To achieve closure in addition, subtraction, and multiplication one
introduces negative whole numbers and zero to construct the set of *integers*. Today we take these things for granted, but in fact the idea
of negative numbers in particular is quite recent. Although they were
*used* earlier, mathematicians only accepted the idea that negative
numbers were legitimate numbers by the latter 19th century! After all,
if you are counting cows, how can you add negative cows to an already
empty field? Numbers were thought of as being concrete properties of
*things*, tools for bookkeeping, rather than strictly abstract
entities about which one could axiomatically reason until well into the
Enlightenment^{5}.

In physics, integers or natural numbers are often
represented by the letters
, although of course in algebra
one *does* have a range of choice in letters used, and some of these
symbols are ``overloaded'' (used for more than one thing) in different
formulas.

Integers can in general also be factored into primes, but problems begin to emerge when one does this. First, negative integers will always carry a factor of -1 times the prime factorization of its absolute value. But the introduction of a form of ``1'' into the factorization means that one has to deal with the fact that and . This possibility of permuting negative factors through all of the positive and negative halves of the integers has to be generally ignored because there is a complete symmetry between the positive and negative half-number line; one simply appends a single -1 to the prime factorization to serve as a reminder of the sign. Second, 0 times anything is 0, so it (and the number ) are generally excluded from the factorization process.

Integer arithmetic is associative, commutative, is closed under addition/subtraction and multiplication, and has lots of nice properties you can learn about on e.g. Wikipedia. However, it is still not closed under division! If one divides two integers, one gets a number that is not, in general, an integer!

This forming of the *ratio* between two integer or natural number
quantities leads to the next logical extension of our system of numbers:
The rationals.