According to the Federal Reserve Bank of Chicago, this is how fractional-reserve banking originated:
Then, bankers discovered that they could make loans merely by giving their promises to pay, or bank notes, to borrowers. In this way, banks began to create money.
There was also the need, however -- as there always is -- of keeping, at any given time, enough money to provide for expected withdrawals: "Enough metallic money had to be kept on hand, of course, to redeem whatever volume of notes was presented for payment."
Hence the name "fractional-reserve banking": commercial banks must hold a fraction of all deposit money as reserves -- which legally (since 1971) are no longer valuable as gold but only as a public debt -- to provide for expected withdrawals: "Under current regulations, the reserve requirement against most transaction accounts is 10 percent."
In the fractional-reserve banking system, on which most of today's international monetary system relies, commercial banks create money by loaning it, hence as a private debt.
Transaction deposits are the modern counterpart of bank notes. It was a small step from printing notes to making book entries crediting deposits of borrowers, which the borrowers in turn could "spend" by writing checks, thereby "printing" their own money.
For example, if a commercial bank receives a new deposit of U$ 10,000.00, then 10% of this new deposit becomes the bank's reserves for loaning up to U$ 9,000.00 (the 90% in excess of reserves), with interest. Likewise, if a loan of that maximum fraction of U$ 9,000.00 does occur and the borrower also deposits it into a bank -- regardless of whether in the same bank or not -- then again 10% of it becomes the latter bank's reserves for loaning now up to U$ 8,100.00 (the 90% now in excess reserves), always with interest. This could proceed indefinitely, adding U$ 90,000.00 to the money supply, valuable only as their borrowers' resulting debt: after endless loans of recursively smaller 90% fractions from the original deposit of U$ 10,000.00, that same deposit would have eventually become the 10% reserves for itself as a total of U$ 100,000.00.
Thus through stage after stage of expansion, "money" can grow to a total of 10 times the new reserves supplied to the banking system, as the new deposits created by loans at each stage are added to those created at all earlier stages and those supplied by the initial reserve-creating action.
Now let us further examine what is happening here. First, we have a deposit. Then, we have a loan of up to a fraction (of 90%) of this deposit. Finally, the borrower can deposit the borrowed money into another bank account, in the same bank or not. Suddenly, the trillion dollar question emerges: is the borrowed money in these two bank accounts the same?
On the one hand, the answer is yes: all borrowed money came from the original deposit -- so it is that same original money.
On the other hand, the answer is no: all money deposited into the borrower's account possibly stays in the original depositor's account -- so it is not that same original money.
How can that be?
Let us consider gold instead of bank accounts. Gold at once is and represents money. It is money by being its own social equivalence to all commodities, and so the generic exchange value in their price. It represents money by being the object in which all commodities must be priced, whether valuable or worthless in itself (independently of being money). Whatever we choose for the representation of money -- whether valuable in itself or not -- it must be socially identical to all commodities in their exchange value, or in their equivalence to it and to each other in it. This general equivalence is monetary identity, which is purely abstract. Yet money must also be an object, like gold, possibly a commodity with its own exchange value, again like gold: the object in which to price all commodities. This object is a monetary representation, which is not only concrete (like gold), but also replaceable -- say, as gold by silver.
So let us go back to fractional-reserve banking. Now, by conceptually distinguishing money from its representation, we can clearly see what is happening in that ambiguous loaning from bank deposits: commercial banks are mistaking bank accounts for the money they represent. This way, when they deposit a loan from any account into any other, they must mistake the same loan for both accounts, hence duplicating its money, rather than subtracting it from the source account. That confusion between monetary identity (deposit money) and its representation (bank accounts) is thus what alone replicates loaned money: two deposits in different accounts must always be different money, even if one is just a loan of money from the other.
The same confusion affects a variety of monetary representations, like paper notes and metal coins. Even when sheer gold represents money, there is no inherent distinction between monetary identity and its representation. Any such inherent indistinction (confusion) is precisely what I call representational monetary identity.
With no representational identity of money, not a single fraction of bank-account balances could belong to both its depositors and their borrowers. As account money, deposits from loans are new money. However, as deposit money, they are just fractions of other account balances. Hence banks lacking up to 90% of all money their clients can withdrawal: bank loans are just bank-account money that vanishes once repaid.
Additionally, because all money created by commercial banks is just a sum of balance fractions borrowed from client accounts, that money must be worth only as credit, or as the corresponding debt principal. This way, except for money not yet in loans nor else reserved -- whether in bank accounts (excess reserves) or not -- but not from loans, bank loans are the whole money supply left for paying their own interest. Consequently, such an interest-paying, self-indebted money supply must grow at least at its own interest rate less any other money off the banks' reserves.
Then, who should create the additional money? Supposedly, governments would do it. Yet historically, central banks have been issuing most of this money in exchange for promises from their governments of paying it back with interest, just like commercial banks replicate it in exchange for promises from their clients of paying it back with interest. So paying the additional interest (that on public money-as-debt) requires even more money: central banks must create -- and are creating -- ever new public money-as-debt for paying interest on both private and old public money-as-debt, thus recursively amplifying the problem.
In both this exposition and the world, we can already see the disastrous consequences of such a monetary system, with its limitless, exponential growth of the money supply as a debt -- first private, then public. We have a problem: debt becoming money. What is the solution? The answer comes from understanding the problem: since to create irrational, self-multiplying money we must confuse monetary identity with its representation, the solution is to disentangle them.
By which not even gold money, for having as much a representational identity as that of bank accounts, is immune to its own self-indebtedness. Indeed, it was by creating proxy representations of monetary gold that fractional-reserve banking originally flourished. The reason is that -- as we will see -- with any monetary proxies of gold, its representational monetary identity must become a debt.
Hence the advent of central banking: because monetary gold proxies are already a debt, all additional such money, even if public, must be borrowed. So any public-debt-free, government-issued monetary proxies of gold, for not solving the money-as-private-debt problem, could only postpone the money-as-public-debt one.
Still, if the only solution to the whole (both public and private) money-as-debt problem is an inherently distinct monetary identity, then how to implement it?
Fortunately, an already existing monetary system inherently distinguishes monetary identity from its representation: the Bitcoin monetary system. It uses public-key cryptography (the same technology of private Internet connections) to implement monetary identity as a private key and its representation as the corresponding public key, so this representation becomes inherently distinct from its represented money. The whole Bitcoin system relies on that distinction: as an essentially decentralized monetary system, it controls the money supply by self-certifying a public chain of monetary transactions, which contains money representations (public keys) alone, and never the money (private keys) they represent. This way, monetary identity remains nonrepresentational, private, possibly anonymous (pseudonymous), and impossible to replicate.
Yet in case Bitcoin eventually fails, any other solutions, not only to the money-as-public-debt problem, but also to the underlying money-as-private-debt one, must also consist in distinguishing monetary identity from its representation.
See Modern Money Mechanics.