# Is zero odd or even?

Dear Straight Dope:

Hi! My name is Sandra Xavier and I teach math in an American school in Rio de Janeiro, Brazil. Last week some of our students came up with a question that led me and the high school math teacher to look for some additional information before answering them.

The question was: Is zero even or odd number?

In formal tests/quizzes usually zero is taken as an even number. But I read something years ago that said zero is neither even nor odd.

Rio, eh? Well, I guess we've made a pretty good start on eradicating ignorance in the U.S. It's time to start on the rest of the world.

Regarding your question, the bottom line is: zero is an even integer, except under a few highly specialized situations.

The definition of even integers is that they can be written in the form 2N where N is some integer. Another way of saying this is that an even integer has no remainder when divided by 2. An odd integer can be written in the form 2N + 1 where N is some integer. In other words, an odd integer leaves a remainder of 1 when divided by 2.

Under this definition, zero is clearly even, since 0/2 = 0 has no remainder. Zero also fits the nice pattern of alternating even/odd integers. Thus, almost every math book you can think of will include zero as an even number.

The exclusion of zero from the even numbers does occur under special circumstances. For instance, if you are defining even numbers to mean even NATURAL numbers, not even INTEGERS. The natural numbers are the set of counting numbers {1, 2, 3, . . .} which excludes zero. Of course, if an even number must be a natural number, then negative numbers are also excluded from being classified as even or odd.

There may be special other circumstances in which you want to deal with a set called "even" that excludes zero. For instance, Goldbach's conjecture (unproven, I think, to date) is that every even integer greater than 2 can be expressed as the sum of two primes. But that conjecture also excludes 2 from the selected set of "even integers" being considered.

On a tangential topic, 1 is usually excluded from the list of prime numbers. One seems to fit the definition of prime number ("having no divisors except itself and 1"); but to include 1 as a prime number would eliminate the unique factorization theorem, that every number can uniquely be expressed as a product of prime numbers. Hence, 1 is not considered to be a prime number.

Any number not prime is called composite, and 1 is also excluded from the composite numbers. So, in effect, you have 1 is neither prime nor not-prime. Maybe that's what stirred your memory when you were thinking of zero as neither even nor not-even.

Not-even, of course, is a mouse.

**NEVER GIVE A SUCKER AN EVEN BREAK**

Since we posted this, Dimitrius has brought to my attention that if you're at the roulette table, you bet "evens" and the li'l ball lands on 0 or 00, you lose. So that's a non-mathematical, real-life situation where zero is neither odd nor even.