Category Archives: maths

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)

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: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

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

A triangular space-filling curve

Yesterday, I showed you a snowflake.  Today, I’m going to try to explain it. [EDIT: There’s a couple of animations at the end, if you’re not interested in the explanation.]

I was inspired by a space-filling curve called the Hilbert curve.  I wondered how to define a space-filling curve where the successive approximations were based on a triangular grid, rather than a square grid.  So, given a grid like this:we want a path from the bottom-left dot to the bottom right dot (as indicated), passing exactly once through all the other dots in the grid. Continue reading A triangular space-filling curve