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".
If Bitcoin didn't have fractions or sats I don't think that it's adoption will be as fast as it's going now, a lot of people won't afford to hold 1 Bitcoin and it will surely affect its value because demand will be very low. Almost everybody can hold some fractions of Bitcoin in Satoshi, the amount that you hold determines the value in your possession. The fractions of your Bitcoin remains what it is in the Blockchain, what changes is the value, it can either dip or rally. Many newbies will not understand this, even when they buy in sats they will panic if they see the value decreasing, proper knowledge is important.