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Topic outline

 

  • Time: 20 hours
    Level: Intermediate

 
 

Introduction

  • Introduction Resource
  • In this unit we look at some different systems of numbers, and the rules for combining numbers in these systems. For each system we consider the question of which elements have additive and/or multiplicative...
 

1 Real numbers

 

2 Complex numbers

  • 2.1 What is a complex number? Resource
  • We will now discuss complex numbers and their properties. We will show how they can be represented as points in the plane and state the Fundamental Theorem of Algebra: that any polynomial equation with...
  • 2.2 The complex plane Resource
  • Just as there is a one-one correspondence between the real numbers and the points on the real line, so there is a one-one correspondence between the complex numbers and the points in the plane. This correspondence...
  • 2.3 Complex arithmetic Resource
  • Arithmetical operations on complex numbers are carried out as for real numbers, except that we replace i2 by −1 wherever it occurs.
  • 2.4 Complex conjugate Resource
  • Many manipulations involving complex numbers, such as division, can be simplified by using the idea of a complex conjugate, which we now introduce.
  • 2.5 Modulus of a complex number Resource
  • We also need the idea of the modulus of a complex number. Recall that the modulus of a real number x is defined by
  • 2.6 Division of complex numbers Resource
  • The second of the conjugate–modulus properties enables us to find reciprocals of complex numbers and to divide one complex number by another, as shown in the next example. As for real numbers, we cannot...
  • 2.7 Arithmetical properties of complex numbers Resource
  • The set of complex numbers satisfies all the properties previously given for arithmetic in . We state (but do not prove) these properties here.
  • 2.8 Polar form Resource
  • You have seen that the complex number x + iy corresponds to the point (x, y) in the complex plane. This correspondence enables us to give an alternative description of complex numbers, using so-called...
  • 2.9 Roots of polynomials Resource
  • We begin by reminding you of what we mean by the word ‘root’. In this unit we use this term in two different, but related, senses, as given below.
  • 2.10 The complex exponential function Resource
  • Consider the real exponential function f (x) = ex (that is, f (x) = exp x). We now extend the definition of this function to define a function f(z) = ez whose domain and codomain are .
  • 2.11 Further exercises Resource
  • Let z1 = 2 + 3i and z2 = 1 − 4i. Find z1 + z2, z1 − z2, z1z2, , , z1/z2 and 1/z1.
 

3 Modular arithmetic

  • 3.1 Division Resource
  • In this section, instead of enlarging the number system , we do arithmetic with finite sets of integers, namely the sets of possible remainders when we divide by particular positive integers. This type...
  • 3.2 Congruence Resource
  • The Division Algorithm tells us that, when we divide any integer by a positive integer n, the set of possible remainders is {0, 1, 2, …, n − 1}. Integers which differ by a multiple of n have the same remainder...
  • 3.3 Operations in modular arithmetic Resource
  • The Division Algorithm tells us that all the possible remainders on division by an integer n lie in the set
  • 3.4 Modular multiplication Resource
  • In the last subsection we stated that, for any integer n ≥ 2, the set n satisfies the same rules for addition modulo n as the real numbers satisfy for ordinary addition. When it comes to multiplication...
  • 3.5 Further exercises Resource
  • Evaluate the following sums and products in modular arithmetic.
 

4 Equivalence relations

  • 4.1 What is a relation? Resource
  • In this final section we look at a method of classifying the elements of a set by sorting them into subsets. We shall require that the set is sorted into disjoint subsets – so each element of the set belongs...
  • 4.2 Equivalence relations Resource
  • Our formal definition of an equivalence relation involves three key properties. A relation that has these three properties partitions the set on which the relation is defined, as we show later in this...
  • 4.3 Further exercises Resource
  • Let be the relation defined on by
 

References and Acknowledgements

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