The idea of transforming a “difficult” problem into an “easier” problem is one that is used widely in mathematics. There are many types of transforms available to mathematicians, engineers and scientists. In this unit we are going to examine one such transformation, the Laplace transform, which can be used to solve certain types of differential equations and also has applications in control theory.

For functions of a single variable there are two types of integration. The first is indefinite integration, which is effectively “differentiation in reverse”, and the second is definite Integration. Before we consider how to integrate functions of two variables, we shall consider the “construction” of definite integrals and extend this idea to double integrals.

A differential equation is an equation that contains an unknown function, which we need to solve for, and its derivatives. Technically they are ordinary differential equations (ODEs) since they contain ordinary derivatives as opposed to partial derivatives. An equation that contains partial derivatives is called a partial differential equation (PDE). In this module we shall only consider ordinary differential equations. Differential equations are extremely important in science and engineering as they can mathematically describe physical processes such as current flow in electrical systems, motion of mechanical systems, fluid flow, chemical reactions, population dynamics, the spread of infectious diseases, and many other natural phenomena.

Functions of a single variable, i.e. y = f(x) , are useful in representing a variety of physical phenomena. However, in many real-world situations quantities depend on more than one variable giving rise to functions of several variables.

This unit introduces the theory and application of mathematical structures known as matrices. With the advent of computers matrices have become widely used in the mathematical modelling of practical real-world problems in computing, engineering and business where, for example, there is a need to analyse large data sets. Applications of matrices occur: • in all areas of science to solve (large) systems of equations. • in computer graphics to project three dimensional images onto two dimensional screens and apply transformations to rotate and move these screen objects. • in cryptography to encode messages, computer files, PIN numbers, etc. • in business to formulate and solve linear programming problems to optimise resources subject to a set of constraints.

The previous unit introduced the term exponent to represent the repeated multiplication of a number by itself. For example, the exponent tells us how many times we need to multiply the number 10 by itself to obtain 1000, i.e. three times as 10 × 10 × 10 = 1000. Here the base is 10 and the exponent is 3. We now consider the closely related topic of what power a number must be raised to in order to obtain another number. The number being raised to the power is called the base and value of the power is called the logarithm.

This section introduces indices, also known as powers or exponents. Indices provide a shorthand method for representing the repeated multiplication of an expression by itself. A good understanding of indices, and the associated laws of indices, is essential when it comes to applying algebraic manipulation to simplify and solve mathematical expressions and equations.

In this section we introduce the concept of an equation and present techniques for solving different types of equations. We firstly look at the algebraic solution of linear equations in one variable before moving on to simultaneous linear equations and then quadratic equations. In all cases a geometric interpretation is presented along with details on how to graph the relevant functions. At appropriate locations throughout the document links are provided to enable access to further resources at the Mathcentre and the Khan Academy websites.

In the previous section we saw how to add and subtract binary numbers provided the numbers and the corresponding results are non-negative. We now look at how negative numbers are represented by computers and how calculations involving negative numbers are performed.

In this unit we provide a general introduction to number systems and discuss how numbers are represented by computers. We start with a look at the three main systems that occur in computing applications; decimal (base 10), binary (base 2) and hexadecimal (base 16) and describe methods for converting between these three bases. A (very) brief discussion is also presented on conversions involving other bases such as octal (base 8). We then apply the basic techniques we use to add and subtract decimal numbers to enable us to perform these operations manually for binary numbers. The discussion moves on to look at how computers store and represent positive and negative numbers and the concept of signed and unsigned binary numbers is introduced. We present different approaches used by computers for storing numbers with the focus on two’s complement representation. The unit closes with a brief look at a selection of bitwise operators, supported in programming languages such as Java and C, to operate on binary numbers at the bit-level by treating them as strings of bits.

The laws of logic, given in the tables of logical equivalences, provide an alternative method for: proving whether or not compound propositions are logically equivalent. proving a proposition is a tautology, or a contradiction, or neither of these. simplifying compound propositions.

In this unit we present an introduction to propositional logic, a branch of science that is fundamental in the study of mathematics and computer science. The origin of logic dates back to the 3rd century BC and the Greek philosopher Aristotle who developed the earliest form of logical theory through rules for deductive reasoning. Modern mathematical logic is generally recognised as having started with the work of German mathematician Gottfried Leibniz in the 17th century. In the 19th century two English mathematicians, George Boole and Augustus De Morgan, are credited with extending the work of Leibniz and introducing symbolic logic. Other notable contributors to the development of propositional logic include the mathematicians, Gottlob Frege in Germany and Charles Pierce in the USA. The unit begins with a brief overview of some of the terminology that features in propositional logic and the main logical operators (connectives) that are used in the construction of propositions are discussed in detail. Two special types of proposition known as tautologies and contradictions that are respectively always true or false are then described. The concept of logical equivalence is presented before we look at translating propositions from English to their corresponding symbolic form and vice-versa. The idea of a truth table, introduced earlier during the discussion on connectives, is then presented in further detail and we demonstrate how these tables can be used to prove properties such as logical equivalence. We then discuss how logical equivalence can be used to simplify propositions, identify tautologies and contradictions and prove identities. Next we look at how to determine whether a mathematical argument is valid or invalid based on how well the premises support the conclusion. To close the unit we briefly look at the role logic in computing, including simplifying expressions in computer programming and system specification.

In this final unit of the current block we introduce the mathematical concept of relations between sets. We look at how relations differ from functions while noting that a function is actually a special type of relation. Different ways in which relations can be represented are discussed and include ordered pairs, arrow diagrams, matrices and directed graphs. Inverse and composite relations are briefly addressed before we investigate different types of relations (reflexive, symmetric and transitive) and methods for their classification. The important concepts of equivalence relations and equivalence classes are then described along with their properties. In closing the unit we look at the connection between equivalence relations and partitions of sets. The text is supported throughout with relevant examples and where appropriate references for further reading are provided.

In this unit, we look at the concept of a function and introduce some important functions that are fundamental in the study of mathematics and computing. The basic idea of a function is illustrated with a simple example before presenting some more formal definitions and terminology. A function defines a relationship between the elements of two sets and we present different ways to express this relationship including arrow diagrams, formulae, graphs and lists of ordered pairs. We look at how to identify whether a relationship is a function before considering whether functions meet specific criteria that classify them as one-to-one and/or onto functions. The idea of an inverse function is then presented and we illustrate how to calculate an inverse function when it exists and interpret the results graphically. The process of combining two, or more, functions through composition is then discussed. The unit closes with a brief look at some functions that commonly occur in computing and mathematics.

In the sets we have seen up to now the elements are not listed in any particular order. An ordered n-tuple is a list of n elements arranged in a specified order and enclosed in parenthesis rather than curly brackets.

Whether we realise it or not we come across sets, in one form or another, on an almost daily basis. It may be the modules you are studying on your course, or the groceries that you bought in the supermarket last night, or even the teams that qualified for the last 16 of the Champions League in season 2016/17! These are all examples of sets. This unit presents an introduction to sets starting with some basic definitions and an overview of the different ways in which sets are represented. The concept of a subset is introduced and conditions for the equality of sets are given. Operations on sets such as union, intersection and complement are described with the aid of Venn diagrams. We then discuss further set operations including partitions and Cartesian products before briefly considering computer representation of sets. The unit closes with a look at the union and intersection of intervals of the real number line when these intervals are represented as sets.

In this unit we continue with our work on matrices. We describe how to calculate the determinant of a 2 x 2 matrix and introduce the condition for the existence of an inverse matrix. A formula for calculating the inverse of a 2 x 2 matrix is presented supported by examples. Some applications of matrices in the real-world are then given, including solving linear systems of algebraic equations, computer graphics, cryptography and the modelling of graphs and networks.

This unit introduces the theory and application of mathematical structures known as matrices. With the advent of computers matrices have become widely used in the mathematical modelling of practical real-world problems in computing, engineering and business where, for example, there is a need to analyse large data sets.

This unit provides an introduction to vectors. We begin by defining what is meant by the term vector and describe how we distinguish vectors from scalars. The main properties of vectors are presented and the concept of a position vector is introduced. We then look at operations on vectors such as addition, subtraction and scalar multiplication both algebraically and graphically. The idea of a unit vector is introduced and we look at how to express the position vector of a point, in two and three dimensions, in Cartesian components using the standard unit vectors in the directions of the coordinate axes. The unit closes with a look at how to calculate the scalar (dot) product of two vectors.

In this final unit of the current block we introduce the mathematical concept of relations between sets. We look at how relations differ from functions while noting that a function is actually a special types of relation. Different ways in which relations can be represented are discussed and include ordered pairs, arrow diagrams, matrices and directed graphs. Inverse and composite relations are briefly addressed before we investigate different types of relations (reflexive, symmetric and transitive) and methods for their classification. The important concepts of equivalence relations and equivalence classes are then described along with their properties. In closing the unit we look at the connection between equivalence relations and partitions of sets. The text is supported throughout with relevant examples and where appropriate references for further reading are provided.

In this unit, we look at the concept of a function and introduce some important functions that are fundamental in the study of mathematics and computing. The basic idea of a function is illustrated with a simple example before presenting some more formal definitions and terminology. A function defines a relationship between the elements of two sets and we present different ways to express this relationship including arrow diagrams, formulae, graphs and lists of ordered pairs. We look at how to identify whether a relationship is a function before considering whether functions meet specific criteria that classify them as one-to-one and/or onto functions. The idea of an inverse function is then presented and we illustrate how to calculate an inverse function when it exists and interpret the results graphically. The process of combining two, or more, functions through composition is then discussed. The unit closes with a brief look at some functions that commonly occur in computing and mathematics.

The previous unit introduced the term exponent to represent the repeated multiplication of a number by itself. For example, the exponent tells us how many times we need to multiply the number 10 by itself to obtain 1000, i.e. three times as 10 × 10 × 10 = 1000. Here the base is 10 and the exponent is 3. We now consider the closely related topic of what power a number must be raised to in order to obtain another number. The number being raised to the power is called the base and value of the power is called the logarithm.

In this section we introduce the concept of an equation and present techniques for solving different types of equations. We firstly look at the algebraic solution of linear equations in one variable before moving on to simultaneous linear equations and then quadratic equations. In all cases a geometric interpretation is presented along with details on how to graph the relevant functions. At appropriate locations throughout the document links are provided to enable access to further resources at the Mathcentre and the Khan Academy websites.

An important application when working with trees is the ability to search them for data they may hold. In this section we describe two algorithms for searching trees: depth first search (DFS) and breadth first search (BFS). These two algorithms have simple variations for searching digraphs and graphs but these are not followed up here.

The graphs that we have met up to now have all been undirected graphs in the sense that the edges have no orientation. In this section we extend the notion of a graph to include graphs in which “edges have a direction”. These kind of graphs are known as directed graphs, or digraphs for short. As shown in the diagram below the direction of an edge is defined so that movement between two vertices is only possible in the specified direction. The terminology for digraphs is essentially the same as for undirected graphs except that it is commonplace to use the term arc instead of edge. Digraphs can be used to model real-life situations such as flow in pipes, traffic on roads, route maps for airlines and hyperlinks connecting web-pages. We have actually encountered the concept of a digraph before in an earlier unit when we looked at relations on sets. In Section 3.3 of that unit, which was optional, we described how a relation R could be represented diagrammatically by a digraph as an alternative to using an arrow diagram or a matrix.

In recent years graph theory has become established as an important area of mathematics and computer science. The origins of graph theory however can be traced back to Swiss mathematician Leonhard Euler and his work on the Königsberg bridges problem (1735)

In this section we draw on the ideas from elementary number theory that were presented in the last two sections to demonstrate how these methods are applied in the field of cryptography. Some well-known ciphers are introduced and the relevant encryption and decryption processes are described.

As the modulus, m, increases in size it quickly becomes impractical to use multiplication tables or trial and error to find inverses. The Extended Euclidean Algorithm provides a significantly more efficient method for determining the inverse of an integer a modulo m, when it exists. We first show how the Extended Euclidean algorithm can be used to write the GCD of two integers a and m as a linear combination of these integers. If we define d = gcd(a, m) we seek integers x and y such that ax + my = d. In the special case when d = 1 we show how the value of x in the linear combination represents the inverse of a modulo m.

In this unit we introduce some elementary concepts from number theory that are used in many modern ciphers and related security systems. We start with some basic definitions before discussing the division algorithm which lies at the heart of the important Euclidean algorithm. The discussion then moves on to look at prime numbers and describes how prime factorisation can be applied to express any integer, greater than one, as a product of primes. The concept of a greatest common divisor (GCD) of two positive integers is described and we discuss how prime factorisation can be used to calculate this quantity when the numbers are relatively small. We then introduce the Euclidean algorithm which provides an efficient method for calculating the GCD of two integers regardless of their size.

In the previous section we saw how to add and subtract binary numbers provided the numbers and the corresponding results are non-negative. We now look at how negative numbers are represented by computers and how calculations involving negative numbers are performed.

In this unit we provide a general introduction to number systems and discuss how numbers are represented by computers. We start with a look at the three main systems that occur in computing applications; decimal (base 10), binary (base 2) and hexadecimal (base 16) and describe methods for converting between these three bases. A (very) brief discussion is also presented on conversions involving other bases such as octal (base 8). We then apply the basic techniques we use to add and subtract decimal numbers to enable us to perform these operations manually for binary numbers. The discussion moves on to look at how computers store and represent positive and negative numbers and the concept of signed and unsigned binary numbers is introduced. We present different approaches used by computers for storing numbers with the focus on two’s complement representation. The unit closes with a brief look at a selection of bitwise operators, supported in programming languages such as Java and C, to operate on binary numbers at the bit-level by treating them as strings of bits.

The laws of logic, given in the tables of logical equivalences, provide an alternative method for: proving whether or not compound propositions are logically equivalent. proving a proposition is a tautology, or a contradiction, or neither of these. simplifying compound propositions.

In this unit we present an introduction to propositional logic, a branch of science that is fundamental in the study of mathematics and computer science. The origin of logic dates back to the 3rd century BC and the Greek philosopher Aristotle who developed the earliest form of logical theory through rules for deductive reasoning. Modern mathematical logic is generally recognised as having started with the work of German mathematician Gottfried Leibniz in the 17th century. In the 19th century two English mathematicians, George Boole and Augustus De Morgan, are credited with extending the work of Leibniz and introducing symbolic logic. Other notable contributors to the development of propositional logic include the mathematicians, Gottlob Frege in Germany and Charles Pierce in the USA. The unit begins with a brief overview of some of the terminology that features in propositional logic and the main logical operators (connectives) that are used in the construction of propositions are discussed in detail. Two special types of proposition known as tautologies and contradictions that are respectively always true or false are then described. The concept of logical equivalence is presented before we look at translating propositions from English to their corresponding symbolic form and vice-versa. The idea of a truth table, introduced earlier during the discussion on connectives, is then presented in further detail and we demonstrate how these tables can be used to prove properties such as logical equivalence. We then discuss how logical equivalence can be used to simplify propositions, identify tautologies and contradictions and prove identities. Next we look at how to determine whether a mathematical argument is valid or invalid based on how well the premises support the conclusion. To close the unit we briefly look at the role logic in computing, including simplifying expressions in computer programming and system specification.

In the sets we have seen up to now the elements are not listed in any particular order. An ordered n-tuple is a list of n elements arranged in a specified order and enclosed in parenthesis rather than curly brackets.

Whether we realise it or not we come across sets, in one form or another, on an almost daily basis. It may be the modules you are studying on your course, or the groceries that you bought in the supermarket last night, or even the teams that qualified for the last 16 of the Champions League in season 2016/17! These are all examples of sets. This unit presents an introduction to sets starting with some basic definitions and an overview of the different ways in which sets are represented. The concept of a subset is introduced and conditions for the equality of sets are given. Operations on sets such as union, intersection and complement are described with the aid of Venn diagrams. We then discuss further set operations including partitions and Cartesian products before briefly considering computer representation of sets. The unit closes with a look at the union and intersection of intervals of the real number line when these intervals are represented as sets.

In this unit we continue with our work on matrices. We describe how to calculate the determinant of a 2 x 2 matrix and introduce the condition for the existence of an inverse matrix. A formula for calculating the inverse of a 2 x 2 matrix is presented supported by examples. Some applications of matrices in the real-world are then given, including solving linear systems of algebraic equations, computer graphics, cryptography and the modelling of graphs and networks.

This unit introduces the theory and application of mathematical structures known as matrices. With the advent of computers matrices have become widely used in the mathematical modelling of practical real-world problems in computing, engineering and business where, for example, there is a need to analyse large data sets.

This unit provides an introduction to vectors. We begin by defining what is meant by the term vector and describe how we distinguish vectors from scalars. The main properties of vectors are presented and the concept of a position vector is introduced. We then look at operations on vectors such as addition, subtraction and scalar multiplication both algebraically and graphically. The idea of a unit vector is introduced and we look at how to express the position vector of a point, in two and three dimensions, in Cartesian components using the standard unit vectors in the directions of the coordinate axes. The unit closes with a look at how to calculate the scalar (dot) product of two vectors.

This section introduces indices, also known as powers or exponents. Indices provide a shorthand method for representing the repeated multiplication of an expression by itself. A good understanding of indices, and the associated laws of indices, is essential when it comes to applying algebraic manipulation to simplify and solve mathematical expressions and equations.

n this section we introduce the concept of an equation and present techniques for solving different types of equations. We firstly look at the algebraic solution of linear equations in one variable before moving on to simultaneous linear equations and then quadratic equations. In all cases a geometric interpretation is presented along with details on how to graph the relevant functions. At appropriate locations throughout the document links are provided to enable access to further resources at the Mathcentre and the Khan Academy websites.