p-adic numbers are a system of numbers used in number theory, which extends the idea of integers and rational numbers. They are defined based on a prime number p and focus on the divisibility of numbers by p. In this system, numbers are represented in a way that emphasizes their behavior with respect to p, allowing mathematicians to study properties of numbers that are not easily visible in the usual decimal or fractional systems. The construction of p-adic numbers involves creating a metric, or a way to measure distance, that is different from the standard one. This metric allows for the definition of convergence and limits in a new way, leading to a rich structure that has applications in various areas of mathematics, including algebraic geometry and cryptography.