0, odd or even?


by repugno
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repugno
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#1
Apr19-07, 02:58 PM
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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?
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matt grime
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Apr19-07, 02:59 PM
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0 is an even number.


But that has nothing to do with writing cosh as a sum of odd and even *functions*.
HallsofIvy
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#3
Apr19-07, 03:15 PM
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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.

alastor
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Apr20-07, 09:58 AM
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Wink

0, odd or even?


0 mod 2 = 0, it means that 0 is even
Jimmy Snyder
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#5
Apr20-07, 11:25 AM
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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]
mathwonk
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Apr20-07, 10:28 PM
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what about 6? is it odd or even?
HallsofIvy
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#7
Apr21-07, 07:26 AM
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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.
mathwonk
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#8
Apr22-07, 10:03 PM
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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.


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