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Number systems and the rules for combining numbers can be daunting. This unit will help you to understand the detail of rational and real numbers, complex numbers and integers. You will also be introduced to modular arithmetic and the concept of a relation between elements of a set.<link rel="canonical" href="http://www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/number-systems/content-section-0" /> First published on Thu, 07 Apr 2011 as <a href="http://www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/number-systems/content-section-0">Number systems</a>. To find out more visit The Open University's <a href="http://www.open.edu/openlearn/ole-home-page">Openlearn</a> website. Creative-Commons 2011
Number systems and the rules for combining numbers can be daunting. This free course, Number systems, will help you to understand the detail of rational and real numbers, complex numbers and integers. You will also be introduced to modular arithmetic and the concept of a relation between elements of a set.<link rel="canonical" href="http://www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/number-systems/content-section-0" /> First published on Thu, 07 Apr 2011 as <a href="http://www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/number-systems/content-section-0">Number systems</a>. To find out more visit The Open University's <a href="http://www.open.edu/openlearn/ole-home-page">Openlearn</a> website. Creative-Commons 2011
<p>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 inverses in the system. We look at solving certain equations in the system, such as linear, quadratic and other polynomial equations.</p><p>In Section 1 we start by revising the notation used for the <b>rational numbers</b> and the <b>real numbers</b>, 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 <i>x</i><sup>2</sup> = 2, have solutions which are real but not rational.</p><p>In Section 2 we introduce the set of <b>complex numbers</b>. This system of numbers enables us to solve <i>all</i> polynomial equations, including those with no real solutions, such as <i>x</i><sup>2</sup> + 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 <b>complex plane</b>.</p><p>In Section 3 we look further at some properties of the <b>integers</b>, and introduce <b>modular arithmetic</b>. This will be useful in the group theory units, as some sets of numbers with the operation of modular addition or modular multiplication form <b>groups</b>.</p><p>In Section 4 we introduce the concept of a <b>relation</b> 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 <b>equivalence relation</b>. An equivalence relation on a set classifies elements of the set, separating them into disjoint subsets called <b>equivalence classes</b>.</p><p>This unit is an adapted extract from the Open University course <span class="oucontent-linkwithtip"><a class="oucontent-hyperlink" href="http://www3.open.ac.uk/study/undergraduate/course/m208.htm"><i>Pure mathematics
</i>(M208)</a></span></p>
<p>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.</p><p>In Section 1 we start by revising the notation used for the <b>rational numbers</b> and the <b>real numbers</b>, 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 <i>x</i><sup>2</sup> = 2, have solutions which are real but not rational.</p><p>In Section 2 we introduce the set of <b>complex numbers</b>. This system of numbers enables us to solve <i>all</i> polynomial equations, including those with no real solutions, such as <i>x</i><sup>2</sup> + 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 <b>complex plane</b>.</p><p>In Section 3 we look further at some properties of the <b>integers</b>, and introduce <b>modular arithmetic</b>. This will be useful in the group theory units, as some sets of numbers with the operation of modular addition or modular multiplication form <b>groups</b>.</p><p>In Section 4 we introduce the concept of a <b>relation</b> 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 <b>equivalence relation</b>. An equivalence relation on a set classifies elements of the set, separating them into disjoint subsets called <b>equivalence classes</b>.</p><p>This OpenLearn course is an adapted extract from the Open Unviersity course <span class="oucontent-linkwithtip"><a class="oucontent-hyperlink" href="http://www3.open.ac.uk/study/undergraduate/course/m208.htm?LKCAMPAIGN=ebook_&MEDIA=ou">M208: <i>Pure Mathematics</i></a></span></p>