The set of nonnegative integers is a specific set, which refers to allNatural numberSet of, common symbolsNexpress.Non negative integers includepositive integerAnd zero, is aCountable set。
wholeNonnegative integerThe set of is usually called the set of nonnegative integers (or the set of natural numbers).The set of nonnegative integers contains natural numbers such as 0, 1, 2, and 3.Use bold capital letters in mathematics“N"Represents a set of nonnegative integers. Nonnegative integers includepositive integerAnd zero.The set of nonnegative integers is aCountable set。
nature
Announce
edit
1. In the set of nonnegative integers, there is a minimum natural number 0;stayNThe number set formed by the remaining natural numbers is called the positive integer set after removing zero from the+Or N*Means that 1 is at N+Is the smallest element;In N and N+Neither of them has the largest natural number;They are all infinite sets.
2. The natural number 1 is usually called the unit.
3. In N and N+The number obtained by adding the unit 1 to any number is called itsSuccessor numberFrom the smallest element, add 1 one by one. If you continue indefinitely, you can get all other elements in the number set. The smallest element is not the successor of any element.
4. 1 can divide any natural number, and its quotient is still the original natural number, so 1 is the divisor of any natural number.
5. 0 plus any natural number, its sum is still the original natural number, and 1 times any natural number, its product is still the original natural number, so natural numbers are multiples of 1.
6. 1 is neither a prime number nor a composite number.
7. If 0 has property P, then any subsequent number of natural numbers with property P has property P.
8. The numbers in the set of non negative integers can be counted one by one in order, so the set of natural numbers is countable.
9. Any two elements in the non negative integer set can be compared in size, so the natural number set is an ordered set.
10. In the set of nonnegative integers, addition and multiplication can always be implemented, that is, the sum and product of nonnegative integers are still nonnegative integers.
11. The addition and multiplication operations in the set of nonnegative integers meet the commutative law, associative law and the distributive law of multiplication to addition.
12. The addition and multiplication operations in the set of nonnegative integers satisfy the cancellation law.
13. Any non empty subset of the set of non negative integers must have a minimum non negative integer. This conclusion is calledLeast principle。[1]
Nonnegative integer
Announce
edit
Natural number is non negative (0 is not listed as a natural number in textbooks)/positive integer (1, 2, 3, 4...).It is considered that natural number does not includeFatal FrameOne of the reasons is that people start learning numbers from "one, two, three..." rather than "zero, one, two, three..." because it is very unnatural.
Natural numbers usually have two functions: they can be used to count (for example, "there are seven apples"). Seebase；It can also be used for sorting (for example, "this is the third largest city in China"). SeeOrdinal number。
The set composed of natural numbers is countable and has noupper boundThe infinite set of.Mathematicians usually express it as N.(N * represents natural numbers other than 0) There areadditionandmultiplicationOperation, the result of adding or multiplying two natural numbers is still a natural number.You can also subtract ordivision, but the results of subtraction and division are not always natural numbers, so subtraction and division operations are not always true in the set of natural numbers.
Natural number is known by peopleNumber systemThe most basic category of.In order to make the number system have a strict logical basis, mathematicians in the 19th century established two theories on natural numbers: the ordinal number theory and the cardinal number theory, so that the concept, operation and related properties of natural numbers are strictly discussed.The addition and multiplication of natural numbers can be performed inOrdinal numberOr cardinal number theory, and the operations under the two theories are consistent.
In the global scope, the debate about whether 0 is a natural number still exists.In mainland China, the textbooks for primary and secondary schools before 2000 generally did not include 0 in natural numbers, or called it an "expanded natural number series".In the new edition of textbooks for primary and secondary schools after about 2000, 0 was generally included in natural numbers.
Equipotential set
Announce
edit
The equipotential sets of and nonnegative integer sets are:
1. A set consisting of finite sequences of natural numbers
2. Integer set
3. Rational number set
4. Algebraic number set
5. Union of countable sets
The potential of nonnegative integer set is strictly less thanSet of real numbersPotential, that is, one-to-one correspondence cannot be established between the two (seeCantor's diagonal argument ）。In fact, the potential of the set of real numbers is 2N0, that is, thePower setPotential.
