First, elementary number theory.
1. 1 Introduction
1. 1. 1 What is number theory? Gauss said that mathematics is the queen of science and number theory is the queen of mathematics.
As a direction in mathematics, number theory is the most popular. God created integers, and people came to study them, so there was number theory. It is basically a simple question of 1+ 1 = 2, but no one has given a proof yet.
What are the characteristics of integers? In primary school, we all learned the concepts of parity, divisibility, composite number and prime number. Based on these basic concepts, it is the content of number theory to analyze some internal relations of integers. For example, the most primitive Goldbach conjecture: all odd numbers greater than 7 are the sum of three odd prime numbers; All even numbers greater than 4 are the sum of two odd prime numbers. For example: 9 = 3+3+3, 8 = 3+5, 2 1 = 7+7,18 = 5+13 = 7+1... At present, Chen Jingrun, a famous mathematician in China, is here. -China is never short of masters. What is lacking is an environment where a master can make a difference.
The application of 1. 1.2 number theory
From physics to art (music), we can see the shadow of number theory.
1. 1.3 algebra preparation knowledge
A set of numbers represents:
(1) natural number set N = {1, 2,3, ...}
(2) Integer set z = {...,-3,-2,-1,0,1,2, 3, ...}
Non-negative integer set z >；; =0 = {0, 1,2,3,...}
Positive integer set Z+ = {1, 2,3, ...} = N
Set of positive integers greater than 1 z >; 1 = {2,3,4,...}
(3) Residual class Z/nZ
Quote the definition of another book:
Let n be a given positive integer, and Cr(r=0, 1, 2, ..., n- 1) represents the set of all integers in the form of qn+r, where q is an integer, then Cr(r from 0 to n- 1) is called the residue class of module n.
Example: If n is 10, the remaining classes of modulus 10 are {0, 1, 2,3,4,5,6,7,8,9}.
(4) rational number set q
Q = {a/b: a, b∈Z and b≠0}
(5) Real number set r
/rational number q
R/ algebraic operations, such as square root operations.
\ irrational number
\ Transcendental, such as π, e, etc.
(6) complex set c
C = {a+bi: a, b∈R, I is the root of-1