Three essential proof techniques, followed by many not-so-essential

Most people may think that mathematics is about doing hard sums with large numbers. Whilst it is true that you can do seriously hard computations all day long (try computing the integral of the square root of tan), the majority of mathematics is based on proofs.

Here I will outline some of the most common methods of proving a statement (and some of the funnier ones). For each method I will try and give a simple example of a proof using that method.

Image for post
Image for post
Photo by Miguel Henriques on Unsplash

Direct Proof


How can Gauss improve your pizza eating technique?

Given some two dimensional surface in R³, a sensible question to ask when getting acquainted with the beast is what the curvature is at any point (this would be sensible if you are approaching it from a differential geometer’s perspective; from a algebraist’s perspective you might as well ask what the colour 5 smells like).

What is Curvature?

Imagine yourself on the surface in question. Each direction you look in will form a cross-section of the…


Part 2: Mixture Models

If you haven’t read part 1, you can find it here. The general gist of it is that we look at our data and have a guess at what sort of model would fit it best, then we find the parameters for that model that are the most likely to have generated our data.

As promised, we will look at something more complicated now.

Consider the histogram of our data is the following:

Image for post
Image for post

It doesn’t look like anything you will have encountered in high school statistics.

Well, sort of.

One could argue that it looks like two Gaussians mixed together.


Part 1: How to best fit a Gaussian

Suppose, as we all do on a Friday evening, you are looking at data of all the heights of people attending a university. You can plot a histogram of all the data and it might look something like this:

Image for post
Image for post

You hear that there will be a new person arriving tomorrow and you want to find out what the probability that their height is within 5 cm of yours. How do you go about doing this?

1. Guess the Shape


Image for post
Image for post
Photo by Daniel Klein on Unsplash

Why the Butterfly Effect is not the whole story

Chaos theory is potentially one of the most controversial fields in mathematics. Not because of the choice of axioms or some of the results, but due to the fact that no-one can agree what chaos actually means.

In Gleick’s book Chaos, the author interviewed a whole bunch of chaos scientists (chaosists?) and not one of them could agree on a definition of chaos.

The definition given in most pop culture media is the butterfly effect:

The flap of a butterfly’s wings in Brazil can set off a tornado in Texas

This means that given a very small change to a…


A side step into Ramsey Theory and a big number

You may have heard of this incredibly large number called Graham’s number, but more often than not when this subject is discussed, Graham’s number is rarely given in context. What does it mean? Did mathematicians decide to write down a really big number one Sunday afternoon because they were bored?

No.

In fact, Graham’s number is related to a very elegant field of mathematics called Ramsey theory. I will give you a beginner’s guide to Ramsey theory before diving into the deep end with Graham’s Number.

Image for post
Image for post
Paul Erdős, a prolific mathematician who contributed a lot to the field — Archives of the Mathematisches Forschungsinstitut Oberwolfach ©Gabriella Bollobas

Theorem

Suppose you are at a party. How many people need to be present such…


Careers, Computer Science, Software Engineering

Did you know that Google has a secret hiring challenge?

Welcome to Google Foobar.

Google has a secret, invite-only series of coding challenges. Here is my experience with it along with some quick tips at the end for anyone that is just getting started.

The Precursor

One Sunday evening, I was sat at my desk doing what any normal person would be doing on a Sunday at 11 pm: Googling TensorFlow documentation.

At this point, the Google search page did something strange. It showed me the following message:

Image for post
Image for post

Feeling like a character straight out of the film WarGames, I instantly click I want to play.


Six Difficult Ways of Becoming a Millionaire

Want to win a million bucks? Just solve one of these problems. No strings attached. Ok maybe one string: the problems are somewhat hard. Scratch that, really hard.

The Gauntlet is Thrown

At the start of the millenium, the Clay Mathematics Institute put forward these seven problems which are deemed as some of the most difficult problems that remain open. Each problem has a one million dollar bounty for the first person to provide a valid proof (or disproof).

Image for post
Image for post

I have been wanting to write this article for quite some time, but struggled to decide at what level I should present the material. After…


Image for post
Image for post
Stefen Banach (Left) and Alfred Tarski (Right). Photo courtesy of El Pais

The Banach-Tarski Paradox

..two in the hand?

There are many counter-intuitive, bizzare theorems in mathematics. One of my favourites has to be the Banach-Tarski Paradox. A result due to Stefen Banach and Alfred Tarski in 1924, the paradox states that given a ball, there is a way you can cut the ball into a finite number of pieces and rearrange these pieces to form two balls that are exactly the same size as the original. This seemingly defies our intuition. I will start with an ‘intuitive’ approach and then dive straight into the deep end with some technical stuff.

I won’t be proving the theorem here as…


I had the pleasure of witnessing this interaction the other day: two kids were locked in a battle of wits to see who could think of the highest number.

Player one served with a million. Player two with an amazing return of a billion. Player one went for the smash: INFINITY. But player two caught it on the volley — INFINITY PLUS ONE.

At this point the umpire (me) intervenes with the ruling that infinity plus one is actually just infinity. …

Francesco Di Lallo

*Graduate mathematician *Pro climber *I like to make higher level mathematics accessible to the proletariat

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store