Balanced base 2+i (and some gratuitous fractals)

Traditional numeral systems, such as binary, decimal, and dozenal, let us represent non-negative numbers in a systematic way. When we want to write down a negative number, we write a $-$ sign followed by a positive number. There are other numeral systems, such as balanced ternary, that let us write down both positive and negative numbers in a seamless, systematic way.

Balanced ternary, like ternary (base three), is based on powers of three, but unlike ternary, whose digits are 0, 1, and 2, the digits used in balanced ternary are 0, 1, and 1, the last of which represents negative one.  As in binary, the product of any two digits is a digit, which can simplify multiplication, but the symmetry around 0 means that to negate a number, you just turn each digit upside down.

Donald Knuth apparently called balanced ternary “perhaps the prettiest number system of all”, but even in balanced ternary, complex numbers have to be written as a real number plus the product of a real number with $i$, the imaginary unit.  Can we represent all complex numbers in a seamless, systematic way? Continue reading Balanced base 2+i (and some gratuitous fractals)

The original Jubilee

In my original article on Jubilee I hinted, by the choice of the name Jubilee (and via a link), that my ideas were inspired by those in the Torah, specifically in Leviticus chapter 25.  On Sunday this week, I spoke at my church about that chapter.  I stayed away from my own ideas about how a modern Jubilee-like system might work (which you can read about here),  focusing instead on some features of the system described in the Bible.  You can listen to what I said, courtesy of my church’s website.

Dice displays to prompt intuitive understanding of probabilities

About a week and a half ago, when I looked, iPredict showed approximately a 63% probability that there would be a National Prime Minister after the 2017 general election in New Zealand, and a 37% probability that there would be a Labour Prime Minister. Here’s what that looks like as a horizontal bar:

The problem with this way of looking at it is that (especially in New Zealand, with a proportional voting system), there’s a temptation to interpret the proportions as vote shares, rather than as probabilities. And even when I’ve got the idea of vote shares out of my mind, I can still be inclined to interpret it as a prediction that National will win the 2017 election. It’s not; it’s an estimate that there is a 63% probability that National will win the election.

How can I encourage myself to understand this intuitively? Continue reading Dice displays to prompt intuitive understanding of probabilities

Efficient allocation of mathoms in a Bad-Santa-like game

Suppose you’re at a mathom party and it turns out that some of the mathoms are popular, and multiple people want to claim the same mathoms. How can you fairly decide who gets each mathom, and ensure the allocation isn’t wasteful?

Well, a standard economic answer is to auction them to the highest bidders, but then the mathoms aren’t really gifts any more. And what if the process of allocating the mathoms could itself be enjoyable? Continue reading Efficient allocation of mathoms in a Bad-Santa-like game

A non-coercive copyleft licence

In the world of free culture licences, copyleft describes licences that require derivative works to be distributed under the same licence (or, sometimes, a similar one that guarantees the same freedoms to the public).

But I noticed that, at least in the case of free software licences, copyleft licences tend to be longer and more complicated than some of the wonderfully brief so-called permissive licences. For example, the ISC licence consists of a copyright notice, a 34-word permission notice, and a 76-word disclaimer of warranties and liabilities (if I’ve counted correctly).

For comparison, I asked on a relevant mailing list several weeks ago what the shortest copyleft licence anyone knew of was. Continue reading A non-coercive copyleft licence

Dozenal numerals with meaningful components

I’m a mild fan of dozenal, also known as base twelve or duodecimal (which is a very base-ten way of naming base twelve, and which Wikipedia helpfully points out should not be confused with the Dewey Decimal system). I haven’t memorized my times tables in dozenal; like most people, I find decimal sufficient for daily life (and much easier to use with existing technology), which is precisely why imminent global change to dozenal is unlikely.

Still, I find it interesting to wonder what arithmetic would be like if we’d ended up with a more sensible base system, like dozenal, or if we one day did manage to switch to it. The particular subject of this article is: What if, instead of adding a couple of symbols to the familiar decimal ones, we designed new symbols specifically for dozenal? And what if, instead of choosing arbitrary symbols, we built them out of components that each conveyed a particular meaning? Continue reading Dozenal numerals with meaningful components

Rationing for libraries

When a library finds that more than one person wants to borrow a particular book, they have to have some way of rationing access to the book — deciding who gets to borrow it first, and for how long. At my local library, they achieve this by lending books for three weeks at a time, and preventing you from renewing the loan if someone else has requested the book. If lots of other people have requested the book, I think they get to borrow the book for up to three weeks each in the order they placed their requests.

But what if the first person to borrow the book reads a few pages, gets bored, and doesn’t look at it again till it’s due back? Perhaps they intend to read the rest of the book, but never get around to it. Everyone else waiting to borrow the book might place a greater value on having earlier access to it than the first borrower places on keeping it for the full three weeks. Could the library arrange for less wasteful rationing of access to their books? Continue reading Rationing for libraries

A responsible transitive transactions protocol

A while ago, I wrote about transferring money through chains of IOUs. I’ve since discovered that such transfers are now sometimes being called “transitive transactions”.

So, for example, if Alice wants to pay Frank \$600, but Frank doesn’t trust Alice’s IOUs, they might find a path through existing trust relationships, so Alice gives Bob an IOU for \$600, Bob gives his own IOU to Charles, Charles gives one to Denise, Denise gives one to Edith, and Edith gives one to Frank. Alice has paid \$600, Frank has received an IOU for \$600 from someone he trusts, and everyone in between has given an IOU for \$600 and received an IOU for the same amount from someone they trust.

A problem

But Charles will be upset if he thinks the transaction is going ahead, and gives his IOU to Denise, only to discover later that Bob didn’t agree to this transaction; and Denise will be equally upset if Charles then demands his IOU back, but she’s already given hers to Edith. Or, Alice will be upset if she gives her IOU to Bob, but discovers later that Frank never received the money, because Denise, who neither Alice nor Frank know or trust, ran off with the money. So how do all these people coordinate, so that they all agree on whether or not the transaction is going through? Continue reading A responsible transitive transactions protocol

An explicit social contract and non-coercive law enforcement

In philosophy, the idea of a social contract is one way of justifying the government’s authority over individuals. The idea is that you’ve somehow agreed to submit to the authority of the state in return for the state’s protection of your rights (or at least, those of your rights that you haven’t surrendered by submitting to the state’s authority).

But where can I find the terms of this social contract? What if I don’t want to agree to it? What happens if I break the contract? What happens if the government breaks the contract?

Could we come up with an explicit social contract that answers these questions? Would most people actually want to sign it? Would other people have a meaningful choice not to sign it? I think so. Continue reading An explicit social contract and non-coercive law enforcement

Barometric music

Sound is the way we perceive rapid fluctuations in air pressure. The lowest notes humans can hear are around 20 Hz — that is, when the pressure fluctuates up, down, and back to normal 20 times per second —, and the highest notes around 20,000 Hz, though the exact range varies with age, and from person to person.

One of the tools used for tracking and predicting weather is the barometer, which measures the ambient pressure of the atmosphere around it, which fluctuates. Unlike pressure associated with sound, these fluctations are very gradual; I’m no expert, but my vague impression is that each cycle — from, say, the peak of one high to the peak of the next — takes about a week, rather than a day or a month (at least here in New Zealand).

But what if we sped it up? Continue reading Barometric music