Book Review - Numbers And Functions, R. P. Burn

This is the first of my book reviews, aimed at current undergraduate mathematicians, especially those at Warwick University.

For my first review, I will be looking at one of the recommended books (nb. link only works for Warwick Mathematicians) for Warwick's 1st year Analysis I & II course - Numbers and Functions: Steps to Analysis by R. P. Burn.

This book covers all areas of basic analysis and presents them in a problems-and-answers manner. That is to say, the majority of this book is written as brief introductory paragraphs followed by a number of questions which both test understanding of concepts, and produce methods of proving many of the Theorems the book gives. All problems have at least outline solutions, so if you get stuck there are always hints to push you in the right direction.

One of the major advantages of this book, at least for me in my first year, was that I felt that I was teaching myself these topics. As any first year student knows, lectures can be initially very daunting, and most people will find that if they are unable to understand one topic from the lecture, very quickly they will begin to slow down and lose understanding of the following concepts. The only way to remedy this is work outside of a lecture, and for me at least this did not happen. This book really saved me come revision time, and I used it to essentially learn the majority of the course at my own pace.

That is not to say that this book can replace an entire Analysis course - I would suggest you do rather more work than I did, especially if you have problems - but it is a very strong learning aid. It gives problems which are more engineered to gaining a greater understanding of a concept than a typical assignment sheet is, and will help you to not only remember the concepts, but to understand them more fully.

Come revision time, each chapter is finished off with a summary of the definitions and results from the chapter, and has question number references to help you pinpoint the area in which it was proven or used. The historical notes at each chapter are nice, but not really of much use (although they can be a welcome break after looking over 70-odd questions!).

On to content, the book is split into two main sections "Numbers" and, unsurprisingly, "Functions." It all starts very simply, harking back to A-level maths, with a more "mathsy" look at Induction and Inequalities. These give a nice slow introduction to some of the rigour of mathematics, and provide the reader with a real bridge to university-level maths. Sequences, Series and Completeness finish the Numbers section (totalling 131 pages and 352 questions!) and provide detailed looks at many concepts and theorems such as the Fundamental Theorem of Arithmetic, the Bolzano-Weierstrass Theorem and Power Series.

The second half of the book - "Functions" - concerns itself with continuity, completeness, calculus and function sequences. Weighing in at almost 200 pages and 417 questions, the majority of the book lies here. There are far too many concepts to really be mentioned here, but they are again presented in a brilliant style in which the reader really teaches themselves these concepts.

This book really, in my opinion, is the leading book for a first year analysis course, and outshines Guide to Analysis (Macmillan Mathematical Guides) by Mary Hart, and was truly a lifesaver for me saving me having to retake my Analysis exams.

This book is available from Amazon.co.uk from the link below - enjoy!

OpenOffice.org - For University Students

So, I’m going to give a brief overview of the OpenOffice software – a free open source office suite for Linux, Windows and MacOSX. This review is aimed at University students.

I first used Open Office in the Linux room of the maths department of my university in my first year, and at the time, although it was adequate for reading and editing Word docs, I found it was underpowered compared to the MS office suite I had back at home. Unfortunately, I didn't know at the time that the software was months out of date.

Last week, I saw a news story on one of my RSS feeds about how open office is now supporting MS VBA – visual basic for applications, a language for writing macros, which I use most of the time I use Excel – and decided to give it another try. Unfortunately, I didn’t fancy the Novell branded version (a full ~500Mb ISO image), and didn’t want to compile my own. However, I was told by a friend that OOo has its own OOo Basic language, similar to VBA, and there are only a few additions to the new version to make it entirely compatible.

So, what does OOo consist of? You are provided with the standard office features – word processor, spreadsheet, presentation designer and database – along with a couple of extra bits (a math design mode, like MS Equation Editor, and a drawing program similar to Visio). Everything is saved in Open Document Format normally, but things can be saved in Microsoft or Lotus formats, PDF and even as LaTeX source.

So going back to my experience, I downloaded Open Office before creating a spreadsheet for some simple programming (calculating a collection of rational approximations to e) since my copy of MS Office is starting to play up these days (strange how we just accept the fact that as computers get older they start breaking, even if they’re exactly the same as before… That’s for another day though). I installed the software (a 93 Mb download for Windows – tiny really), and jumped straight into Calc – the Open Office spreadsheet.

Moving around Calc felt immediately familiar, with all control buttons in a similar place to in MS Excel. Going straight to the Tools menu, I found the Macros… option, and chose to organise my macros in the Open Office BASIC window. Finding where to put my macro wasn’t immediately obvious – the layout is slightly different to in Excel – but was really obvious enough. Most everything in OOoB is the same as VBA, however there are a few significant differences. You cannot just call “Cells(x,y)” to read or write, you have to perform a rather longer set of commands. Bear in mind, however, that OOoB can be used in any of the Office tools, not just calc. Writing my macro was as easy as in VBA really, and the Help menu is decent (better PDF documentation is available from the OOo website). One issue I did have was pressing the play button in the IDE to run the macro didn’t work, but this turned out to be my fault for not reading the documentation – I simply needed to save somewhere else or run from Calc instead. Finally, on macros, OOo also appears to support Python and something else – unfortunately this is not really my area so I can’t go any futher on that.

So, what about the rest of Calc? Well, it seems to provide all the usability of Excel and then some, from all the in-cell functions to being able to create simple commands with words (eg “profits – losses” given correct titles – this is like Excel for Dummies). Obviously, these for Dummies-esque bits do not really get in the way of everything else, and can be switched off in a jiffy.

The 2 random bits – Math and Draw – work well at what they do, creating Math equations (albeit not with quite as easy to use a GUI as equation editor, although a lot quicker) or creating Hierachical diagrams or even logos quite easily. I have to say though, I will not be using these a lot – if I need to write maths I use LaTeX, if I need to draw things, I’ll use the GIMP.

Writer (the word processor) is powerful and fast, with everything you could need (like mail merge and autocorrect) and some stuff most people would never touch (Python Macros anyone?). It’s a similar story with Impress (Powerpoint) and Base (Access). In fact, being like MS Office and then some seems to be a recurring theme throughout OOo, and with the ability to export to PDF being a big draw to many people (given that the only other way would be through Adobe Acrobat Standard, costing extra money over MS Office), I really feel this is something which could potentially replace MS Office in the home and workplace.

So, in conclusion:

Get OpenOffice.org!