In my lecture, the lecturer said:

Let $K$ be the key generation algorithm. Given a security parameter represented in unary, $1^k$, $K(1^k)$ will output a keypair $(pk; sk)$, known as the public key and the private (or secret) key, respectively.

My question is: When I want to implement the cryptosystem, how should I represent this $1^k$? When I implement the key generation procedure, do I literally require callers to pass me a string of $k$ one-bits that the procedure will never use? And, if so, why is it there in the first place?