# Maths Personal Statement Example 19

I want to carry on studying mathematics for three reasons: first, the enjoyment I have derived from doing problems has grown as I attempt harder problems; second, I like being challenged intellectually, whatever the subject, and the challenges I have met in maths have been tougher and thus more satisfying than any others; third, although I spend a great deal of time working with maths, I feel I have only just begun. I am conscious of how much I don't yet know but, at the same time, really excited at the prospect of pursuing maths at university.

An aspect of my maths studies which has preoccupied me has been the concepts of proofs. Fermat's last theorem made me aware of the difficulty of proving seemingly simple theorems.

Having attended a talk about Fermat's last theorem, I read the book by Simon Singh on this topic. As is well understood, Fermat stated that the sum of two integers, both raised to the power of n, cannot equal a third integer also to the power of n (where n is greater than 2). I found the problem itself intriguing, because the idea that it is impossible to find three numbers to fit the equation when n is greater than two seems counterintuitive when it so clearly holds when n is two in Pythagoras' theorem.

I recently attempted a question based on Fourier's proof that e is irrational, which led me to realise that individual steps, (for example summing the geometric series with the ratio 1/(n+1) and realising that this is larger than the sum of 1/n!), are often quite simple but that piecing them together can produce an elegant proof.

I was intrigued to find out if there were other methods of achieving this result, which, of course, there are - from Euler's well-known proof using a continuous fraction to newer and lesser-known proofs such as Sondow's, which shows e's irrationality through geometry by constructing e from line segments.

This is just one of many examples where there are numerous ways of visualising a problem and often in class, when I find one approach to be the obvious one, I will be surprised at someone having found a completely different one. I have really enjoyed reading how other mathematicians have found proofs, for example in Martin Aigner's Proofs from the Book, and I am often left wondering what leads someone down a particular mathematical path.

I have also become interested in asking whether maths is a human construct or something we have discovered, and gave a talk on this subject. When I started researching this question, I assumed that maths was, like other sciences, something that humans are discovering. However, an opposing argument that at least some part of maths is a human construct leads many to doubt that this is the case and it seems the general consensus is that it is indeed a human construct.

In physics and chemistry, maths has both facilitated my studies, and enabled me to dive deeper into topics, such as quantum mechanics and the laws of thermodynamics. An example of this is Heisenberg's uncertainty principle, which ties mechanics and statistics together nicely. I also read into Schrodinger's equation which models matter as waves rather than particles and the equation itself ties in complex numbers, calculus and concepts that I had never come across before, like a Hamiltonian operator.

I have enjoyed participating in competitions such as the BMO and MOG as this introduced me to writing full, coherent proofs. I also volunteer at a local school, where I teach maths to year seven students. I saw an impact on both myself and the children: I watched them become more confident and able in performing basic arithmetic functions, whilst also encouraging them to ask questions.

Having to explain things I've taken for granted for years, like why there are 180 degrees in a triangle, has helped me to communicate my ideas more clearly.

I am excited by the prospect of studying maths at a higher level and look forward to it greatly.

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