Integers are whole numbers including zero and negatives
An integer is 0, a positive natural number, or the negation of one—so … −2, −1, 0, 1, 2 …. They form a countably infinite set inside the rationals and reals, writable without a fractional part: 21 and −2048 qualify; 9.75 and √2 do not.
Latin integer meant whole, or more literally not touched, joining the negative prefix in to tangere, to touch; entire shares the root through French entier. Originally the term covered only positive whole numbers, making it a synonym for the naturals, and negatives joined as their usefulness became clear. Talk of the set of integers had to wait for Georg Cantor's set theory in the late 19th century. The letter Z, from German Zahlen, numbers, is attributed to David Hilbert, and its first known textbook use is Bourbaki's Algèbre of 1947, though rival letters such as J lingered into the early 1960s. American New Math teaching in the late 1950s redefined whole numbers to exclude negatives, and that phrase remains ambiguous today.
Sums, products and differences of integers are always integers, which is where they improve on the naturals, but division fails, since 1 divided by 2 is a fraction, and so does raising to negative powers. Under addition they form an abelian group that is also cyclic, generated by repeatedly adding 1 or minus 1, and every infinite cyclic group is essentially this one. Under multiplication they form only a commutative monoid, because 2 has no integer inverse.
As a ring, the integers are the most basic example: every ring receives exactly one homomorphism from them, and any ring of characteristic zero contains a copy. The rationals are the smallest field that contains them. Division with remainder does work: for any a and nonzero b there are unique q and r with a equal to q times b plus r and r between zero and the size of b, and repeating it gives the Euclidean algorithm for greatest common divisors. That makes the integers a principal ideal domain in which factorization into primes is essentially unique, the fundamental theorem of arithmetic.
Ordered from minus infinity to infinity with no bounds, they have zero as neither positive nor negative.
Source: Integer
