# Inequality of reciprocal functions

Simplify[1/(1/a + 1/b)]


{ (a b)/(a + b) }

That doesn't look right. The given function has an infinite term for a = 0 and one for b = 0, while the result has one for (a + b) = 0. So that's probably the reason why:

a b/(a + b) === 1/(1/a + 1/b)


{ False }

though only a minute ago it Simplified it that way! Also

Simplify[1/(1/x)]


{ x }

seems to disregard the 1/0 case. Can anybody get me out of my confusion?

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a b/(a + b) === 1/(1/a + 1/b) returns False because they are not structurally the same (which is what SameQ[] checks for). As for your actual question, Mathematica returns what is called a "generically correct" answer; see this for details. – J. M. Jul 12 '12 at 12:47
You seem to be confused about the math itself a bit. The general notion you are missing is that 1/inf=0 , so 1/(1/0)=1/inf=0. There is absolutely no problem in simplifying 1/(1/x)=x. The simplification in the first case are also equivalent when you plug in zeros, for a=0, the first is 1/(inf+1/b)=1/inf=0, and the second is 0*b/(0+b)=0. So you are simply reasoning wrongfully about these "boundary cases". – jVincent Jul 12 '12 at 13:19
@jVincent - I know that, thanks. I was just wondering if MMA keeps track of every element in the equation to determine the domain of the function as a whole. – stevenvh Jul 12 '12 at 13:33

Use FullSimplify when testing equality:
x = 1/(1/a + 1/b)

It's circular in this case, but the conclusion is use Simplify when testing equality. – Chris Degnen Jul 12 '12 at 13:33