Imagine you’re bored one day and decide to start flipping a coin just to pass the time. The first flip lands on heads. The second flip also lands on heads. Then the third. That’s a little unlikely. To your surprise, the fourth flip also lands on heads. At this point, you pause to consider the…
Tag: Math
This post contains no assertions of fact
(not clickbait) Many of my longer blog posts take a very, very long time to write, as in both the amount of time spent working on it and also the amount of time between when I start the first draft and when I hit publish. One thing that delays me significantly is doing research and…
Why 0.999… = 1
0.999… repeating (or 0.9‾0.\bar{9}) is equal to 1. It’s a fact. To some, this is highly counterintuitive, unbelievable even. I’ve seen various explanations for how it works, from informal explanations to formal proofs. I recently found an interesting (not original) proof, but first I’ll give a couple less rigorous explanations. Informal arguments Tripling a third…
The pigeonhole principle
Let me tell you about one of my favorite things. The pigeon and dove family, Columbidae, is a group of widespread and common birds. Members of this family were domesticated thousands of years ago and they have had symbolic importance throughout history. Many species continue to live closely with humans. People who keep pigeons are…
Let’s talk about the singularity
The concept of the technological singularity is a confused one. Here’s the thing about “general AI” or “self-aware AI”—it will be indistinguishable from really good mimicry of self-aware intelligence. When the singularity happens, we won’t notice. This is an excerpt of a YouTube comment on a video about AI technology. I thought about responding to…
The barber shop pole illusion explained with math
This is a well known optical illusion in which rotation appears to be vertical motion. There are several aspects of the barber pole design which make the illusion more effective, including having multiple stripes, making the stripes different colors, and using rotation. We will strip these things away in order to see what is happening…
Pedagogy recapitulates history (sort of)
Because of the way mathematical ideas build on other mathematical ideas, the order in which these ideas were discovered/invented is often the same as the order in which they are taught. All humans typically start where humanity (is thought to have) started, with counting objects and identifying simple shapes. From there, we begin making calculations….
Operations
You’re certainly already familiar with several mathematical operations, such as addition, subtraction, multiplication, and division. You may also be familiar with the logical operations AND, OR, NOT, XOR, and IF…THEN. More generally, an operation can be defined on any nonempty set. Operations have the following properties. An operation is a function that takes some number…
Metric space
We take for granted that we can measure distances. Most people take for granted that we can represent physical space mathematically and calculate distances between pairs of points. We generally think of ourselves as living in R3, or 3D Euclidean space. Since this is how we represent physical space, it’s generally the most useful (though…
Stories of math notation (part 2): The worst notation in mathematics
Sometimes notation is confusing. One way this can happen is when the same notation is used for multiple different concepts. Usually, if the same notation is used for different things, they occur in different contexts which allows them to be distinguished. In at least one case, the same notation is used for different things in…
