My old high school has a program where they invite primary school kids to the school to do science and maths activities – and this year, they asked me to help run one of these, the Mini Mathematicians class. This problem actually came out of a class we did there, on card shuffling.

Continue reading “Card shuffling – it’s actually really mathematical”

I want to have a look at a very famous (and bizarre) result, the infinite sum

At first, it doesn’t seem to make any sense. If you add all these numbers up, the result keeps getting larger, so how does it approach a finite number – and a negative one for that matter?

Continue reading “1+2+3+4+5…=-1/12”

In this post, I want to have a look at a really neat part of calculus, Taylor series. It’s essentially a process you can use to approximate some graph, e.g. e^{x}, using a polynomial – the idea is, the higher order the polynomial, the more accurate the approximation gets.

## Let’s dive in! (This gets a bit tricky)

How does it work? First, some terminology – the first order approximation is a linear approximation (so ax+b), second order is quadratic (ax^{2} + bx + c), and so on. So let’s start with a first order approximation of, say, e^{x}.

Continue reading “Taylor Series”

(The above photo is from Philippe Petit’s famous tightrope walk across the WTC)

This is a really nice question about limits from a uni assignment I was given. Say you are at the start of a stretchy tightrope, a kilometre long. You walk along it at 1 metre per second – but at the end of every second, the rope stretches another kilometre longer. Do you ever reach the end?

Continue reading “Infinitely stretchy ropes and the Harmonic Series”

Here’s a paper I promised some time ago, about creating Pythagorean triples.I’m not really too sure about how maths papers should be constructed, so it’s probably a bit rubbish, but oh well.

When we start learning calculus, we tend to be given a nice understanding of where derivatives come from, and how the process links to tangents. But how are integrals linked to area?

I found a nice explanation of this here, and I thought I might have a go at rehashing their explanation.

So, say you want to find the area under some curve, from x to (x+dx).

Continue reading “Why do integrals work?”

Ok, I’ll admit I was a bit late with this one – this was something I worked on over 2 years ago; it started of as a school maths question that kinda got out of hand. The question was something like this:

If you set up a billiards table, of dimensions MxN, and fire a ball at a 45 degree angle from the bottom left, which pocket will it fall into? (assume that the side pockets are taken out so there are only corner ones)

Continue reading “Holy Bouncing Billiard Balls, Batman!””

In graph theory, a graph is a bunch of points (nodes) and lines connecting them (edges).

You can use it to represent all kinds of information – cities, where the edges are roads, for example. Ways around a city block. It’s a pretty versatile system for representing data, and has all kinds of useful applications, but here I’m going to look at the graphs themselves, and some interesting properties they possess.

Proof by induction is one of the most prominent forms of proof. It’s often likened to mathematical dominoes – set the dominoes up, knock one down, and the rest fall as well. How does it work?

It’s quite hard to explain without an example, so I will look at it through the lens of finding a general formula for adding natural numbers.

Say I want to add up 1+2+3+4+5+6…+99+100. That’s a lot of adding – is there a shortcut formula? as it turns out, there is. adding up all the integers from 1 to “n” is equivalent to:

Continue reading “Proof by Induction, and more Fibonacci”