Modulonarithmetik

Modulonarithmetik, also known as modular arithmetic or clock arithmetic, is a system of arithmetic for integers where numbers "wrap around" upon reaching a certain value—the modulus. It is a fundamental concept in number theory and abstract algebra. The core idea is to consider the remainder of an integer after division by a fixed integer, the modulus. For example, in modulo 12 arithmetic, which is commonly used for timekeeping, 13 is equivalent to 1, 14 is equivalent to 2, and so on. This equivalence is denoted by the congruence relation, for instance, 13 ≡ 1 (mod 12).

The basic operations of addition, subtraction, and multiplication can be performed in modular arithmetic. To add two numbers modulo n, you add them as usual and then find the remainder when the sum is divided by n. The same principle applies to subtraction and multiplication. Division in modular arithmetic is more complex and involves the concept of modular multiplicative inverses.

Modular arithmetic has numerous applications. It is crucial in cryptography, particularly in algorithms like RSA, where it is used for secure communication. It also finds use in computer science for tasks such as hashing and error detection/correction codes. In pure mathematics, it is a building block for understanding more abstract algebraic structures like rings and fields. The set of integers modulo n, denoted as Z_n or Z/nZ, forms a ring under modular addition and multiplication.