Mathematically, operation is a kind of behavior, through the possible combination of known quantities, new quantities can be obtained. The essence of operation is the mapping between sets.
Generally speaking, operation refers to algebraic operation, which is a correspondence in a set. For a pair of elements a and b taken out in order in set A, there is a unique third element c in set A corresponding to them. It is called that an operation is defined in set A.
From this operation, two operations can be obtained, that is, one of a and b is taken as the desired operation, and c is taken as the known operation. The operation thus obtained is called the inverse operation of the original operation.
For example, addition is the operation of finding a+b=c when a and b are known, so the operation of finding b when a and c are known, or the operation of finding a when b and c are known, is the inverse operation of addition, called subtraction.
For example, the addition 5+3=8 in arithmetic, where 5 and 3 are inputs, 8 is the result, and the plus sign "+" indicates that this is an addition operation. This is a common Binary operation , essentially A×B → C Mapping of forms. Other common operations include absolute values trigonometric function 、 Inverse trigonometric function 、 Logical negation Wait, these are Unary operation , essentially A → B Mapping of forms. Algebraic operation Both are binary operations. There are many examples of binary operations. Addition, subtraction, multiplication, division Power , Prescription logarithm ； Intersection, union, complement, difference and Cartesian product between sets; Logical And Logical OR Etc. Mathematically, binary operations are defined as follows: Assumptions S and T namely aggregate ， S On T The value operation R refers to Descartes direct product S × S reach T A mapping of, that is mapping ： R: S × S → T
According to the traditional writing, for S Two elements in a , b , we use a R b To represent this operation.
When S = T We say that the operation is closed.
such as S = T Is a set of real numbers. At this time, we can define addition, subtraction, multiplication and division operations respectively. [1]
Another example S yes n Dimensional reality vector Set, T Is a set of real numbers, we can define inner product Operation. In addition to the above common Algebraic operation In addition, there are many other operations, such as square root operation, derivatives Operation, integral operation, convolution Operations, rounding operations, etc. These operations can be regarded as“ operator ”Role of. The so-called operator can be seen as acting on the operation element Function symbol 。 For example, the operator of subtraction operation is minus sign -, the operator of square root operation is root sign √￣, and the operator of derivative operation is d/d x The operator of integral operation is the integral sign. According to mathematical rules, the process of substituting or transforming quantities (or numbers) to obtain the expression results. It is the main content of mathematical research. Mathematics is a subject that studies quantity and its operation, graph and its transformation. The most basic operation of number is the four arithmetic operations, namely, addition, subtraction, multiplication and division. A number multiplies itself several times, which is called the power operation; A number is raised to the nth power (n is a positive integer), which is called the square root operation. Four arithmetic operations, together with power and square root operations, are collectively referred to as algebraic operations.
In higher mathematics, in addition to algebraic operations, there are limit operations, derivatives, integrals and other operations. The most basic operation is limit operations. Operations related to limits are called "analytical operations".
Each kind of operation has its own suitable algorithm, such as associative law, commutative law, distributive law, etc.
The original meaning of operation in Chinese is to move arithmetic or move arithmetic beads. Now it generally refers to any transformation in mathematics.
Addition is written with the plus sign "+" between terms; The result is indicated by an equal sign. For example, [2]
In some cases, even if no symbol appears,
A fraction immediately following a number represents a mixed number. For example,
This symbol can be controversial because in most other contexts, two numbers together represent multiplication.
The sum of a series of related numbers can be represented by the sigma symbol to indicate iteration. For example,
In general addition, numbers are called addends, and the result is called sum; Adding is to transfer so many addends to the sum. This is distinguished from the factors that need to be multiplied. In fact, in Renaissance Many authors did not consider the first plus sign at all. Today, due to the exchange of property with markups, "farmers" are rarely used, and these two terms are usually called addends. All the above terms are derived from Latin. "Add" and "add" are English words derived from the Latin verb "add", which in turn are "ad" and "" of "original" - European root * deh3 "give"; therefore, the complement is to give. Using the gerundive suffix - nd leads to "add", "what to add". Similarly, from the perspective of "increase", one is "strengthen", "increase things".
"Sum" and "summary" come from the Latin noun "highest, highest" and related words. Because the ancient Greeks and Romans often increased upward, which is contrary to the modern downward practice, making a number higher than the addend. The plus sign "+" (Unicode: U+002B; ASCII:+) is the abbreviation of Latin "et", meaning "and". It appears in mathematical works dating back to at least 1489.
When subtraction is used, there is a minus sign "−" between two items, and the result is represented by an equal sign. For example,
In some cases, subtraction is "to be understood", even if no sign appears:
For a column with two numbers, the smaller number is indicated in red, which usually indicates that the smaller number in the column is to be subtracted. The difference from the following is below a row. This is very common in accounting.
Formally, the number that is subtracted is called the minus, and the number that is subtracted from it is the minuend.
All these terms are derived from Latin. "Subtraction" is an English word derived from the Latin verb subtrahere. It is a compound word of "from under" and "to pull", so it should be extracted from below. The result of using gerun diving suffix - nd in "subtrahend", "subtracted things". Similarly, from the minute "reduce or decrease", one gets "small" and "things decrease".
Countable is the world in which the genus rules the whole quantity, while the four principles of counting operation It can be regarded as all the equipment of mathematicians—— maxwell [3] The working intensity operation Speed and memory function can surpass humans, but it is impossible to surpass humans in consciousness, reasoning, etc—— Zhou Haizhong [4] Life is like solve equations ， operation Each step of the seems to be irrelevant, but it is necessary for the final solution. The result is often fascinating, but I prefer the process itself, which is the secret of the result—— Feng Ding [5] 
