Irrational numbers are somewhat difficult to work with. Unfortunately, they’re also quite useful and crop up both in pure and applied mathematics, and tons of places you may not expect. When written in decimal form, they result in an infinite sequence of numbers with no apparent pattern. If we round or truncate this number, we lose accuracy and introduce some level of error into any calculation.
Let’s get started with perhaps the most fundamental mathematical object, the set.
In math, we often talk about collections of things. A set is just that: a collection of things.
The elements of a set could be anything: numbers, equations, lines, shapes, fruit or any other thing you can dream up. Let’s be a little more specific and give a more precise definition of a set.
Procedural generation is everywhere—you’ve probably encountered it without even realising. It’s what gives in-game worlds their rolling hills, jagged cliffs, and winding cave systems. And at the heart of it all is Perlin noise: a special kind of randomness that isn’t entirely random at all. It’s smooth where it needs to be, rough when we want it to be, and endlessly customizable.
You’ve swiped through dozens of profiles, but why does it feel like there’s no perfect match for you? What if there was another way where we could keep everyone happy?
I think it’s time to play Cupid! (But be warned: it’s not all sunshine and rainbows; there’s some tough decisions ahead)
The sun dipped low over the bustling City of Königsberg, casting golden reflections over the Pregel River. Its waters divided the town into four distinct land masses, connected by seven foot-bridges that had become a curious point of both pride and frustration for its residents. By day, the bridges bustled with merchants and townsfolk, but as the evenings drew in, they became the source of a mysterious puzzle. Could someone, starting from any point on land, cross each of the seven bridges exactly once and return to where they began? It may sound simple, almost trivial, yet no matter how the townspeople tried, no one could find a solution.
Prime numbers are among the most intriguing puzzles in mathematics — seemingly random yet deeply significant.
Their elusive pattern defies easy detection, making them both a source of fascination and frustration. The challenge grows exponentially with larger numbers, where determining if a number is prime becomes immensely time-consuming.
However, some clever shortcuts can speed up the search and help us swiftly eliminate impostors.
Have you ever climbed a staircase that just didn’t seem to be made for human legs?
The steps are too small to comfortably take one at a time, but too large for an easy double-stair.
You either end up shuffling along with baby-steps, or stretching so far that it feels like you’re trying out for the gymnastics team.
Suppose you’ve just been hired as the new manager of Hilbert’s Infinite Hotel.
On your first day, you arrive at work only to be greeted by an infinitely long line of people in the lobby each expecting a room.
The Problem? All of the infinite number of rooms are already occupied by an infinite number of guests. The Infinite Hotel is full.
You’ve probably seen this puzzle before.
You could probably solve it without too much hassle.
But let’s ask a more interesting question…
What is the fewest number of moves required to solve the puzzle?
Can we prove we’re correct?
Luckily, studying for a degree in mathematics is very different from high school.
Math class didn’t exactly get the best reputation in school — and I can see why. It tells students to learn about seemingly pointless techniques and memorise different formulas that 99% of them will never use again… unless they go on to become math teachers.
Things change, however, when you decide to study maths at a higher level.
