Analytic number theory is the application of methods from analysis to the study of integers, in particular primes. This may seem paradoxical: at the heart of analysis lie notions of continuity and differentiability — and what could be more discrete and discontinuous than the set of primes?
Everyone knows Euler’s famous identity, linking the imaginary unit to trigonometric functions:
$$ \mathrm{e}^{\mathrm{i} \pi} = -1 $$
But another remarkable identity is also due to Euler, this one linking the set of prime numbers to an analytical function:
Linear Algebra is one of the foundational topics for all applied mathematics. But compared to Analysis, it initially often feels stranger and less familiar. Although technically not hard, the level of abstraction is higher, making it hard to see what all the formalism is supposed to achieve.