1, intuitive definition method
Intuitive definition is also called primitive definition. The original concept produced by intuition cannot be explained by other concepts. The meaning of the original concept can only be described by other terms and their respective characteristics, such as points, lines, surfaces, elements of sets, correspondence, etc. In geometry, primitive concept is the result of people's generalization and abstraction of a class of things in long-term practical activities, and it is the product of primitive abstract thinking activities. Intuition is rarely defined.
2. Definition method of "species+class difference"
Definition method of "species+taxonomic difference": defined concept = nearest species concept (species)+taxonomic difference. This is a commonly used definition of connotation. The "closest concept" is the latest concept of the defined concept, and the "class difference" is the essential attribute that distinguishes the defined concept from other concepts in the latest concept.
For example, the class concepts with parallelogram as the closest concept include rectangle, diamond and so on. The equality of adjacent sides of a diamond is the essential attribute that distinguishes it from a rectangle, and the equality of adjacent sides is the class difference of a diamond. Let's look at a few examples defined by "species+class differences":
An isosceles trapezoid is a trapezoid with two equal waists.
A right-angled trapezoid is a trapezoid with a right-angled base.
An isosceles triangle is a triangle with equal sides or angles.
Logically, it can also be defined by summarization and extension, such as "rational numbers and irrational numbers are collectively called real numbers".
As can be seen from the above example, to define a concept with the method of "species plus class difference", we must first find out the latest concept of the defined concept, then compare the objects reflected by other concepts in the same concept of the defined concept to find out the "class difference", and finally add the class difference to the latest concept to form a defined concept and give a definition. The definition of species plus class difference is also called substantive definition in formal logic, which belongs to deductive definition, and the order is from general to special. This definition not only reveals the particularity of the object reflected by the concept, but also points out the generality, which is an effective definition method. Because of the different category characteristics and category differences of the concept itself, the narrative forms are also different.
This definition method can reveal the connotation of the defined concept with the connotation of the known concept. Revealing the connotation of concepts is accurate and clear, which is helpful to establish the relationship between concepts and systematize knowledge. Therefore, it is widely used in the definition of mathematical concepts in middle schools.
3. Definition method of occurrence
Occurrence definition method (also called constructive definition method): The definition method of revealing the essential attributes of the defined concept by describing the occurrence process of the object or the characteristics formed by the defined concept is called occurrence definition method. This definition is a special form of the definition of "species+class difference". The class difference in definition is to describe the occurrence process or formation characteristics of the defined concept, rather than to reveal the unique essential attributes of the defined concept.
For example, the locus of a point equidistant from a fixed point on a plane (space) is called a circle (sphere). In addition, the concepts of cylinder, cone, frustum of a cone, differential, integral and coordinate system are also defined by the generation method in middle school mathematics.
Another example is:
The locus of a point whose sum of the distances from a point to two fixed points on a plane is equal to a fixed length is called an ellipse.
The trajectory of a moving point that rotates around a central point or axis and gradually moves away from the central point or axis is called a helix.
The straight rod is tangent to the circle and rolls without sliding. The locus of a fixed point on this straight bar is called the involute of a circle.
Let it be an event in test E. If E is repeated for n times, and A appears for times, it is called the frequency of occurrence of event A in n tests.
Under certain conditions, when the number of experiments increases, the frequency of event A gradually stabilizes at a fixed constant p, which is called the probability of event A. 。
Therefore, as long as there is human mathematical activity, there is a generative definition of the concept.
4. Inverse definition method
This is a definition that gives the extension of concept, also called inductive definition. For example, integers and fractions are collectively called rational numbers; Sine, cosine, tangent and cotangent functions are called trigonometric functions; Ellipse, hyperbola and parabola are called conic curves; Logical sum, negation, product operations are called logical operations and so on, which are all defined in this way.
5, the conventional definition method
Mathematical concepts designed for the needs of practice or the development of mathematics itself. Actually,
People find some concepts very important, so they point them out and use them in mathematical activities, such as some specific numbers: pi, the base e of natural logarithm, etc. Some important values: average, frequency, variance, etc. Generalization of some mathematical activities: for example, algebra refers to the mathematical activities of studying finite multivariate finite operations; Geometry refers to the mathematical activity of studying the structure and form of objects in space and spatial structure; Random events refer to things that may or may not occur under the same conditions of society and nature, but their frequency is stable in a large number of repeated experiments; Probability refers to the mathematical measurement of the probability of random events; Wait a minute.
At the same time, in the development of mathematics, it is sometimes necessary to reach an agreement on mathematical concepts. For example, in a zero-power protocol, a vector with a modulus of zero is defined as a zero vector, and a vector with a modulus of 1 is defined as a unit vector. For example, the direction of vector product is defined by the right-hand rule. In mathematics teaching, it is necessary to instill in students that mathematical concepts can be agreed (its deeper meaning is that mathematics can be created). Consistency is the result of simple thinking. Because of this convention, mathematics is easy to operate. Conventional practice is not unique, but it should be reasonable or conform to the laws of objective things. For example, it is not impossible to specify the direction of vector product according to the left-handed rule. Convention is not arbitrary, but generally only those important concepts. For example, the limit when convention n tends to infinity is the base e of natural logarithm, because this number is very important for calculation.
6. Descriptive definition
Descriptive definition is also called descriptive definition. The concepts of movement, change and relationship in mathematics are strictly expressed (beyond the stage of intuitive description), and these concepts belong to descriptive definitions, such as equality function, sequence limit, function limit and so on.
Function concept: let d be a subset of the set of real numbers. If, according to the given rules, each of D has a unique corresponding real number Y, it is called a unary real function defined on D, which describes the relationship between the variable Y and the variables in the concept.
Concept of sequence limit: for sequence {} and a number, if any given positive number has a natural number, for all natural numbers, n is the limit of sequence {} when n tends to infinity, which is recorded as. The concept describes the degree of "as close as possible (as long as possible)", which makes the intuitive expression of "infinite approach" rise to a strict level.
The concept of function limit: for a function and a number A defined nearby, if there is a positive number for any given positive number, as long as X is established within the defined domain, this number is the limit when approaching, marked as "as close as possible to A", which is a strict mathematical concept.
7. Process definition
Some complex mathematical concepts are created based on practical mathematical activities, and such concepts are guided by processes. For example:
Derivative: Let y= be defined near the point. When the independent variable obtains the variation (≠0), the function obtains the corresponding variation and ratio. When the limit exists, this limit value is called derivative, which is obtained through the process of "making change-making quotient-finding limit".
A boundary function is defined on definite integral: []. When the point is inserted into [], the limit of sum exists, and this limit value is called the definite integral on []. The concept of definite integral is obtained through the process of "dividing [] (insertion point) into one work and one limit".
Besides, there are other ways to give concepts in mathematics. For example, the definition of N-dimensional vector space: "N is the whole of an ordered real array (), and give it the operations of addition and number multiplication ()+". It is an analogy extension of two-dimensional vector space {0}. For example, the concepts of "group" and "distance space" are defined by a set of axioms. Axiomatic methods are mostly used in advanced mathematics and middle schools.
In addition, there are the definitions of recursion in middle school mathematics (such as "order determinant, n-order derivative, n-fold integral"), the definition of another object (such as using exponential concept to define logarithm) and so on.
The above classification is relatively rough. The definition of learning concept is not to distinguish which definition it belongs to, but to understand the connotation of the concept, grasp the extension of the concept, and apply them to learn mathematical knowledge and solve related problems.
In order to correctly define this concept, the definition should meet the following basic requirements:
The definition of (1) should be commensurate. That is, the extension of the defined concept must be the same as that of the defined concept, which can neither be expanded nor reduced. It is to be suitable, not wide or narrow. For example, an infinitely acyclic decimal is called an irrational number. An irrational number is defined as an infinite decimal (too wide), or an irrational number is defined as a number that is not divisible by the square root (too narrow). Obviously,
(2) The definition cannot be circular, that is, in the same scientific system, the concept of B cannot be defined by the concept of A,
At the same time, we use the concept of B to define the concept of A. For example, our angle is called a right angle, and one-ninetieth of the right angle is called 1 degree, thus creating a cycle.
(3) The definition should be clear and concise, generally without negative forms and unknown concepts. For example, a straight line is called a straight line (unclear); Two groups of planar parallelograms with parallel opposite sides (not concise); Numbers that are not rational numbers are called irrational numbers (negative numbers); For junior high school students, in the complex number a+ i, the number of imaginary part 6-0 is called real number (applying unknown concept), which is inappropriate.