Quotes from Timothy Gowers

the second derivative conveys just the idea we want—a comparison between the value at x and the average value near x. It is worth noting that if f is linear, then the average of f(x - h) and f(x + h) will be equal to f(x), which fits with the familiar fact that the second derivative of a linear function f is zero.

The reason special names are given to these quadratic irrationalities is that any quadratic algebraic integer is a linear combination (with ordinary integers as coefficients) of 1 and one of these fundamental quadratic algebraic integers.

there are only four units in the ring R-1 of Gaussian integers, namely ±1 and ±i; multiplication by any of these units effects a symmetry of the infinite square tiling

Fundamental to understanding the arithmetic of Rd is the following question: which ordinary prime numbers p are irreducible elements of Rd and which ones factorize as products of irreducible elements in Rd? We will see shortly that if a prime number does factorize in Rd, it must be expressible as the product of precisely two irreducible factors.

if, for each t, we write u(t) for the function from to that takes x to u(x, t), then it describes how the function u(t) "evolves" over time. The Cauchy problem for an evolution equation is the problem of determining this evolution from knowledge of its initial value u(0).

A3 0 is an additive identity: 0 + a = a for any number a. That is all you need to know about 0. Not what it means – just a little rule that tells you what it does.