Does that mean an infinite amount of numbers are added to the end of the coin? I doubt this to be true but why isn't it?
Coins are not directly represented in the system - instead what is stored is the chain of transactions beginning from newly mined coins up until the present, so yes in a sense this transaction record (blockchain) goes on indefinitely, and from this your wallet balance is determined. Currently the blockchain is about 4.5 GB and growing. You can see a lot of statistics at
blockchain.info.