Maths, Music, and More



Card shuffling – it’s actually really mathematical

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

\sum_{n=1}^\infty n = 1+2+3+4+5+... = -\frac{1}{12}

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?
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Taylor Series

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. ex, 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 (ax2 + bx + c), and so on. So let’s start with a first order approximation of, say, ex.
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Infinitely stretchy ropes and the Harmonic 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?
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Pythagoras paper

Here, have a paper

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.


Why do integrals work?

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).
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Holy Bouncing Billiard Balls, Batman!”

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)
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Tidbits of graphs

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

graph.PNGYou 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.

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Proof by Induction, and more Fibonacci

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:
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