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Number systems

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M208_6Pure mathematics
Number systems

About this free course

This free course is an adapted extract from the Open Unviersity course M208: Pure Mathematics www3.open.ac.uk/study/undergraduate/course/m208.htm

This version of the content may include video, images and interactive content that may not be optimised for your device.

You can experience this free course as it was originally designed on OpenLearn, the home of free learning from The Open University: www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/number-systems/content-section-0.

There youll also be able to track your progress via your activity record, which you can use to demonstrate your learning.

The Open University, Walton Hall, Milton Keynes, MK7 6AA

Copyright 2016 The Open University

Intellectual property

Unless otherwise stated, this resource is released under the terms of the Creative Commons Licence v4.0 http://creativecommons.org/licenses/by-nc-sa/4.0/deed.en_GB. Within that The Open University interprets this licence in the following way: www.open.edu/openlearn/about-openlearn/frequently-asked-questions-on-openlearn. Copyright and rights falling outside the terms of the Creative Commons Licence are retained or controlled by The Open University. Please read the full text before using any of the content.

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978-1-4730-1383-4 (.kdl)
978-1-4730-0615-7 (.epub)

Contents
Introduction

In this course 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 inverses in the system. We look at solving certain equations in the system, such as linear, quadratic and other polynomial equations.

In Section 1 we start by revising the notation used for the rational numbers and the real numbers, and we list their arithmetical properties. You will meet other properties of these numbers in the analysis units, as the study of real functions depends on properties of the real numbers. We note that some quadratic equations with rational coefficients, such as x2 = 2, have solutions which are real but not rational.

In Section 2 we introduce the set of complex numbers. This system of numbers enables us to solve all polynomial equations, including those with no real solutions, such as x2 + 1 = 0. Just as real numbers correspond to points on the real line, so complex numbers correspond to points in a plane, known as the complex plane.

In Section 3 we look further at some properties of the integers, and introduce modular arithmetic. This will be useful in the group theory units, as some sets of numbers with the operation of modular addition or modular multiplication form groups.

In Section 4 we introduce the concept of a relation between elements of a set. This is a more general idea than that of a function, and leads us to a mathematical structure known as an equivalence relation. An equivalence relation on a set classifies elements of the set, separating them into disjoint subsets called equivalence classes.

This OpenLearn course is an adapted extract from the Open Unviersity course M208: Pure Mathematics

Learning outcomes

After studying this course, you should be able to:

  • understand the arithmetical properties of the rational and real numbers
  • understand the definition of a complex number
  • perform arithmetical operations with complex numbers
  • explain the terms modular addition and modular multiplication
  • explain the meanings of a relation defined on a set, an equivalent relation and a partition of a set.
1 Real numbers
1.1 Rational numbers

In OpenLearn course M208_5 Mathematical language you met the sets

  • Picture 1 = {1, 2, 3, }, the natural numbers;

  • Picture 2 = {, 2, 1, 0, 1, 2, }, the integers;

  • Picture 3 = {p/q : pPicture 4Picture 5 , qPicture 6Picture 7 }, the rational numbers.

Picture 8 is the set of numbers that can be written as fractions, such as Number systems - image 9 and 1.7 = Number systems - image 10 .

Notice that each set in this list is a subset of the succeeding one; that is,

Number systems - image 11

To represent the rational numbers geometrically, we use a number line. We begin by drawing a line and marking on it points corresponding to the integers 0 and 1. If the distance between 0 and 1 is taken as a unit of length, then the rational numbers can be arranged on the line in a natural order. For example, the rational 4/3 is placed at the point which is one third of the distance from 1 to 2.

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