Finite (modular) arithmetics

Finite (modular) arithmetics

Finite (Modular) Arithmetic

Introduction to Finite (Modular) Arithmetic

  • Finite (Modular) Arithmetic is a system of arithmetic for integers, which considers only the remainder when a number is divided by a positive integer called the modulus.
  • The integer in concern in the modulus is sometimes referred to as being ‘taken mod’ the modulus.
  • It is called “modular” because it involves a modulus, and “finite” because there are a finite number of possible outcomes for any given calculation.

Understanding Modulo Operations

  • To perform a modulo operation, find the remainder after division.
  • “a mod n” is the remainder when a is divided by n. For instance, 9 mod 5 = 4 because 4 is the remainder when 9 is divided by 5.
  • The modulo operation is often represented using the symbol %. For instance, in many programming languages, 9%5 will yield 4.

Properties of Finite (Modular) Arithmetic

  • Finite (modular) arithmetic exhibits many of the same properties as regular arithmetic, including commutativity, associativity, and distributivity.
  • These properties mean that, in many cases, calculations in modular arithmetic can be done in any order without changing the result, just like in normal arithmetic.
  • Nevertheless, remember that division is not always possible in finite (modular) arithmetic.

Equivalence Classes and Congruences

  • In finite (modular) arithmetic, a set of integers that give the same remainder when divided by the modulus is termed an equivalence class.
  • Two numbers are said to be congruent modulo n if they’re in the same equivalence class for modulus n. It’s represented as ‘ ≡ ‘.
  • For Instance, 7 and 2 are congruent modulo 5, as for both the remainder is 2 when divided by 5. This will be written as 7 ≡ 2 (mod 5)

Finite (Modular) Arithmetic and Number Theory

  • Modular arithmetic is a fundamental concept in number theory, and is used in several areas, including cryptography, computer science, and coding theory.
  • In particular, it plays a key role in public key cryptography algorithms such as RSA.
  • It also has applications in areas like music theory and visual art, where it can be used to create mathematical patterns.

Problems and Exercises

  • Apply the concepts and principles of modular arithmetic in various problems and exercises. This will help to consolidate understanding of the topic.
  • Carry out calculations using modulo arithmetic and interpret equivalence classes and congruencies to build fluency with the concepts.
  • Working with finite (modular) arithmetic problems will require both computational skills and theoretical understanding encompassing their relationship with number theory.