Is 0 an odd or even number? The reason why I ask is this: I need to write cosh(x) as the sum of an even and odd function. I could only come up with cosh(x) = cosh(x) + 0, where cosh(x) would be the even and 0odd. However, this doesn't make any sense since 0 is exactly divisible by 2 with no remainder, hence it is even. So which one is it?
0 is an even number. But that has nothing to do with writing cosh as a sum of odd and even *functions*.
The definition of an "even" function is that f(-x)= f(x). The definition of "odd" function is f(-x)= -f(x). If f(x)= 0 for all x then f(-x)= 0= -0= -f(x) but also f(-x)= 0= f(x) so f(x)= 0, the constant function, is both even and odd. However, as matt grime said, that has nothing to do with the fact that 0 = 2(0) is an even number. cosh(x) is already an even function. sinh(x) is an odd function. In fact, e^{x}= cosh(x)+ sinh(x). cosh(x) and sinh(x) are the even and odd "parts" of e^{x}.
In general, given a function f, you can write it as the sum of an even function and an odd function as follows: [itex]f_{even}(x) = (f(x) + f(-x))/2[/itex] [itex]f_{odd}(x) = (f(x) - f(-x))/2[/itex]
Okay, I'll bite: even? Actually, a more interesting question would be whether 5 is odd or even. The number 5 is obviously odd. The constant function (which is what this thread is really about), f(x)= 5, is even.
good point, so the answer to the OPs question is "yes". i.e. all constant functions are even and one of them is also odd.