Algorithmic Mathematics

Written by: Leonard Soicher and Franco Vivaldi (Queen Mary, University of London)

Professors Leonard Soicher and Franco Vivaldi of the School of Mathematical Sciences at Queen Mary, University of London have written this web book for the corresponding course MAS202 Algorithmic Mathematics.

Students should have some exposure to discrete mathematics but do not need computing experience. Students will be introduced to basic algorithms for computing exactly with integers, polynomials and vector spaces which will teach them how to think algorithmically and be able to design and analyze algorithms.

This online algorithmic mathematics textbook will also provide a constructive approach to abstract mathematics with a focus on algebra. In their introduction of the elements of ring and field theory, the authors will show how algorithms offer concrete tools, constructive proofs and a crisp environment where the benefits of rigor and abstraction become tangible.

Table of Contents for Algorithmic Mathematics

  1. Basics
    1. The language of algorithms
      1. Expressions
      2. Assignment statement
      3. Return statement
      4. If-structure
      5. While-loops
    2. Boolean calculus
    3. Characteristic functions
  2. Arithmetic
    1. Divisibility of integers
    2. Prime numbers
    3. Factorization of integers
    4. Digits
    5. Nested algorithms
      1. Counting subsets of the integers
    6. The halting problem
  3. Relations and partitions
    1. Relations
    2. Partitions
  4. Modular arithmetic
    1. Addition and multiplication in Z/(m)
    2. Invertible elements in Z/(m)
    3. Commutative rings with identity
  5. Polynomials
    1. Loop invariants
    2. Recursive algorithms
    3. Greatest common divisors
    4. Modular inverse
    5. Polynomial evaluation
    6. Polynomial interpolation
  6. Algorithms for vectors
    1. Echelon form
    2. Constructing an echelon basis
    3. An example
    4. Testing subspaces
  7. Some Proofs
    1. A note on ring theory
    2. Uniqueness of quotient and remainder
  8. Hints for exercises
   

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Algorithmic Mathematics

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