Is money a conserved quantity? Excuse the naïve question, but I have a rather limited understanding of economics, and this question has been bothering me since my school economics teacher couldn't answer it satisfactorily many years ago.  It is also a question that is a prerequisite to many others I have.

Is money a conserved quantity, in the same sense as we have conserved quantities in physics (e.g. energy)?

In daily transactions that I have experience with, it obviously is. If I buy something from Joe, and give him \$10, I will have \$10 less and he will have \$10 more.  But the total amount we have together has not changed.

Does this apply to all transactions?  If not, when is it violated?  In what situations is money created, and when does it disappear (other than physically destroying paper based money)?

This then leads to a closely related question: Where can I find out how much is there of a given currency?  How much USD exists in total, if we combine the dollars owned by everyone in the world?  

If this question is considered too basic, I would appreciate suggestion about where a mathematically minded person might read up on this topic.
Νominal money could be seen as a "conserved quantity" in a similar/analogous sense that we have in physics, under some qualifications and restrictions. 

I stress that we are talking about the nominal value of money, which is an indisputable quantity and legal tender (the concept of "real" value is an estimated value and hence subject to disagreements etc.)

Then if we exclude
1) The central bank/government with the authority of the law
2) The commercial banking system as regards Transactions that have to do with taking or repaying loans
3) Actions that just destroy the physical carrier (say, burn the paper bill)

...then transactions do not alter the amount of nominal money, and in that sense, the nominal value/amount of money is conserved.

We exclude 1) because the central bank / government have the power by law to create new money (and destroy existing money, for example "cutting zeros" after a hyperinflation episode).  

We exclude 2) because commercial banks create money under the "fractional reserve banking" system, while money is destroyed whenever a loan is repaid (the part that has to do with the principal, not the interest). The OP should ask a different question if the OP wants to know about that.

We exclude 3) because we can argue that these are not "normal economic transactions".

So the system in which nominal money can be considered a "conserved quantity" is a subset of a real world economic system.