P-adic numbers are a type of number in mathematics that were first introduced by German mathematician Kurt Hensel in the early 20th century. They are an extension of the rational numbers and are important in number theory and algebraic geometry.
P-adic numbers have many applications in mathematics, including in number theory, algebraic geometry, and representation theory. They also have practical applications in cryptography and coding theory. P-adic numbers also have connections to other areas of mathematics, such as topology and complex analysis.
The main difference between p-adic numbers and real numbers is the way they are constructed. Real numbers are based on the decimal representation of numbers, while p-adic numbers are based on the p-adic representation, which uses a different base. Additionally, p-adic numbers have different properties than real numbers, such as the fact that they are not totally ordered.
P-adic numbers are used in cryptography because they have different properties than real numbers, making them useful for creating secure codes and encryption methods. P-adic numbers are also used in coding theory, which is important for error correction in data transmission.
Yes, there are several books about p-adic numbers that are suitable for beginners, including "p-adic Numbers: An Introduction" by Fernando Quadros Gouvea and "A Course in p-adic Analysis" by Alain M. Robert. Additionally, many introductory books on number theory or algebraic geometry also cover p-adic numbers.