In the daily learning process, did you wake up immediately when you heard the knowledge points? Knowledge points are not necessarily words. In addition to the definition of mathematical knowledge points, equally important formulas can also be understood as knowledge points. Still worried that there is no knowledge point summary? The following is a summary of my collection of compulsory mathematics knowledge points in senior high school. Welcome to read the collection.
The first chapter is the preliminary algorithm.
1. 1. 1
The concept of algorithm
Features of the algorithm:
(1) finiteness: The sequence of steps of the algorithm is finite, and it must stop after a finite number of operations, not an infinite number.
(2) Determinism: Every step in the algorithm should be deterministic, can be effectively executed and get certain results, and should not be fuzzy.
(3) Sequence and correctness: The algorithm starts from the initial step and is divided into several definite steps. Each step can only have one definite subsequent step, and the former step is the premise of the latter step. Only when the previous step is carried out can the next step be carried out, and each step is accurate to complete the problem.
(4) Uniqueness: the solution of a problem is not necessarily unique, and a problem can have different algorithms.
(5) Universality: Many specific problems can be solved by designing reasonable algorithms, such as mental calculation and calculator calculation, which must be solved through limited and pre-designed steps.
1. 1.2
flow chart
The concept of (1) program composition: the program block diagram, also known as the flow chart, is a graph that accurately and intuitively represents the algorithm with specified graphics, flow lines and words.
(2) Graphical symbols and their functions that constitute the program box.
When learning this part of knowledge, we should master the shape, function and usage rules of each graphic. The rules for drawing program block diagram are as follows:
1. Use standard graphic symbols.
2. The block diagram is generally drawn from top to bottom and from left to right.
3. Except for the judgment box, most flow chart symbols have only one entry point and one exit point. The decision box has a unique symbol and multiple exit points.
4. The judgment box is divided into two categories, one is the judgment of "yes" and "no", and there are only two results; The other is multi-branch judgment, which has several different results.
5. The language described in graphic symbols should be very concise and clear.
(3) Three basic logical structures of the algorithm: sequence structure, conditional structure and cyclic structure.
1, sequence structure: sequence structure is the simplest algorithm structure. Reports and boxes are made from top to bottom. It consists of several processing steps that are executed in turn. It is the basic algorithm structure that any algorithm can't do without.
The embodiment of sequence structure in the program block diagram is that the program blocks are connected from top to bottom by pipeline, and the algorithm steps are executed in sequence. For example, in the schematic diagram, box A and box B are executed sequentially, and the operation specified in box B can only be executed after the operation specified in box A is executed.
2. Conditional structure:
Conditional structure refers to the algorithm structure of choosing different flow directions according to whether the conditions are true or not by judging the conditions in the algorithm.
Whether the condition P is true or not, choose to execute box A or box B. No matter whether the condition P is true or not, only one of box A or box B can be executed. It is impossible to execute box a and box b at the same time, and it is impossible to execute both. A judgment structure can have multiple judgment boxes.
3. Circular structure: In some algorithms, it is often that a processing step is repeatedly executed from a certain place according to certain conditions. This is the cycle structure, and the repeated processing steps are the cycle body. Obviously, the loop structure must contain the conditional structure. Circular structures can be subdivided into two categories:
(1), one is the current circulation structure, as shown in the left figure below. Its function is to execute block a when the given condition p holds. After the execution of block A is completed, it will be judged whether condition P is established. If it is still true, block A is executed again, and so on until condition P is not true once. At this point, block A will not be executed and the loop structure will be left.
(2) The other is the until cycle structure, as shown in the right figure below. Its function is to execute first, and then judge whether the given condition P is true or not. If p still does not hold, continue to execute block A until the given condition P holds, then stop executing block A and leave the loop structure.
Note: the loop structure of 1 will terminate the loop under certain conditions, which requires conditional structure to judge. Therefore, the loop structure must contain conditional structure, but "infinite loop" is not allowed.
There is a count variable and an accumulation variable in the loop structure. The counting variable is used to record the number of cycles, and the cumulative variable is used to output the results. Counting variables and accumulating variables are generally executed synchronously, accumulating once and counting once.
1.2. 1
Input, output statements and assignment statements
3. Assignment statement
General format of (1) assignment statement;
(2) The function of the assignment statement is to assign the value expressed by the expression to the variable;
(3) The "=" in the assignment statement is called the assignment number, which is different from the equivalent number in mathematics. The left and right sides of the assignment number cannot be interchanged, it assigns the value of the expression on the right side of the assignment number to the variable on the left side of the assignment number;
(4) The left side of the assignment statement can only be a variable name, not an expression, and the expression on the right side can be data, constant or formula;
(5) A variable can be assigned multiple times.
Note: ① The left side of the assignment number can only be a variable name, not an expression. 2=X is wrong. ② The left and right distribution numbers cannot be interchanged. For example, "A = B" and "B = A" have different meanings. ③ Algebraic calculus cannot use assignment statements. (such as simplification, factorization, solving equations, etc.). (4) The assignment symbol "=" has different meanings from the equal sign in mathematics.
Analysis: In the if-then-else statement, "condition" indicates the condition of judgment, and "statement 1" indicates the content of operation to be executed when the condition is met; "Statement 2" indicates the content of the operation to be performed when the conditions are not met; END IF indicates the end of the conditional statement. When the computer executes, the condition after IF is judged first, and if the condition is met, the statement after THEN is executed1; If the condition is not met, execute the statement 2 1.3. 1 after ELSE.
1, toss division. Also known as Euclid algorithm, the steps of finding the greatest common divisor by alternating division are as follows:
(1): divide the larger number m by the smaller number n to get a quotient ≠0, then divide the divisor n by the remainder, and use the divisor RRS0 and a remainder r0;
(2): If 0=0, then n is the greatest common divisor of m and n; If 0R0 gets the quotient S 1 and the remainder r1; RRR；
(3): If 1=0, then 1 is the greatest common divisor of m and n; If 1≠0, R0 is divided by the remainder R 1 to get the quotient S2 and the remainder R2; Calculate in tuRn until Rn=0, and the rn obtained at this time? 1 is the greatest common divisor.
2. More phase subtraction
China also had an algorithm for finding the greatest common divisor in the early days, that is, the subtraction technique. The steps to find the greatest common divisor with more subtraction skills in "Nine Chapters Arithmetic": What is half, what is not half, and what is the denominator? The fewer the number of children, the more they decrease and the more they lose, and so on, the number is about equal.
Translated as: (1): Give two positive numbers at will; Determine whether they are all even numbers. If so, reduce by 2; If not, proceed to the second step. (2): Subtract the smaller number from the larger number, then compare the smaller number with the difference, and subtract the number from the larger number. Continue this operation until the numbers obtained are equal, then this number (equal number) is the greatest common divisor. Example 2 Find the greatest common divisor of 98 and 63 by polyphase subtraction.
3, the difference between division and subtraction:
(1) are all methods to find the greatest common divisor. In calculation, division is the main method and subtraction is the main method. The number of calculations of division is relatively small, especially when the size of two numbers is very different.
(2) From the form of the result, the result of division is obtained when the remainder of division is 0, while subtraction is obtained when subtraction equals difference.
1.3.2
Qin's algorithm and sorting
1, Qin algorithm concept:
Evaluation of f (x) = anxn+an-1xn-1+...+a1x+A0.
f(x)= anxn+an- 1xn- 1+…。 +a 1x+A0 =(anxn- 1+an- 1xn-2+…。 +a 1)x+A0 =((anxn-2+an- 1xn-3+…。 +a2)x+a 1)x+a0
=......=(...(anx+an- 1)x+an-2)x+...+a 1)x+a0
To require the value of polynomial, first calculate the value of the sequence polynomial in the innermost bracket, that is, v 1=anx+an- 1.
Then calculate the value of the polynomial step by step from the inside out, that is, v2 = v1x+an-2v3 = v2x+an-3 ... VN = VN-1x+A0.
In this way, the evaluation problem of polynomial of degree n is transformed into the problem of finding the value of polynomial of degree n.
Chapter II Statistics
2. 1. 1
simplerandom sampling
1. Population and sample
In statistics, the whole research object is called population, each research object is called individual, and the total number of individuals in the population is called overall capacity. In order to study the related properties of the population, we randomly select a part: research, which we call samples. The number of individuals is called sample size.
2. Simple random sampling, also called pure random sampling. In other words, the survey units are randomly selected from the population, without any grouping, classification, queuing, etc. The characteristics are: the probability of each sample unit being extracted is the same (the probability is equal), and each unit of the sample is completely independent, and there is no certain correlation and exclusion between them. Simple random sampling is the basis of other sampling forms. This method is usually only used when the difference between the whole units is small and the number is small.
3. Common methods of simple random sampling:
(1) draw lots; (2) Random number table method; ⑶ Computer simulation method; ⑷ Direct extraction with statistical software. In the sample size design of simple random sampling, the main considerations are: ① population variation; ② Allowable error range; ③ Degree of probability assurance.
4. Draw lots:
(1) Number each object in the investigation team;
(2) Prepare the lottery tool and implement it;
(3) Measure or investigate each individual in the sample.
Please investigate the favorite sports activities of students in your school.
5. Random number table method: For example, use random number table to select 65,438+00 students from their class to participate in an activity.
2. 1.2
systematic sampling
1. systematic sampling (equidistant sampling or mechanical sampling): sort the units of the population, then calculate the sampling distance, and then sample according to this fixed sampling distance. The first sample was selected by simple random sampling. K (sampling distance) =N (population size) /n (sample size)
Prerequisite: For the variables studied, the arrangement of individuals in the group should be random, that is, there is no regular distribution related to the variables studied. You can start sampling from different samples and compare the characteristics of several samples under the conditions allowed by the investigation. If there are obvious differences, it shows that the distribution of samples in the population follows a certain cycle law, and this cycle coincides with the sampling distance.
2. Systematic sampling, namely equidistant sampling, is one of the most commonly used sampling methods in practice. Because it has low requirements for sampling frames and simple implementation. More importantly, if there are some auxiliary variables related to the survey indicators available, and the whole unit is queued according to the size of the auxiliary variables, systematic sampling can greatly improve the estimation accuracy.
2. 1.3
group sampling
1. stratified sampling (type sampling): first, all units in the crowd are divided into several types or levels according to certain characteristics or signs (gender, age, etc.). ), and then extract a sub-sample from each type or level by simple random sampling or systematic sampling. Finally, these sub-samples are combined to form a total sample.
Two methods:
(1). Firstly, the population is divided into several layers by stratification variables, and then extracted from each layer according to the proportion of each layer in the population.
(2) Firstly, the population is divided into several layers by stratification variables, and then the elements of each layer are arranged in order of stratification. Finally, samples are extracted by systematic sampling.
2. Stratified sampling is to divide people with strong heterogeneity into sub-populations with strong homogeneity, and then draw samples from different sub-populations to represent sub-populations, and all samples represent people again.
Stratification standard:
(1) Take the main variables or related variables to be analyzed and studied in the investigation as the standard of stratification.
(2) Ensure that the variables with strong homogeneity in each layer, strong interlayer heterogeneity and outstanding overall internal structure are used as stratified variables.
(3) Take those variables with obvious stratification as hierarchical variables.
3. The proportion of stratification:
(1) Proportional stratified sampling: a method to extract sub-samples according to the proportion of units of various types or levels to the total units.
(2) Non-proportional stratified sampling: If the proportion of some levels in the population is too small, the sample size will be very small. At this time, this method is mainly used to facilitate special research or comparison of different levels of subpopulations. If we want to infer the population from the sample data, we need to first weight the data of each layer, adjust the proportion of each layer in the sample, and restore the data to the actual proportion structure of each layer in the population. 2.2.2 Use the digital features of the sample to estimate the digital features of the population.
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