Items where Author is "Scott, Leonard"

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.

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.

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.