2 NCR 2)3(a+b)^3(3 NCR 0)(3 NCR 1)(3 NCR 2)(3 NCR 3)。 .

When b is 1) [Y X refers to the x power of y above, the discovery of Yang Hui's triangle is a very wonderful page. Yang Hui was born in Hangzhou in the Northern Song Dynasty.

In his book "Detailed Explanation of Algorithms in Nine Chapters" written in 126 1, he compiled a triangle table as shown above, which is called "Root of Roots Root of Roots Root of Roots Root of Roots". And such triangles are often used in our Olympic Games. The simplest thing is to ask you to find a way.

The specific usage will be taught in the teaching content, and the remaining number is equal to the sum of the two numbers on its shoulder. In fact, ancient mathematicians in China were far ahead in many important fields of mathematics, and compiled the triangle table as shown above.

In his book "Nine Chapters of Algorithms" written by 126 1, Yang Hui Triangle is a numerical table of triangles arranged by numbers. The general form is as follows, the word is Qian, and its two hypotenuses are composed of the number 1 Yang Hui, and the discovery of Yang Hui Triangle is a very wonderful page.

The history of ancient mathematics in China once had its own glorious chapter; (a nCr b) refers to the combination number] In fact, therefore, the Y term of the X layer of Yang Hui Triangle is directly (y nCr x), and it is not difficult for us to get that the sum of all the terms of the X layer is 2 X (that is, A is in (A+B) X, Chinese ancient mathematicians were in the leading position in many important fields of mathematics:112113338161520156/kloc-. 35 35 2 1 7 1 ....................................................................................................