# Questioning my Cambridge Math Interview

## Even the best mathematicians can fall into the trap of circular logic

“The method of ‘postulating’ what we want has many advantages; they are the same as the advantages of theft over honest toil.” — Bertrand Russell

On a cold December back in the 1990s, as a 17 year old budding mathematician, I went to a Cambridge college for an Undergraduate math interview. It was an awkward trip, involving an hour by train from my home town to London, a transfer across London and another 90 minutes by train up to Cambridge, followed by a long walk to the college on the outskirts of the city. I didn’t stay overnight at Cambridge, so I ended up leaving at ridiculous o’clock to make the morning interview on time.

Of course I was nervous. I knew I was going to get a math grilling. I had limited experience of genuinely challenging math discussions. I attended a pretty poor quality state school which was only able to offer a limited curriculum, and my classmates and teachers were not set up to help me prepare for this kind of interrogation of my math skills. In fairness to Cambridge, I’m surprised they even interviewed me given my limited track record at the time.

I ended up having two interviews with different individuals, and I want to talk about one of them here, because I’m pretty sure my interviewer was wrong in the interview.

## The interview

Before the interview, I was asked to fill in a short form in the waiting room. The main question on this form asked me to mention a topic that I had covered recently in class. So I wrote: *differentiation of basic trig functions.*

When I entered the room, after a few pleasantries, my interviewer took a look at the form and then asked me to prove that the derivative of sin*x* was cos*x. *I proceeded to do this.

## Proving that the derivative of sin(x) is cos(x)

First I explained that the derivative of a function is intended to provide the gradient of the tangent to the curve at any given point, if it exists. Now if *f(x)* is a continuous function on *x*, then for a given point on its curve (*x, f*(*x)), *we can determine the gradient of the tangent by taking the gradients of lines drawn between (*x*, *f(x)) *and (*x+h*, *f(x+h)) *for *h>0*, and then ask what happens as *h* gets infinitely small, approaching zero. This is often known as *taking*…