One of the numbers 1, 2, 3, ... (OEIS A000027), also called the counting numbers or natural numbers. 0 is sometimes included in the list of "whole" numbers (Bourbaki 1968, Halmos 1974), but there seems to be no general agreement. Some authors also interpret "whole number" to mean "a number having fractional part of zero," making the whole numbers equivalent to the integers.Due to lack of standard terminology, the following terms are recommended in preference to "counting number," "natural number," and "whole number."setnamesymbol..., , , 0, 1, 2, ...integersZ1, 2, 3, 4, ...positive integersZ-+0, 1, 2, 3, 4, ...nonnegative integersZ-*0, , , , , ...nonpositive integers, , , , ...negative integersZ--
The positive integers are the numbers 1, 2, 3, ... (OEIS A000027), sometimes called the counting numbers or natural numbers, denoted .They are the solution to the simple linearrecurrence equationwith .A plot of the first few positive integers represented as a sequence of binary bits is shown above. The top portion shows to , and the bottom shows the next 510 values.
An odd number is an integer of the form , where is an integer. The odd numbers are therefore ..., , , 1, 3, 5, 7, ... (OEIS A005408), which are also the gnomonic numbers. Integers which are not odd are called even.Odd numbers leave a remainder of 1 when divided by two, i.e., the congruence holds for odd . The oddness of a number is called its parity, so an odd number has parity 1, while an even number has parity 0.The generating function for the odd numbersisThe product of an even number and an odd number isalways even, as can be seen by writingwhich is divisible by 2 and hence is even.
The set of natural numbers (the positive integers Z-+ 1, 2, 3, ...; OEIS A000027), denoted , also called the whole numbers. Like whole numbers, there is no general agreement on whether 0 should be included in the list of natural numbers.Due to lack of standard terminology, the following terms are recommended in preference to "counting number," "natural number," and "whole number."setnamesymbol..., , , 0, 1, 2, ...integersZ1, 2, 3, 4, ...positive integersZ-+0, 1, 2, 3, 4, ...nonnegative integersZ-*0, , , , , ...nonpositive integers, , , , ...negative integersZ--
One of the numbers ..., , , 0, 1, 2, .... The set of integers forms a ring that is denoted . A given integer may be negative (), nonnegative (), zero (), or positive (). The set of integers is, not surprisingly, called Integers in the Wolfram Language, and a number can be tested to see if it is a member of the integers using the command Element[x, Integers]. The command IntegerQ[x] returns True if has function head Integer in the Wolfram Language.Numbers that are integers are sometimes described as "integral" (instead of integer-valued), but this practice may lead to unnecessary confusions with the integrals of integral calculus.The ring of integers has cardinal number of aleph0. The generating function for the nonnegative integers isThere are several symbols used to perform operations having to do with conversion between real numbers and integers. The symbol ("floor ") means "the largest integer not greater than ,"..
An integer is said to be highly cototient if the equationhas more solutions than the equations for all , where is the totient function.The first few highly cototient numbers are 2, 4, 8, 23, 35, 47, 59, 63, 83, 89, ...(OEIS A100827).The first few prime highly cototient numbers are 2, 23, 47, 59, 83, 89, 113, 167,269, 389, 419, 509, ... (OEIS A105440).
An even number is an integer of the form , where is an integer. The even numbers are therefore ..., , , 0, 2, 4, 6, 8, 10, ... (OEIS A005843). Since the even numbers are integrally divisible by two, the congruence holds for even . An even number for which also holds is called a singly even number, while an even number for which is called a doubly even number. An integer which is not even is called an odd number.The oddness of a number is called its parity, so an odd number has parity 1, while an even number has parity 0.The generating function of the even numbersisThe product of an even number and an odd number isalways even, as can be seen by writingwhich is divisible by 2 and hence is even.
Consecutive numbers (or more properly, consecutive integers) are integers and such that , i.e., follows immediately after .Given two consecutive numbers, one must be even and one must be odd. Since the product of an even number and an odd number is always even, the product of two consecutive numbers (and, in fact, of any number of consecutive numbers) is always even.
The doublestruck capital letter Z, , denotes the ring of integers ..., , , 0, 1, 2, .... The symbol derives from the German word Zahl, meaning "number" (Dummit and Foote 1998, p. 1), and first appeared in Bourbaki's Algèbre (reprinted as Bourbaki 1998, p. 671). The ring of integers is sometimes also denoted using the double-struck capital I, .
The ring of integers is the set of integers ..., , , 0, 1, 2, ..., which form a ring. This ring is commonly denoted (doublestruck Z), or sometimes (doublestruck I).More generally, let be a number field. Then the ring of integers of , denoted , is the set of algebraic integers in , which is a ring of dimension over , where is the extension degree of over . is also sometimes called the maximal order of .The Gaussian integers is the ring of integers of , and the Eisenstein integers is the ring of integers of , where is a primitive cube root of unity.