A number (more specifically, an integer) is odd if it is not congruent to zero, modulo 2. In other words, when you divide the number by two, you will get some-number-and-a-half (not a whole number), or a number x is odd if (and only if) it can be written as x = 2n + 1 for some integer n.

If x and y are odd numbers, and z is an even number, then:

  • x + y is even
  • x * y is odd
  • x + z is odd
  • x * z is even

A function f is odd if for every value x in its domain, -x is also in the domain and f(-x) = -f(x). A graph of an odd function is symmetrical about the origin.

If f and g are odd functions, and h is an even function, then:

  • nf(x) is odd for any real number n
  • f(x) + g(x) is odd
  • f(x) * g(x) is even
  • f(x) * h(x) is odd
  • f(g(x)) is odd
  • f(h(x)) is even
  • h(f(x)) is even
  • If the domains of f and h overlap, then f(x) + h(x) is not even, but it is not odd either unless h(x) = 0

A polynomial of x is odd if the power of x in every term is an odd number. (Note that (-x)2n + 1 = (-x)*(-x)2n = (-1)*(x)*(x)2n = -((x)2n + 1); every term in an odd polynomial is an odd function.)

Odd polynomials are odd functions, and continuous odd functions can be approximated to any desired accuracy by odd polynomials. However, there are functions which are neither even nor odd, and there are polynomials which are neither even nor odd. This is different than for integers; every integer is either even or odd. (In fact, the function f(x) = 0 is both odd and even, thanks to ariels for setting this straight for me.)

BrianShader reminds me: You might like to point out that any function can be written as an even function plus an odd function.