Natural numbers use n

R stands for real number

Positive integers are N+ or N*

The negative integer is N-

Q of rational number

There are many definitions of 0, and here are just the most common ones. (Many properties of 0 are listed upstairs, but they are generally undefined. )

First, the definition of natural number 0 and its generalization.

1. According to Peano's axiomatic system of natural numbers, 0 is the first number in natural numbers. Piano axiom 1 Yes: 0 belongs to natural number set.

2. natural number set's definition can also take 1 as the first natural number, so the axiom 1 becomes: 1 belongs to natural number set. At this time, 0 does not belong to natural number set. Therefore, 0 is an extension of a natural number. You can define "extended natural number set", that is, 0 is the difference between any two equal natural numbers (subtraction is defined first, of course), or you can use the general definition of 0 in later algebra to merge 0 into this extended natural number set.

3. Integer, rational number, real number and complex number all come from 0 in the set of natural numbers. In the extension theory of number sets, smaller number sets are embedded into larger number sets in the form of ordered pairs or equivalent sequence classes of larger number sets. For example, the equivalence class of any two ordered pairs with the same natural numbers is defined as an integer (meaning the difference between these two natural numbers), and the equivalence class of an ordered pair composed of two identical natural numbers is 0.

4. In piano's axiom, only natural numbers are defined abstractly. Natural numbers in set theory can also be constructed by construction. So the natural number 0 is equivalent to an empty set, and 1 is {empty set}, 2 is {empty set, {empty set}}, and so on.

Second, 0 in general algebraic theory.

In the general algebraic structure, if the addition operation is defined (general addition is interchangeable), then the definition of 0 is the element that satisfies the requirement that any element in the set can still get the property of the element (that is, the property of x+0=x) by adding it. If there is a 0 element in any field, the 0 in the real number field can also be defined in this way.

If an algebraic structure does not define addition, but only multiplication, then sometimes it can be said that any element in the set can still get 0 by multiplying it (that is, 0*x=0 or x*0=0). Because there is no commutative law in multiplication here, there are "left 0 yuan" and "right 0 yuan". If the n-order square matrix on the number field k forms a group in terms of multiplication, it can be said that it has left and right 0 yuan.

By the way, 0 is another symbol in Boolean algebra, which follows the law of logical operation.

Attachment: piano's axiom of natural numbers (that is, the axiomatic definition of natural numbers)

PA 1: Zero is a natural number.

PA2: Every natural number has a successor (also a natural number).

PA3: Zero is not the successor of any natural number.

PA4: Different natural numbers have different successors.

PA5: (inductive axiom) Let the set composed of natural numbers contain zero, and whenever the set contains a natural number, it also contains its successors, then the set must contain all natural numbers.