How is the monetary part of bitcoin stored? How is the monetary aspect of a bitcoin stored? 
I understand that the blockchain is a set of records documenting transactions. I've also seen that blockchains in general can store arbitrary information (such as contracts, small notes, etc).

What I don't understand is how monetary values are recorded, exchanged, and divided.

Is a literal number value encrypted into the transaction? Or is it something more abstract?
Within the Bitcoin protocol, each Bitcoin is represented as 100000000 satoshis. Therefore, you can divide bitcoin up with a precision of 0.00000001.

In a transaction, the number of satoshis being delivered to each output is recorded, along with a record of which previous output is being spent.

The fee is then derived from sum(Inputs) - sum(Outputs). The actual representation of the amount itself is encoded using the Variable Length Integer representation