Introduction to analysis
Introduction to analysis

Start this free course now. Just create an account and sign in. Enrol on the course to track your learning.

Free course

Introduction to analysis

Introduction to analysis

Introduction

Please note: a Statement of Participation is not issued for this course.

This free OpenLearn course, Introduction to analysis, is an extract from the Open University course M208 Pure mathematics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , a second level course that introduces the three main branches of pure mathematics, namely group theory, analysis and linear algebra. Proofs are a vital part of pure mathematics. Students studying M208 are expected to read through a number of proofs to improve understanding of the course material and to develop mathematical skills, including producing convincing arguments and problem solving.

Introduction to analysis consists of material from M208 Unit AA1, Numbers, and has five sections in total. You should set aside between three to four hours to study each of the sections; the whole extract should take about 16 hours to study. The extract is a small part (around 5%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.

This extract looks at real numbers and their properties, with a particular emphasis on inequalities, which play a crucial role in analysis. It is relatively self-contained and should be reasonably easy to understand for someone who has not studied any of the previous texts in the course. However, a few techniques and definitions taught in earlier units in M208 are present in the extract without explanation. 

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the M208 video material (although please note that the PDF may contain references to other parts of M208). In this extract, some illustrations have also been removed due to copyright restrictions.

Regrettably, mathematical and statistical content in PDF form is not accessible using a screenreader, and you may need additional help to read these documents.

Section 1 starts by revising rational numbers and their decimal representations. Then, real numbers are introduced as infinite decimals.

Section 2 looks at rules for manipulating inequalities. You will see how to find the solution set of an inequality.

Section 3 looks at various techniques for proving inequalities.

Section 4 introduces the concept of a least upper bound, which is of great importance in analysis.

Section 5 looks at how least upper bounds can be used to define arithmetic operations on the set of real numbers.

This OpenLearn course is an adapted extract from the Open University course M208 Pure mathematics.

M208_9

Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to University-level study, we offer two introductory routes to our qualifications. You could either choose to start with an Access module, or a module which allows you to count your previous learning towards an Open University qualification. Read our guide on Where to take your learning next for more information.

Not ready for formal University study? Then browse over 1000 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus371