Let us build the first number instead of announcing it. Take n goods. For a given good — say bread — there are n − 1 other goods it can be exchanged against: that many rates to know. Now do it for EACH of the n goods: that gives n times (n − 1) rates, i.e. the product n × (n − 1). There is where the multiplication comes from. But each exchange has been counted twice: “one loaf for two fish” and “two fish for one loaf” are the same rate, stated both ways round. So we divide by 2. With 5 goods: 5 × 4 = 20 ordered pairs, hence 20 / 2 = 10 genuinely distinct rates. Check it by writing them out, they are all there.

Back to the double coincidence, the one the poll made you feel. You produce bread and want fish. A person met at random produces fish with a one in n chance; and given that they produce fish, they want your bread with a one in n − 1 chance, since they desire one of the n − 1 goods other than their own. So the probability that both conditions fall together is 1 / [n × (n − 1)]. And from there to the number of meetings the step is short: if one attempt in p succeeds, it takes p on average to succeed once — as with rolling a 6 on a die, where a one in 6 chance is paid for with 6 throws on average. So it takes about n × (n − 1) meetings to bring off one exchange. Look carefully: it is the same product as for the prices, simply not divided by two. A single count governs both obstacles — the prices are half of it, the meetings the whole.

Now let us bring in a common unit, outside the n goods. On the rates side: each good gets a price in that unit, i.e. n prices instead of n(n − 1)/2 — and every ratio between goods follows by a division. On the meetings side: you no longer need two that coincide, but two separate moments. You sell your bread to whoever wants it, without caring what they produce; you buy fish from whoever has it, without caring what they desire. The coincidence problem is not solved, it is CUT IN TWO — and each half solves itself. Let us put a figure on it, because it is not free either: a person at random wants your bread with a one in n chance, so it takes about n meetings to sell, then as many again to find the fish, i.e. 2n in all. Compare that with n(n − 1): the ratio is n(n − 1) / 2n, that is (n − 1)/2 — exactly the same ratio as for the rates. A single number measures both gains. Click each term:

An attentive reader will object: those 4,950 rates are not independent. If I know a loaf is worth two fish and a fish is worth three apples, I deduce that a loaf is worth six apples — I did not need the third quotation. Strictly speaking, n − 1 well-chosen rates are enough to determine all the others, i.e. 99 for 100 goods. The objection is right, and here is the answer: to DEDUCE a rate through that chain of multiplications, you must already have chosen a reference good to relate everything to — a coherent system of references. At a pinch, a simple chain would do — bread to fish, fish to apples, and so on — with no good at the centre. But then each conversion means going back up the chain, and the further apart the goods, the more the multiplications pile up. The most economical solution is to relate everything to ONE single good, which is called a NUMERAIRE: a single multiplication then suffices between any two goods. That is an efficiency argument, not a logical necessity — but it is the only solution actually observed, and it consists precisely in inventing what money does. The 4,950 count the quotations a decentralised market must DISPLAY and keep up to date, pair by pair; the 99 count the logically irreducible information once the reference has been adopted. The real gain from money is read in that gap.