A mathematical-social puzzle

[hudebnik]

Last week Governor Chris Christie imposed even/odd-day gas rationing in much of New Jersey, elaborating that "for purposes of this rule, 0 is treated as an even number." Today Mayor Bloomberg imposed even/odd-day gas rationing in New York City, saying among other things

Vehicles with license plates ending in an even number or the number “0” can make purchases of motor fuel on even numbered days.

Why is it necessary to single out 0 and "treat" it as an even number? Is there some planet on which 0 is not an even number? Is there some doubt in somebody's mind, somewhere, that 0 is an even number? Is the evenness of zero somehow culture-specific?

Comments

[freetrav.livejournal.com]

On a slightly different topic, why is 1 treated as special when discussing primality? A prime number is defined as having no integer factors other than itself and 1; 1 has no integer factors other than itself and 1, therefore, 1 is prime.

primes

[hudebnik.livejournal.com]

That always puzzled me too until I took a course in ring theory. The definition of "prime number" is a special case of the definition of "prime elements in a ring" (the integers being a very well-known example of a ring).

A unit is any element of a ring that has a reciprocal: in the integers, the only units are 1 and -1. An element p is prime if, whenever p divides rs, either p divides r or p divides s. But since units divide everything, it's sort of a cheap shot to count them as prime.

By contrast, an element p is irreducible if p cannot be written as the product of two non-units. In other words, if p = rs, then either r or s is a unit. Again, units satisfy this as a sort of "cheap shot" -- if p = rs is a unit, then both r and s are units -- and units are normally excluded from counting as irreducible. In the integers, primes and irreducibles are the same; in some other rings, they're not.

What about 0? It's clearly not irreducible, since it can always be written as a product of two non-units -- 0 and any non-unit you care to choose. OTOH, 0 would be prime, since the "whenever 0 divides rs" part never happens unless rs is itself 0, which (at least in an integral domain) implies that either r or s is zero. Except that the definitions of both "irreducible" and "prime" explicitly exclude both 0 and units.