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How did three geometric problems lead to the emergence of modern algebra
Three geometric problems

The Challenge of Classical Difficult Problems —— Three Geometric Difficult Problems and Their Solutions

Greece, located in southern Europe, is a famous ancient European country and the hometown of geometry. The three geometric problems put forward by the ancients here left a deep impression in the history of science. This worldwide problem, which lasted for more than two thousand years, was probably unexpected by the ancient Greeks who put forward the three major problems.

First, put forward three major issues

There are various geometric shapes in practice, and curves and straight lines are the most basic graphic features. Accordingly, the first basic geometric figures that humans can draw are straight lines and circles. Draw a straight line with a tool with a straight edge, and draw a circle with a tool with one end fixed and the other end rotatable, thus producing rulers and compasses.

When the ancient Greeks said rulers, they meant rulers without scales. They feel from a lot of drawing experience that they can draw all kinds of geometric figures that meet the requirements with only two drawing tools: ruler and compass. Therefore, the ancient Greeks stipulated that rulers and compasses can only be used for a limited number of drawings, which is called ruler-compass drawing method.

In the long-term drawing practice, people have made a large number of drawings that meet the given conditions according to the requirements of ruler drawing, and even some complex drawing problems can be made through limited steps. From the 6th century BC to the 4th century BC, the ancient Greeks encountered three painting problems that troubled them.

1. Angle trisection problem: divide any given angle into three equal parts.

2. Cubic product problem: Find the side length of a cube so that the volume of the cube is twice that of the known cube.

3. Turn a circle into a square: find a square and make its area equal to that of a known circle.

These are three famous problems in ancient geometric drawing, which were put forward before the publication of Geometry Elements. With the spread of geometric knowledge, they were later widely spread around the world.

2. It's hard to see but simple.

On the surface, these three problems are simple, and their drawing seems to be possible. Therefore, there are many people who have been engaged in the study of the three difficult problems of geometry for more than 2000 years. People also put forward various solutions, such as Archimedes, Pappus and others all found a good method of angle bisection and Boluo method to solve the cubic product problem. But all these methods, either do not conform to the ruler drawing method or approximate solution, can not be regarded as the solution of the problem.

During this period, mathematicians also reformed the problem and found many problems closely related to the three major problems, such as finding the line segment to be equal to the circle, dividing the circle equally, inscribed the circle with a regular polygon and so on. But no one can think of a solution to the problem. In this way, the three painting problems have racked many people's brains, and countless people have tried countless times without success. Later, some people realized that the positive results were hopeless, and turned to the opposite side to wonder if these three problems were impossible for the rulers to do.

Mathematicians began to think about which figures can be drawn with rulers and which can't. What is the standard? Where is the boundary? But this is still a very difficult problem.

Three. The discovery of gauss

The wheel of history turned to17th century. Descartes, a French mathematician, founded analytic geometry, which provided an algebraic research method for judging the possibility of drawing ruler and ruler, and made a new turning point for solving three major problems.

The first breakthrough was the German mathematician Gauss. He was born in a poor family in Brunswick on April 30th, 1777. His grandfather is a farmer, his father is a day laborer and his mother is a mason's daughter. Neither of them has a school education. Because of his poor family, in winter nights, in order to save fuel and lamp oil, his father always lets his children go to bed after dinner. Gauss climbed into the attic, secretly lit the homemade radish oil lamp and read in the dim light. His childhood cleverness won the love of a duke. /kloc-was sent to Caroline College by the Duke when he was 0/5 years old, and 1795 came to study at the University of G? ttingen. Because of Gauss's diligence, in the second year after he entered school, he made a polygon of positive 17 according to the ruler drawing method. Then Gauss proved an important theorem of drawing with a ruler: If an odd prime P is fermat number, then a positive P- polygon can be drawn with a ruler, otherwise it cannot be drawn.

It can be concluded that polygons with 3 sides, 5 sides and 17 sides can be made, but polygons with 7 sides, 1 1 sides and 13 sides cannot be made.

Gauss not only made many outstanding achievements in mathematics, but also made important contributions in physics and astronomy. He is known as the "Prince of Mathematics". After Gauss died, according to his wishes, people carved a positive 17 polygon on his tombstone to commemorate his outstanding mathematical discovery when he was young.

Four. Final victory

After the birth of analytic geometry, people know that straight lines and circles are the trajectories of linear equations and quadratic equations respectively. In algebra, the problem of finding the intersection points of straight lines, straight lines and circles, and circles is only a problem of solving linear equations or quadratic equations. Through finite addition, subtraction, multiplication, division and square root, the final solution can be obtained from the coefficients (known quantities) of the equation. Therefore, the question whether a geometric quantity can be made with a ruler compass is equivalent to whether a known quantity can be obtained by adding, subtracting, multiplying, dividing and squaring. In this way, on the basis of analytic geometry, combined with the experience of Gauss and others, people have a deeper understanding of the possibility of drawing with a ruler, and come to the conclusion that the line segments or points that can be drawn with a ruler can only be drawn through finite addition, subtraction, multiplication, division and square root (square correction and positive value).

With standards, it is time to explore boldly and demonstrate cautiously. Whoever can avoid many dangerous beaches and connect his thoughts will be the final winner. 1837, 23-year-old Wan Zhi realized her dream with her wisdom and perseverance, which proved that the cubic product and bisector at any angle could not be solved by a ruler and a ruler drawing, and declared the great victory of mankind in overcoming three geometric problems for more than 2000 years.

His proof method is this:

Assuming that the side length of a known cube is a and the calculated side length of a cube is x, there should be a relationship of x3=2a3 according to the requirement of cubic product. So the cubic product is actually a line that satisfies the equation x3-2a3 = 0.

Section x, but some equations have no rational roots. If a= 1, it is necessary to make a line segment with a cubic root of length 2, but the cubic root of 2 is beyond the operation range of addition, subtraction, multiplication, division and square root of rational numbers, and beyond the range of numbers mentioned in the ruler drawing criterion, so the problem cannot be solved.

With similar ideas, he proved that bisection of an angle is also an impossible problem. In fact, Wanzier also proved a Gauss-Wanzier theorem: If the number of edges n can be written in the following form: n = 2t P 1 P2...pn, where p 1, p2, ... pn are all different prime numbers in the form of 22k+ 1, then the circumference can be divided into n equal parts with a ruler, and

1882, the German mathematician Lin Deman proved the transcendence of π by means of eiπ=- 1, thus solving the problem of turning a circle into a square. Assuming that the radius of a circle is R and the side length of a square is X, the root of an algebraic equation with a circle as the square cannot be represented by addition, subtraction, multiplication and division, so it is impossible to draw with a ruler and a ruler method.

Since then, the three difficult problems of classical geometry have been answered.

For more than 2000 years, generation after generation has overcome three major problems, and some people can't help but ask, is it worth it? If you really need angle trisection, cubic product and rounding into a square in practice, as long as it is effective, why use a ruler and a ruler drawing method to solve it? In fact, mathematical research does not have to be practical. Mathematicians should find out every unknown mystery. This persistent pursuit is the scientific spirit. More importantly, the study of the three difficult problems, in turn, promoted the development of mathematics, and new mathematical ideas and methods emerged, such as the angle bisection method discovered by Archimedes and Pappus, Bob Lott's use of two triangular plates to solve the cubic product problem (which was proved to be true when I was in junior high school), bisecting circles, making regular polygons, Gauss's great discovery on the drawing standards of rulers and straightedges, and so on. Every breakthrough is not only the victory of human wisdom, but also the development of science and technology.

It is particularly worth mentioning that while solving three geometric problems, French mathematician Galois studied the impossibility problem from a universal perspective. 1830, 19-year-old Galois put forward a systematic theory and method to solve this kind of problem, thus establishing the group theory. Group theory is the basis of modern abstract algebra. It is a mathematical model of many practical problems and is widely used. The three major geometric drawing problems are only inferences, examples or exercises of this theory. Therefore, it is generally believed that the solution of the three major problems is attributed to Galois theory, but Galois theory was not published until 14 years after his death, and it was not until 1870 that Galois theory was first introduced comprehensively and clearly.

Group and equality connection

Elementary algebra begins with the simplest one-dimensional linear equation. On the one hand, we discuss binary and ternary linear equations, on the other hand, we study quadratic equations and those equations that can be transformed into quadratic equations. Along these two directions, algebra discusses the linear equations with any number of unknowns, also known as linear equations, and also studies the higher-order unary equations. This stage is called advanced algebra.

Advanced algebra is a general term for the development of algebra to an advanced stage, including many branches. Higher algebra offered by universities now generally includes linear algebra and polynomial algebra.

On the basis of elementary algebra, advanced algebra further expands the research object and introduces many new concepts and quantities which are completely different from usual ones, such as set, vector and vector space. These quantities have operational characteristics similar to numbers, but the research methods and operational methods are more complicated.

A set is the sum of things with certain attributes; Vector is a quantity with both direction and value; Vector space, also called linear space, is a collection of many vectors, which conforms to the rules of some specific operations. The object of operation in vector space is not just a number, but a vector, and its operation properties are also very different.

A brief history of the development of higher algebra

The history of algebra tells us that many mathematicians have gone through a rather uneven road and made painstaking efforts in solving higher-order equations.

People have long known the solution methods of linear and quadratic equations in one variable. As for the cubic equation, in the 7th century, China also got a general approximate solution, which was described in the Classic of Ancient Calculations compiled by the mathematician Wang Xiaotong in the Tang Dynasty. /kloc-in the 3rd century, Qin, a mathematician in the Song Dynasty, fully studied the method of finding the positive roots of digital higher-order equations in his book Shu Shu Jiu Zhang, that is to say, Qin got the general solution of the higher-order equations at that time.

In the west, it was not until the Renaissance at the beginning of16th century that Italian mathematicians discovered the Catan formula for the solution of cubic equation with one variable.

In the history of mathematics, it is said that this formula was first obtained by Italian mathematician Nicolo Tartaglia, and later deceived by Milan mathematician cardano (150 1 ~ 1576) and published in his own works. So now people still call this formula cardano formula (or Cardan formula), in fact, it should be called Nicolo Tartaglia formula.

After solving the cubic equation, the general quartic equation was solved by Ferrari Italy (1522 ~ 1560) soon. This naturally urges mathematicians to continue their efforts to find solutions to higher-order equations with five or more degrees. Unfortunately, although this problem has consumed many mathematicians' time and energy, it has lasted for more than three centuries and has not been solved.

1At the beginning of the 9th century, Abel, a young Norwegian mathematician (1802 ~ 1829), proved that an equation with five or more degrees cannot have algebraic solutions. Even the roots of these equations cannot be expressed by algebraic operations such as addition, subtraction, multiplication, division, multiplication and roots. Abel's proof is not only difficult, but also does not answer the question whether every specific equation can be solved by algebraic method.

Later, the problem that an equation with five or more degrees cannot have algebraic solution was completely solved by a young French mathematician, Galois. At the age of 20, Galois was arrested and imprisoned twice for taking an active part in the French bourgeois revolutionary movement. 1April, 832, died in a private duel shortly after he was released from prison, at the age of 2 1.

Before dying, Galois predicted that he could not get rid of the fate of death, so he wrote a letter to his friends overnight, sketched out his life's mathematical research experience, and attached a manuscript. In a letter to his friend Chevalier Ye, he said, "I have made some new discoveries in my analysis. Some are about equation theory; Some are about the overall function. The public demand for Jacobian or Gauss is not about the correctness of these theorems, but about the importance of these theorems. I hope that some people will find it beneficial for them to eliminate all this confusion in the future. "

After Galois died, according to his last wish, Chevalier Ye published his letter in the Encyclopedia Review. It took 14 years for his manuscript to be edited and published by joseph liouville (1809 ~ 1882) and recommended to the mathematical community.

With the passage of time, the significance of Galois Galois' research results is more and more recognized by people. Although Galois is very young, his contribution to the history of mathematics is not only to solve the algebraic solution problem of higher-order equations that has not been solved for centuries, but more importantly, he put forward the concept of "group" when solving this problem, and thus developed a whole set of theories about groups and fields, opening up a brand-new algebraic world and directly affecting the reform of algebraic research methods. Since then, algebra is no longer centered on equation theory, but instead studies the structural properties of algebra, which promotes the further development of algebra. Among the classic works of mathematical masters, Galois's paper is the thinnest, but his mathematical thought is brilliant.

Basic contents of higher algebra

Algebra started from the general problems of higher algebra and developed into a large-scale mathematical discipline including many independent branches, such as polynomial algebra and linear algebra. The research object of algebra is not only numbers, but also the transformations of matrix, vector and vector space, which can be operated. Although it is also called addition or multiplication, the basic arithmetic of numbers is sometimes no longer effective. Therefore, the content of algebra can be summarized as studying some sets with operations, which are mathematically called algebraic systems. Such as groups, rings, domains, etc.

Polynomial is the most common and simplest function, and it is widely used. Polynomial theory is based on the calculation and distribution of roots of algebraic equations, which is also called equation theory. The study of polynomial theory mainly lies in discussing the properties of algebraic equations, so as to find a simple method to solve the equations.

The research contents of polynomial algebra include divisibility theory, greatest common factor, multiple factors and so on. These are basically the same as middle school algebra. The divisibility of polynomials is very useful for solving algebraic equations. Solving an algebraic equation is nothing more than finding the zero of the corresponding polynomial. When the zero point does not exist, the corresponding algebraic equation has no solution.

We know that a linear equation is called a linear equation, and the algebra that discusses linear equations is called a linear algebra. Determinants and matrices are the most important contents in linear algebra.

The concept of determinant was first put forward by Japanese mathematician Guan Xiaohe in17th century. 1683, he wrote a book called Method of Solving Problems, the title of which means "Determinant Method of Solving Problems". The concept of determinant and its development have been clearly stated in the book. The concept of determinant was first put forward by German mathematician Leibniz in Europe. Jacoby, a German mathematician, summarized and put forward the system theory of determinant in 184 1.

Determinant has certain calculation rules, and the solution of a linear system of equations can be expressed as a formula by determinant, so determinant is a tool for solving linear systems of equations. Determinant can represent the solution of linear equations as a formula, that is, determinant represents a number.

Because the determinant requires the same number of rows and columns, the arranged table is always square. Through the study of it, the theory of matrix is discovered. A matrix is also a table of numbers. The numbers are arranged in rows and columns. The number of rows and numbers can be equal or different.

Matrix and determinant are two completely different concepts. A determinant represents a number, while a matrix is just an ordered arrangement of some numbers. Using the tool of matrix, the coefficients in linear equations can be formed into vectors in vector space; In this way, a series of theoretical problems such as the solution of a multivariate linear equation system and the relationship between different solutions can be completely solved. Matrix is widely used in many aspects, not only in the field of mathematics, but also in the fields of mechanics, physics, science and technology.

Algebra studies not only numbers, but also the transformation of matrix, vector and vector space. For these objects, you can perform operations. Although it is also called addition or multiplication, the basic arithmetic of numbers is sometimes no longer effective. So the content of algebra can be summarized as some sets with operations. In mathematics, such sets are called algebraic systems. The more important algebraic systems are group theory, ring theory and domain theory. Group theory is a powerful tool to study the symmetry law of mathematical and physical phenomena. Now the concept of group has become the most important and universal mathematical concept in modern mathematics, and it has been widely used in other departments.

The relationship between advanced algebra and other disciplines

Algebra, geometry and analytical mathematics are the three basic disciplines of mathematics, and the occurrence and development of each branch of mathematics are basically around these three disciplines. So what's the difference between algebra and the other two subjects?

First of all, algebraic operations are limited and lack the concept of continuity, which means that algebra mainly focuses on discreteness. Although continuity and discontinuity are dialectical unity in reality, in order to understand reality, it is sometimes necessary to divide it into several parts, then study and understand it separately, and then combine them to have a general understanding of reality. This is a simple but important scientific means for us to know things, and it is also the basic idea and method of algebra. Algebra pays attention to discrete relations, but it cannot explain its shortcomings at this time. Time has proved that this characteristic of algebra is effective in many times and directions.

Secondly, algebra is not only of direct practical significance to physics, chemistry and other sciences, but also plays an important role in mathematics itself. Many new ideas and concepts in algebra have greatly enriched many branches of mathematics and become the common foundation of many disciplines.

Elementary algebra begins with the simplest one-dimensional linear equation. On the one hand, we discuss binary and ternary linear equations, on the other hand, we study quadratic equations and those equations that can be transformed into quadratic equations. Along these two directions, algebra discusses the linear equations with any number of unknowns, also known as linear equations, and also studies the higher-order unary equations. This stage is called advanced algebra.

Advanced algebra is a general term for the development of algebra to an advanced stage, including many branches. Higher algebra offered by universities now generally includes linear algebra and polynomial algebra.

On the basis of elementary algebra, advanced algebra further expands the research object and introduces many new concepts and quantities which are completely different from usual ones, such as set, vector and vector space. These quantities have operational characteristics similar to numbers, but the research methods and operational methods are more complicated.

A set is the sum of things with certain attributes; Vector is a quantity with both direction and value; Vector space, also called linear space, is a collection of many vectors, which conforms to the rules of some specific operations. The object of operation in vector space is not just a number, but a vector, and its operation properties are also very different.

A brief history of the development of higher algebra

The history of algebra tells us that many mathematicians have gone through a rather uneven road and made painstaking efforts in solving higher-order equations.

People have long known the solution methods of linear and quadratic equations in one variable. As for the cubic equation, in the 7th century, China also got a general approximate solution, which was described in the Classic of Ancient Calculations compiled by the mathematician Wang Xiaotong in the Tang Dynasty. /kloc-in the 3rd century, Qin, a mathematician in the Song Dynasty, fully studied the method of finding the positive roots of digital higher-order equations in his book Shu Shu Jiu Zhang, that is to say, Qin got the general solution of the higher-order equations at that time.

In the west, it was not until the Renaissance at the beginning of16th century that Italian mathematicians discovered the Catan formula for the solution of cubic equation with one variable.

In the history of mathematics, it is said that this formula was first obtained by Italian mathematician Nicolo Tartaglia, and later deceived by Milan mathematician cardano (150 1 ~ 1576) and published in his own works. So now people still call this formula cardano formula (or Cardan formula), in fact, it should be called Nicolo Tartaglia formula.

After solving the cubic equation, the general quartic equation was solved by Ferrari Italy (1522 ~ 1560) soon. This naturally urges mathematicians to continue their efforts to find solutions to higher-order equations with five or more degrees. Unfortunately, although this problem has consumed many mathematicians' time and energy, it has lasted for more than three centuries and has not been solved.

1At the beginning of the 9th century, Abel, a young Norwegian mathematician (1802 ~ 1829), proved that an equation with five or more degrees cannot have algebraic solutions. Even the roots of these equations cannot be expressed by algebraic operations such as addition, subtraction, multiplication, division, multiplication and roots. Abel's proof is not only difficult, but also does not answer the question whether every specific equation can be solved by algebraic method.

Later, the problem that an equation with five or more degrees cannot have algebraic solution was completely solved by a young French mathematician, Galois. At the age of 20, Galois was arrested and imprisoned twice for taking an active part in the French bourgeois revolutionary movement. 1April, 832, died in a private duel shortly after he was released from prison, at the age of 2 1.

Before dying, Galois predicted that he could not get rid of the fate of death, so he wrote a letter to his friends overnight, sketched out his life's mathematical research experience, and attached a manuscript. In a letter to his friend Chevalier Ye, he said, "I have made some new discoveries in my analysis. Some are about equation theory; Some are about the overall function. The public demand for Jacobian or Gauss is not about the correctness of these theorems, but about the importance of these theorems. I hope that some people will find it beneficial for them to eliminate all this confusion in the future. "

After Galois died, according to his last wish, Chevalier Ye published his letter in the Encyclopedia Review. It took 14 years for his manuscript to be edited and published by joseph liouville (1809 ~ 1882) and recommended to the mathematical community.

With the passage of time, the significance of Galois Galois' research results is more and more recognized by people. Although Galois is very young, his contribution to the history of mathematics is not only to solve the algebraic solution problem of higher-order equations that has not been solved for centuries, but more importantly, he put forward the concept of "group" when solving this problem, and thus developed a whole set of theories about groups and fields, opening up a brand-new algebraic world and directly affecting the reform of algebraic research methods. Since then, algebra is no longer centered on equation theory, but instead studies the structural properties of algebra, which promotes the further development of algebra. Among the classic works of mathematical masters, Galois's paper is the thinnest, but his mathematical thought is brilliant.

Basic contents of higher algebra

Algebra started from the general problems of higher algebra and developed into a large-scale mathematical discipline including many independent branches, such as polynomial algebra and linear algebra. The research object of algebra is not only numbers, but also the transformations of matrix, vector and vector space, which can be operated. Although it is also called addition or multiplication, the basic arithmetic of numbers is sometimes no longer effective. Therefore, the content of algebra can be summarized as studying some sets with operations, which are mathematically called algebraic systems. Such as groups, rings, domains, etc.

Polynomial is the most common and simplest function, and it is widely used. Polynomial theory is based on the calculation and distribution of roots of algebraic equations, which is also called equation theory. The study of polynomial theory mainly lies in discussing the properties of algebraic equations, so as to find a simple method to solve the equations.

The research contents of polynomial algebra include divisibility theory, greatest common factor, multiple factors and so on. These are basically the same as middle school algebra. The divisibility of polynomials is very useful for solving algebraic equations. Solving an algebraic equation is nothing more than finding the zero of the corresponding polynomial. When the zero point does not exist, the corresponding algebraic equation has no solution.

We know that a linear equation is called a linear equation, and the algebra that discusses linear equations is called a linear algebra. Determinants and matrices are the most important contents in linear algebra.

The concept of determinant was first put forward by Japanese mathematician Guan Xiaohe in17th century. 1683, he wrote a book called Method of Solving Problems, the title of which means "Determinant Method of Solving Problems". The concept of determinant and its development have been clearly stated in the book. The concept of determinant was first put forward by German mathematician Leibniz in Europe. Jacoby, a German mathematician, summarized and put forward the system theory of determinant in 184 1.

Determinant has certain calculation rules, and the solution of a linear system of equations can be expressed as a formula by determinant, so determinant is a tool for solving linear systems of equations. Determinant can represent the solution of linear equations as a formula, that is, determinant represents a number.

Because the determinant requires the same number of rows and columns, the arranged table is always square. Through the study of it, the theory of matrix is discovered. A matrix is also a table of numbers. The numbers are arranged in rows and columns. The number of rows and numbers can be equal or different.

Matrix and determinant are two completely different concepts. A determinant represents a number, while a matrix is just an ordered arrangement of some numbers. Using the tool of matrix, the coefficients in linear equations can be formed into vectors in vector space; In this way, a series of theoretical problems such as the solution of a multivariate linear equation system and the relationship between different solutions can be completely solved. Matrix is widely used in many aspects, not only in the field of mathematics, but also in the fields of mechanics, physics, science and technology.

Algebra studies not only numbers, but also the transformation of matrix, vector and vector space. For these objects, you can perform operations. Although it is also called addition or multiplication, the basic arithmetic of numbers is sometimes no longer effective. So the content of algebra can be summarized as some sets with operations. In mathematics, such sets are called algebraic systems. The more important algebraic systems are group theory, ring theory and domain theory. Group theory is a powerful tool to study the symmetry law of mathematical and physical phenomena. Now the concept of group has become the most important and universal mathematical concept in modern mathematics, and it has been widely used in other departments.

The relationship between advanced algebra and other disciplines

Algebra, geometry and analytical mathematics are the three basic disciplines of mathematics, and the occurrence and development of each branch of mathematics are basically around these three disciplines. So what's the difference between algebra and the other two subjects?

First of all, algebraic operations are limited and lack the concept of continuity, which means that algebra mainly focuses on discreteness. Although continuity and discontinuity are dialectical unity in reality, in order to understand reality, it is sometimes necessary to divide it into several parts, then study and understand it separately, and then combine them to have a general understanding of reality. This is a simple but important scientific means for us to know things, and it is also the basic idea and method of algebra. Algebra pays attention to discrete relations, but it cannot explain its shortcomings at this time. Time has proved that this characteristic of algebra is effective in many times and directions.

Secondly, algebra is not only of direct practical significance to physics, chemistry and other sciences, but also plays an important role in mathematics itself. Many new ideas and concepts in algebra have greatly enriched many branches of mathematics and become the common foundation of many disciplines.