Money matters

I think we accept without questioning a lot of things. And some of these things are mostly worthless spending time on and therefore the acquiescence is justified. Now, just for the heck of it, I ask one such question.

First, the background. I have been travelling a bit recently and this has given me the opportunity to observe a few different currencies. They are nice mostly. The diversity is actually what makes it intriguing. For some people---the numismatists among us---it is intriguing enough to start a hobby. I have also indulged in some such acts : the American quarters, for example. There are quarters that have names of states (and other US territories I have come to learn) inscribed on them, and I have all of the 50 states except Michigan (If you have my evasive MI quarter, I can offer you a profit of 100% on it). The colours of the notes are actually nice, which is why I feel that the dollar is the dullest I have seen thus far. Colours, sizes, faces, watermarks, Braille symbols aside, what intrigued me was probably what is most to do with the currencies---the numbers. No, not the note numbers. The denominations.

Observation All currencies I have seen, have a subset of the following set as the denominations {0.01, 0.02, 0.05, 0.10, 0.20, 0.50, 1.00, 2.00, 5.00, 10.00, 20.00, 50.00, 100.00, 200.00, 500.00, 1000.00, ...}. You see a pattern here, that {1, 2, 5} play a key role and are repeated in all values---in ones, tens, hundreds etc.

Question There must be a reason for this pattern. If so, what? If not, why?

I now try to answer this question. I should at this point mention that I have done no research into this question. I have not attempted to look it up anywhere, not even a Google search. So, whatever follows, is what I think is most reasonable.

In trying to answer the question, I make a few assumptions and definitions that help me subsequently.

Assumption Everyone works in the decimal system. It seems to me that this assumption is logical. Even if it is disputed, the final answer here will have a counterpart that can be obtained with a similar analysis.

Definition Values are synonymous to decimal places. E.g. Units, Tens, Hundreds, Thousands etc.

Definition Practical values are those that would be useful. E.g. {Hundredths, Tenths, Ones, Tens, Hundreds, Thousands} are the practical values for many currencies. Some (like the South Korean dollar) might choose other practical values. This depends on the exchange rate / buying power of the currency.

Definition A denomination is called consistent if it repeats at every value. Thus, a currency denomination {0.01, 0.02, 0.05, 0.10, 0.20, 0.50, 1.00, 2.00, 5.00, 10.00, 20.00, 50.00, 100.00, 200.00, 500.00} is consistent, because the same denominations of {1, 2, 5} repeat at each value. We can denote a consistent denomination by simply specifying the denomination at ones value, and we call this set the basis of the denomination. For the above example, the basis is {1, 2, 5}.

Some currencies have slightly different denominations (like the 25 paise coins in India), but these exceptions are not consistent (i.e. they are not repeated in all values. e.g. you might find 25 paise coins, but not 2.50 rupee coins / notes, or 25 rupee notes). I will disregard such inconsistent denominations.

Assumption The choice of the denominations should be made keeping the convenience of the people using them in mind. Convenience will be qualified and quantified next.

Definition A consistent denomination is convenient if it allows for the paying of all practical numbers. That is to say that {2} is an inconvenient choice for the basis of a consistent denomination because you cannot make a payment of, e.g. 0.03 using this denomination.

The convenient denomination choice problem can be broken down to a similar, but simpler problem.

Lemma A consistent denomination is convenient if it allows every number from 1 through 9 to be payable through its basis.
Proof This is a straightforward claim, which I will explain through an example. Since we are using the decimal system, every practical number will consist of digits 0 through 9. All these numbers can therefore be split into different values, i.e. $245.86 can be split as $200 + $40 + $5 + $0.8 + $0.06. If all numbers from 1 through 9 are payable using the basis, any number at any other value will be payable through the basis at that value. Since $4 is payable using the basis, $40 will be payable using the corresponding denominations of Tens value etc.

The choice of a convenient, consistent denomination is thus reduced to the choice of a basis set that allows paying of all number from 1 to 9. We have thus far only qualified convenience. I now quantify it. For the purpose of illustration, we will assume, unless otherwise stated, that the basis is {1, 2, 5} since it is the most common basis. Strings of the basis are concatenations of the elements of the basis, e.g. 12 is a string. A string is a payment of a number if the sum of the symbols in the string is equal to that number, e.g. 12 is a payment of 3. I will assume that all permutations of the symbols in a string give the same string, so that the strings 12 and 21 are the same. The length of a payment is the number of symbols in it, e.g. the length of 12 is 2. For each number n from 1 to 9, let C(n) be the set of all payments of n. Here is an example of C(n) with the above basis.

C(1) = {1}
C(2) = {11, 2}
C(3) = {111, 12}
C(4) = {1111, 112, 22}
C(5) = {11111, 1112, 122, 5}
C(6) = {111111, 11112, 1122, 222, 15}
C(7) = {1111111, 111112, 11122, 1222, 115, 25}
C(8) = {11111111, 1111112, 111122, 11222, 2222, 1115, 125}
C(9) = {111111111, 11111112, 1111122, 111222, 12222, 11115, 1125, 225}

Let l(n) be the least length of the length of all payments in C(n). The following lists l(n) for the continuing example.

l(1) = 1
l(2) = 1
l(3) = 2
l(4) = 2
l(5) = 1
l(6) = 2
l(7) = 2
l(8) = 3
l(9) = 3

Let l(B) be the average of l(n) from 1 through 9 for the basis B. From the above, l({1, 2, 5}) = 17 / 9. In order for the comparison of l(B) between different bases B to be fair, we must consider different bases with the same cardinality. Let |B| = k. k = 3 for our example. Let l*(k) be the minimum of l(B) over all bases B of cardinality k, and let B*(k) be a basis (not necessarily unique) achieving this minimum.

Definition B*(k) is an optimally convenient consistent denomination of cardinality k.

Here's the rationale for this definition. With any basis B, l(n) denotes the least number of currency notes (or coins) necessary to make a payment of n. l(B) therefore is the average number of notes necessary to make a payment with the denomination being B, assuming that each number from 1 to 9 is equally likely (a reasonable assumption, I believe. Perhaps the tenths and the hundredths places might not satisfy this assumption, but let us disregard this). A denomination basis that minimizes this number is optimal in the sense of requiring the least number of notes for any transaction on an average, and therein lies the convenience of the chosen denomination.

Given this, it is natural to ask if {1, 2, 5}, the basis for the universal consistent denomination, is an optimally convenient consistent denomination. The problem of finding optimally convenient consistent denominations seems tough in general---it involves the "hard" problem of optimizing a set that will give the least average length of partitions of numbers based on that set. But for the case of the decimal number system, which we assume in our scenario, we can easily deduce a few results.

Claim B*(1) = {1} and l*(1) = 5.

In fact, {1} is the only convenient consistent denomination of cardinality 1. And in this case l(n) = n, and hence the claim is true. We note that 1 has to belong to every convenient consistent denomination.

Claim B*(9) = {1, 2, ..., 9} and l*(9) = 1.

Straightforward claim. Every number has a note so that l(n) = 1 for all n.

Claim l*(k) is non-increasing in k.
Proof Let m be a number from 1 to 9 not contained in B*(k). Let B' be the union of B*(k) and {m}. Then l*(k + 1) is not larger than l(B'), by definition. Further, l(B') is not larger than l*(k) because l*(k) can be achieved with B' by never using m.

Claim l*(k) lies between 1 and 5 for k between 1 and 9.
Proof This follows as a corollary to the above three claims. l*(k) is strictly larger than 1 for k smaller than 9 because there is at least one number n for which there is no note in the basis, i.e. for which l(n) is strictly larger than 1. l*(k) is strictly smaller than 5 for k larger than 1 because there exists at least one number n other than 1 for which there is a note in the basis, i.e. for which l(n) = 1.

Let us come back to the main question. Is {1, 2, 5} optimally convenient? Let us check some other bases of cardinality 3. Perhaps powers of 2 are good? We have l({1, 2, 4}) = 17 / 9, the same as that of {1, 2, 5}! So we know there is no sacredness to the universally accepted denomination. Odd numbers might be better? l({1, 3, 5}) = 17 / 9, again. But no better. Consider the peculiar basis {1, 3, 4}. Here are the details:

The optimal payments are 1, 11, 3, 4, 14, 33, 34, 44, 144 and hence, l({1, 3, 4}) = 16 / 9, better than the others! From a few other bases I checked quickly, nothing seemed to do better. I believe {1, 3, 4} is optimal, and uniquely so. At least one conclusion is clear.

Claim {1, 3, 4} is more convenient than {1, 2, 5}.

So then, at least in terms of convenience of the users defined as is done here, {1, 2, 5} is not optimal. Is the choice of {1, 2, 5} then a misguided one, forced upon us by history than anything else? Or is there another way to look at this?

Coming back to the notion of convenience introduced here, is the choice of k = 3 a good choice? How would one otherwise choose k, and in turn, fix a basis for an optimal convenient consistent denomination?

Inception

This is a compilation of the flaws/inconsistencies I found in Inception. Also included are some apparent inconsistencies and my explanations resolving them. They are listed randomly, and are titled with geeky names inspired by the names of the episodes of The Big Bang Theory.

For those who haven't yet seen what is arguably one of the best movies ever made (#3 on IMDb at the time of writing this post, #2 after TDK on my list of Nolan movies), this would be the place to stop reading.

The suicide anachronism Cobb and Mal, after having spent 50 years in limbo, kill themselves by sleeping on a railway track. If indeed they had been there for 50 years (and as portrayed in one of the closing scenes), they must have been old when they died. The scene however shows them still being young.

The imbalanced parentheses Mismatch of the order of dreams. I would think that the order in which the dream layers are shown must be a LIFO stack, i.e. when person B is dreaming a dream inside person A's dream, B would have to wake up from the inner dream before A wakes up from the outer one. This is not the case, at least (not necessarily the only case), for Ariadne dreaming within Eames's dream. One possible explanation is that all people wake up at the same time, but each is shown sequentially in the film. Another is that when the person dreaming the outer dream wakes up, those within the inner dream are trapped in limbo, and since they do eventually die there, they wake up from the outer dream also. But the question remains that if they are indeed waking up from limbo, they should get back to reality and not still be in any other dream level. But, this is not the case and hence the inconsistency.

The dreamer ambiguity The dreamer of the third level dream---the one where they are in the snow capped mountains---is not clear. I believe there are two dialogue sequences where it is claimed that the dreamers are different. When Ariadne asks "Whose dream are we getting into?", Cobb replies saying it is Fischer's dream, and Arthur adds that he will be helping them break into his own dream. This is just before they enter the third level dream. However, in the third level dream, Cobb says "Eames, this is your dream..."

The weightlessness paradox The premise is that if the dreamer is experiencing a state of weightlessness, so does everyone in his dream. This is true because when Arthur is dreaming the second level dream sitting in the freely falling van in the first level dream of Yusuf, everyone in Arthur's dream in the hotel is weightless. If that is the concept, then everyone involved in the third level dream, including Eames and Fischer who are the two possible dreamers, are experiencing weightlessness. Why does this not carry forward to the dream in the snow capped mountains?

The Browning disappearance In the second level dream, when Cobb, Arthur and Ariadne are with Fischer in the room, Fischer's projection of Browning (It is surely Fischer's projection because Eames is shown entering the room later) is shown to be entering the room and being caught by the team, showing him to be the one conducting the kidnap in dream level one to annul the will that would dissolve the Fischer-Marrow empire. Then, both Browning and Fischer will be plugged to the dream sharing machine along with the others, but somehow everyone except Browning appear in the third dream level, i.e. Browning is conspicuously missing.

The Cobb deficiency Another concept is that of the architect. There are dialogues which hint that the architect is someone who builds the worlds in the dreams at different levels and "explains" them to the people who will be sharing that dream. There is another dialogue where Cobb tells Ariadne, who is the architect, not to explain him the specifics of the plan for the world in the dreams. I found these to be contradictory. But this might be explained as Cobb not wanting to know the specifics himself because of his condition. He would still want others to know the plans well. This is justified by a scene where Yusuf is shown working with Ariadne on the plans, and another where Eames is said to have added a "shortcut" through a maze in the dream level three. Also, Cobb constantly asking Ariadne to explain things to Saito and Fischer also corroborates this explanation.

The Saito time-dilation The fact that only Saito was old in the dream when Cobb came to convince him to kill himself to honour the agreement between them can be explained. Conceptually, the age of a person in a certain dream level depends on the time spent in that dream by that person. When Saito dies, he goes into some unknown, but deep, dream level. Now when Cobb is searching all the dream levels to get to Saito, it can be said that he spends not too much time in each level. So, in the first scene of the film, Cobb can be assumed to have just arrived into Saito's dream whereas Saito himself has been there for decades. But there is a dialogue where Cobb, while asking Ariadne to kill herself and get back and explaining why he will stay back, says "Saito must be here somewhere." If that dialogue is taken literally to conclude that Saito was trapped in the same level dream, the age difference is difficult to explain. By "somewhere here", I think Cobb meant "in some dream deeper than here."

The limbo explanation Cobb also gets stuck in limbo because he does die in a dream---not in level three, but in level two, when everyone comes out of the van--- Cobb stays behind, drowns and dies.

The bullet repulsion What is perhaps the most unrealistic part of the film is the fact that the entire team suffers only one bullet shot after being barraged by bullets almost everywhere---in reality, with a lot of bullets that seem to be dodging Cobb in Mombasa; dream level one, with infinite bullets shot at the van; two, with bullets shot at Arthur; three, with the bullets from a whole army of militants not finding their targets. But I am willing to consider this Nolan's V-effect and disregard it.

Kudos to Gandhe for coming close to the main concept of the film!