Bitcoin is represented as a digital ledger. It does not "split" like you are thinking. It is divisible. One Bitcoin can be divided into 100 million smaller pieces called satoshis. Therefore, if I had 1BTC and sent someone half of it, I am sending them 50 million satoshis. Ultimately, it is still just digits on a digital ledger. There is nothing physical about it, so there is neither "splitting" nor "joining".