Wu Guoping: What is a function? What kind of legend does it have?
Speaking of functions, no matter how well you do in math, you will know a little bit about it. The knowledge content of functions can be said to run through the entire mathematical learning, especially after entering junior high school, we have to learn a function (positive proportional function), inverse proportional function, quadratic function and so on. Entering high school and college, not to mention, there are various types of functions, so that many people call math learning is not easy.
The learning of functions is very logical and systematic, coupled with the fact that the abstraction of the entire piece of knowledge content is very strong, there are many knowledge points, etc., it is necessary for everyone to have a good mathematical foundation and improve the comprehensive learning ability in order to calmly face the learning of functions. For example, based on the image of the function, the property, the expression, etc., if we describe a function, that is, there are generally four ways to represent the function: the analytical method, the list method, the image method and the language description method.
Analytic is a method of expressing the relationship between two variables in terms of an equation containing mathematical relations.
List method refers to the method of using a list to represent the functional relationship between two variables.
The image method refers to the value of the independent variable x of a function and the corresponding dependent variable y as the abscissa and ordinate coordinates of the points respectively, and to depict its corresponding points in the Cartesian coordinate system, and the graph composed of all these points is called the image of the function.
The analytic method, the list method, and the image method are the three methods we most commonly use to represent functions, each with its own advantages and disadvantages, such as the advantage of the analytic method is that it can concisely, accurately, and clearly represent the quantitative relationship between the function and the independent variable; the disadvantage is that the corresponding value often has to go through more complex operations, and the functional relationship in the actual problem may not be expressed by expressions.
Therefore, based on the advantages and disadvantages of these three methods, there is also a method of representing functions called the language narrative method, that is, the use of language words to describe the relationship of functions.
Many people can feel the complexity of function learning when they see this, which is mainly based on the particularity of the function itself. Let's take a look at the definition of functions in modern mathematics, and you can see some clues.
Let A, B be a non-empty set of numbers, if, according to some definite correspondence f, so that for any number x in set A, there is a uniquely determined number y in set B and its correspondence, then it is called mapping f:A→B as a function from set A to set B, denoted as y=f(x), x∈A or f(A)={y丨f(x)=y,y∈B}.
where x is called the argument, y is called the function of x, set A is called the domain of the function, y corresponding to x is called the function value, and the set of function values {f(x)丨 x∈A} is called the value range of the function, and f is called the corresponding law.
Among them, the definition domain, value domain and corresponding law are called the three elements of the function, which are generally written as y=f(x), x∈D, if the definition domain is omitted, it generally refers to the set that makes the function meaningful.
From the definition of the function, we can see that the function is a correspondence that occurs between the collections. At the same time, according to the sentence "If according to some definite correspondence f", we must deeply understand that there are more than one functional relationship between A and B.
Therefore, when the corresponding law of a function can be expressed analytically, we use an analytic representation; if a functional relationship cannot be expressed in an analytical way, then it needs to be expressed in images, lists, and other forms.
Mathematics learning we pay attention to the logical relationship between cause and effect, only by mastering each link, you can really understand the meaning of a certain knowledge point, in order to understand how important it is to master the basic knowledge for mathematical learning. Just like the concept of functions, it did not arise out of thin air, its development history is an abbreviation of a mathematical history, let's briefly understand it.
In the early 17th century, the Italian mathematician Galileo Galileo, in his book Two New Sciences, expressed the relationship between functions in the language of words and proportions, which is an early description of the concept of variables or functions.
Around 1637, the French mathematician Descartes, in his analytic geometry, already mentioned the dependence of one variable on another. Unfortunately, perhaps due to the limited mathematical knowledge at the time, Descartes did not further refine the concept of functions.
In the late 17th century , although the British physicist and mathematician Newton and the German philosopher and mathematician Leibniz established calculus , it established milestones for the development of mathematics. Unfortunately, neither of them nor mathematicians of the same period knew the general meaning of functions, and most functions were studied as curves.
For example, in 1673, Leibniz first used "function" (function) to represent "power", but he only used this word to represent the relevant geometric quantities of points on the curve, such as abscissa, ordinate, tangent length, etc. In his calculus discussion, Newton used "flow" to represent the relationship between variables.
In 1718, the Swiss mathematician Johann Bernoulli defined the concept of functions on the basis of the Leibniz concept of functions: "a quantity composed of any one variable and any form of constant." He meant that all the equations of the variable x and the constant were called x functions, and stressed that functions should be represented by formulas.
Euler's mathematical masterpiece "Introduction to Infinite Analysis"
In 1748, the Swiss mathematician Euler defined a function in his book Introduction to Infinite Analysis as " A function of a variable is an analytic expression consisting of some number or constant of that variable and any one of them." ”
Euler's greatest advance was to refer to the definition of the function given by John Bernoulli as an analytic function, and further distinguish it between algebraic and transcendental functions, and also considered "arbitrary functions".
In 1755, Euler gave another definition: "If certain variables depend in a certain way on other variables, that is, when the latter variables change, the previous variables also change, we call the preceding variables a function of the latter variables." ”
It can be seen that Euler's definition of functions was more general and more extensive than John Bernoulli's, contributing to the development of mathematics at that time.
In 1821, the French mathematician Cauchy combined the knowledge of functions of his predecessors to give the definition of the function from the definition of variables: there is a certain relationship between some variables, when given the value of one of the variables, the value of other variables can be determined with it, the initial variable is called the independent variable, and the other variables are called functions.
Cauchy's greatest contribution was the first appearance of the term argument, while pointing out that there is no need for analytic expressions for functions. Unfortunately, Cauchy believes that functions can be represented by multiple function analytics, which limits the development of functions.
In 1822, the French mathematician Fourier found that some functions can be represented by curves, can also be represented by a formula, or by multiple equations, thus ending the debate over whether the concept of functions is represented by a unique equation and pushing people's understanding of functions to a new level.
In 1837, the German mathematician Dirichlet boldly proposed that how to establish the relationship between x and y was not an important matter. On this basis, Dirichlet broadened the concept of functions, arguing that for every definite value of x on an interval, y has a definite value, then y is called a function of x.
The biggest feature of Dirichlet's definition of functions is that it avoids the description of dependencies in function definitions and is accepted by all mathematicians in a clear way, which is what we often call the classical function definition.
The definition of functions really changed qualitatively after the German mathematician Cantor founded set theory.
The American mathematician Oswald Wibrun uses the concepts of "set" and "correspondence" to give modern functions definitions, and uses the set concept to further concretize the correspondence, domain and value domain of functions. This breaks the limit of "variables are numbers", and variables can be numbers or other objects.
In 1914, the German mathematician Hausdorff defined functions with the ambiguous concept "ordinal even" in the Compendium of Set Theory, which avoided the concepts of "variables" and "correspondences" with unclear meanings.
In 1921, the Polish mathematician Kuratowski used the concept of sets to further define "ordinal couples", making Hausdorff's definition more rigorous.
In 1930 , modern mathematics formally defined a function as follows : if for any element x of the set M , there is always an element y determined by the set N corresponding to it , then a function is defined on the set M , denoted f.
Element x is called the independent variable, and element y is called the dependent variable.
From the discovery to the establishment of the functional relationship, it has gone through hundreds of years before and after, and many mathematicians have invested in it before and after, spending a lot of time and energy to study. Seeing this, do you think you have worked hard enough to learn math?
The English name of the function is function, when translated as Chinese, why is it a function?
Late Qing Dynasty famous mathematician, astronomer, botanist, and mechanician Li Shanlan (sitter)
In 1859, Li Shanlan, a famous mathematician in China's Qing Dynasty, translated "function" into the "function" of Chinese when translating the book "Algebra".
Li Shanlan believes that the ancient Chinese word "letter" and the word "contain" are common, and both have the meaning of "containing", so "function" refers to the meaning of the formula containing variables, specifically: where the formula contains the variable x, the formula is called the function of x.
A brief history of the development of functions is a microcosm of the history of the development of mathematics, and every mathematical term that seems very simple to us today does not know how many mathematicians and mathematical workers have spent their lives investing in it to have today's mathematical achievements.
Therefore, I hope that in the process of mathematical learning, we must work hard, pay attention to methods, persevere, reflect more, think more, etc., so that we can slowly learn mathematics