Integers

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 Enlightenment⁵.

In physics, integers or natural numbers are often represented by the letters $i,j,k,l,m,n$ , 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 $-1 * -1 = 1$ and $1 * -1 = -1$ . 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 $\pm 1$ ) 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.