### A mathematical-social puzzle

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?

rufiniahudebnikrufiniaAlso, I will say that I have three post-secondary degrees. I cannot explain to you if 0 is even or odd. I would look at the scheme and go "well, 0 goes on the even days, THAT MAKES SENSE" but the "why" is not something I udnerstand.

hudebnikThe even numbers are defined as the numbers which are 2 times an integer. 0 is an integer, so 2 times 0 is an even number.

Alternatively... the integers are partitioned into even numbers and odd numbers in alternation. If you start with an even number, the integers before and after it are odd; if you start with an odd number, the integers before and after it are even. 2 is even (I hope there's no dispute about that!), so 1 is odd, so 0 is even.

thatpotteryguy(Deleted comment)cvirtueshalmesteregalingalehudebnikgalingaleunclrashidhudebnikBut once you've accepted that 0 is a number, there's no conceivable justification for saying it's not even.

freetravhudebnikprimesA 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

bothr 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.

Edited at 2012-11-09 11:49 pm (UTC)minstrlmummr