# 0, odd or even?

by repugno
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 P: 80 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?
 HW Helper Sci Advisor P: 9,395 0 is an even number. But that has nothing to do with writing cosh as a sum of odd and even *functions*.
 PF Patron Sci Advisor Thanks Emeritus P: 38,395 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, ex= cosh(x)+ sinh(x). cosh(x) and sinh(x) are the even and odd "parts" of ex.
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## 0, odd or even?

0 mod 2 = 0, it means that 0 is even
 P: 2,162 In general, given a function f, you can write it as the sum of an even function and an odd function as follows: $f_{even}(x) = (f(x) + f(-x))/2$ $f_{odd}(x) = (f(x) - f(-x))/2$
 HW Helper Sci Advisor P: 9,371 what about 6? is it odd or even?
 PF Patron Sci Advisor Thanks Emeritus P: 38,395 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.
 HW Helper Sci Advisor P: 9,371 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.