# If a language is undecidable, then its complementary language must also be undecidable?

Reference from here If a Language is Non-Recognizable then what about its complement? There exist complementary languages of unrecognizable languages that are recognizable, and there exist complementary languages of unrecognizable languages that are undecidable.

If a language is unrecognizable, then it must be undecidable.

If a language is undecidable, then its complementary language must also be undecidable?

If $$A(x)$$ is an algorithm deciding $$x\in L^c$$, then $$\lnot A(x)$$ is an algorithm deciding $$x\in L$$.
To be more pedantic, by $$\lnot A(x)$$ I mean the algorithm that executes $$A(x)$$ and if that accepts (resp. rejects) then it rejects (resp. accepts). In a programming language it's likely something like bool not_a(string x) { return not a(x); }.
Hence, if $$L^c$$ is decidable then $$L$$ is such.
Using a similar argument you can prove that $$L^c$$ is decidable if and only if $$L$$ is such.