Elementary number theory is a compulsory course for elementary education majors, especially for science students, and it is also a refresher course for teachers engaged in elementary mathematics teaching. Including divisibility, congruence, congruence equation, indefinite equation, indefinite equation, simple continued fraction of integer and so on. These contents not only conform to the teaching thinking of primary school mathematics teachers, but also help learners accumulate the necessary ability and knowledge to engage in primary school mathematics education.
Some people say: "Mathematics is the gymnastics of thinking, the crown of science, and number theory is the jewel in the crown." This pearl has long been shining in primary school mathematics-the number theory we learned in primary school mainly includes the following categories:
The problem of divisibility: (1) the nature of divisibility; (2) The divisibility of numbers (the content of junior high school entrance examination) Remainder problem: (1) Application of divisor with remainder = divisor × quotient+remainder. (The remainder is always less than the divisor) (2) The nature and application of congruence.
Parity problem: (1) parity and addition and subtraction; (2) Parity, multiplication and division of prime numbers: the key point is the decomposition of prime factors.
Factor multiple: (1) The greatest common divisor and the least common multiple theorems (2) The law of determining the number of factors.
It can be seen that the application of elementary number theory is closely related to elementary mathematics education. For elementary number theory, all I have learned is nine Niu Yi hairs, not to mention any constructive questions. I can only talk about the core content of elementary number theory-congruence, and explain the relationship between them through its application in elementary mathematics.
Congruence was first put forward by German mathematician Gauss and systematically studied. It is the core part of elementary number theory. It contains many unique ideas, concepts and methods of number theory, and its appearance makes number theory a symbol of an independent branch of mathematics. This content includes its properties, residue classes and residue systems, Euler.
Theorem and cyclic decimals. Before learning elementary number theory, we were not familiar with the concept of congruence. In fact, congruence has been deeply used in our primary school mathematics study and Olympic Mathematics. In primary school, it is mainly reflected in the use of remainder, which is an important concept in primary school mathematics and a hot topic in mathematics competitions, among which there are many related concepts and strong methods.
In elementary school, we knew that if an integer A is divided by a positive integer M, the quotient is Q and the remainder is R, then a=qm+r, where both Q and R are natural numbers, and 0 ≤ R < M, now we have learned the knowledge of congruence. If two positive integers A and B are divided by a non-zero natural number M, the remainder is the same, and A = QM+R, and B = PM.
Below, I use an example to show the application of congruence in primary school mathematics teaching:
Example: A divided by 5 equals 1, and B divided by 5 equals 4. If 3A > B, what is 3A-B divided by 5? This problem appears in primary school Olympic mathematics, and the general solution of primary school students is:
Method 1: Make up the numbers. Let A be 6 and B be 9, so that A and B satisfy the conditions that A is divided by 5 1, B is divided by 5 by 4, 3a-B = 9, and the remainder of 9/5 is 4.
Method 2: let a = 5x+1b = 5y+43a-b =15x-5y-15x-5y-5+4 = 5 (3x-y-1. 5 = (15x+3-5y-4)/5 = 3x-y-1/5 = (3x-y-1)+4/5 According to x, y is a positive integer, 3a >；; So the remainder is 4. And the solution in elementary number theory: solution: ∵a≡ 1(mod5), ∴3a≡3(mod 5), or 3a ≡ 8 (mod 5). (1) and ∵ b ≡.
Therefore, 3 A-B divided by 5 equals 4.
We can see that the two methods, especially the second one, are based on congruence knowledge to deal with problems, but the formal expression is simpler than the elementary number theory practice in universities. In the training of Olympic mathematics thinking in primary schools, there are countless applications of congruence thought, such as "pigeon hole principle".
The most typical example of congruence application can be said that congruence theory is a very important mathematical model in modern algebra. In addition, many other mathematical knowledge involves congruence, such as Euler function, which is also one of the important functions in elementary number theory and embodies the idea of congruence in the process of proof.
Anyone who has studied elementary number theory should know that the biggest difference between elementary mathematics and elementary number theory lies in how elementary mathematics applies theorems and laws, and elementary number theory should understand why it is applied like this. Obviously, elementary number theory is a deeper knowledge, which has made a leap in difficulty. So what position does number theory occupy in primary school mathematics test questions? It can be said that when you open any math tutorial book, the problems of number theory occupy a prominent position. Some experts found in all kinds of primary school mathematics competitions that the score of directly using number theory knowledge to solve problems accounts for about 30% of the total score of the whole test paper, and in the final test questions of the competition, this score ratio is higher. Teachers like to use math questions as the basis to distinguish top students from ordinary students, and the quality of this part of learning will directly determine the performance of students in the selection of exams.
To sum up, elementary number theory, as a course for students majoring in elementary education, not only cultivates students' solid mathematical foundation, but also helps normal students to apply the theory of elementary number theory to elementary education flexibly and further cultivate scientific outlook on life and values.