Extended data:

I. Z as the representation of integer set

Definition of 1 and z

The integer set z is the set of all integers, including positive integers, negative integers and zero.

2, the nature of z

The integer set z is infinite, that is, it contains an infinite number of elements. At the same time, the integer set is countable, which means that there is a mapping from natural number set n to integer set Z.

3. the structure of z

Integer set z has some important subsets, such as positive integer set N*, negative integer set Z- and zero set 0. These subsets have special positions in the integer set and form the hierarchical structure of the integer set.

Second, the application of Z in the field of mathematics

1. In group theory, integer set z is often used as the basis of groups, such as defining the group structure of modular operations.

2. In algebra, Z is the basis of algebra, and all integers can be regarded as an element in Z. At the same time, improving Z is also the basis of many other mathematical objects, such as rational numbers and polynomials.

3.z plays an important role in number theory. For example, it can represent odd and even numbers, and it can also represent prime numbers and composite numbers. In addition, Z is also widely used to study some properties of integers, such as the unique decomposition theorem of integers.

4. In geometry, z is often used to represent the coordinates of points. For example, on a two-dimensional plane, a point can be represented by its x and y coordinates, both of which are integers.

Comparison between integer set and three large number sets

1, natural number set n

Natural numbers include all positive integers, excluding zero. Unlike natural numbers, integers include zero and negative numbers. Therefore, natural numbers can be regarded as a subset of integers.

2, rational number set q

Rational number noise scale is a rational number that can take any value between two integers, so rational numbers can be regarded as a subset of integers. But some fractions (such as irrational numbers) can't be expressed in finite decimal system. These numbers belong to the set r of real numbers.

3. Real number set r

Real numbers are rational or irrational numbers that can take any value between two rational numbers. Real numbers can be regarded as a subset of rational numbers. However, Liang La, the range of real numbers is wider than rational numbers, including those numbers that cannot be expressed in finite decimal system.