Mathematics For Computer Science

This article is a brief introduction to categories of mathematics a developing computer scientist is likely to need to progress (depending upon their field). The content is comprehensive by breadth, gets technical and often goes beyond A level capabilities.

The article is primarily descri ptive vocabulary. I attended a short compulsory module called "mathematics for programming" during my postgraduate involvement in computing and A.I at Southbank University (SBU).

From the syllabus (SBU) I quote "essential ideas of discrete mathematics and logic provide a working vocabulary of basic concepts for programming. This course aims to introduce such a vocabulary to students...."

I will name the category (CONCEPT) and give a general outline of content associated with that category.

NUMBER SYSTEMS - types of numbers can be placed into SETS called number sets. These include Natural numbers, Integers, Rational numbers, Real numbers, Complex numbers.

ALGEBRA - The study of variables and the rules for manipulating these variables.

MATHEMATICAL LOGIC - Consider the overlap between the fields of logic and computer science. These include the semantics of programming languages, program verification, lambda calculus, comminatory logic, computational theory (formally recursion), modal logic. There has always been a strong influence from mathematical logic in the field of Artificial Intelligence (A.I).

TRIGONOMETRY - concerned with the relationship between angles and ratio lengths (TAN, SINE, COSINE), useful for computer graphics.

COORDINATE SYSTEMS - a system in geometry that uses 1 or more numbers (or coordinates) to uniquely determine the position of points or other geometric elements on a manifold (topological space). 1 dimension includes lines and circle, 2 dimensions are called manifolds and are called surfaces and include the plane and sphere.

DETERMINANTS - These include MATRICES, the scaler value that is a function of the entries of a square matrix - MATRIX = ad - bc. ARRAYS (data structure) an arrangement of equally spaced items in computer memory. LISTS (an abstract data type) that represents a finite number of ordered values. VECTORS, geometric matrix transforms (transforming aiding the study of geometry). Differential and Integral Calculus - the mathematical study of continuous change in the same way that geometry is the study of shape and algebra and a generalisation of arithmetic operations. Differential Calculus (rate of change, slopes of curves) Integral Calculus (accumulations of quantities, area under or between curves).

ALGORITHM - a finite set of rigorous instructions to solve a class of specific problems or to perform a calculation.

DISCRETE MATHEMATICS - the study of mathematical structures that can be considered DISCREET rather than continuous. DISCREET maths includes INTEGERS, GRAPHS, STATEMENTS IN LOGIC.

COMBINATORICS - combinatorial logic - eg predicate logic, lambda calculus.

NUMERIC ANALYSIS - the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) - distinguished from discrete mathematics. Numerical methods that find approximate solutions rather than exact ones.

ARITHMATIC - an elementary part of mathematics that considers the properties of traditional operations on numbers - addition, subtraction, multiplication, division, extraction of roots.

PROBABILTY THEORY - attempted prediction of random events and comments about their behaviour, includes terms like probability space, which assign measures taking values between 0 and 1 termed probability measure to a set of outcomes called sample space.

Consider now MATHS IN THE NATIONAL CURRICULUM 11 - 16 (see my article in ESSENTIAL PRINCIPLES)

1. NUMBER

2. ALGEBRA, RATIO, PROPORTION and LOTS of CHANGE, GEOMETRY and MEASURE.

3. PROBABILITY

4. STATISTICS

By KS3 (age 11 years) ideas in mathematics that are useful to computer scientists are being developed. My next article in this series traces the development of how these more specific concepts are integrated into the NATIONAL CURRICULUM at KS3/KS4 or the 11 - 16 age range including GCSE (and a little beyond).