Category Archives: Analysis

Analysis

0.\bar{9} = 1.0

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Lara Alcock wrote in her book "How to Study for a Mathematics Degree" a warning to all mathematics students to be prepared to adjust their intuition.

Very quickly after starting with "A First Course in Mathematical Analysis" by David Brannan (see my earlier post), I have come across something that very much breaks my intuition. It turns out that 0.\bar{9}=1.0.

 

    \[Let\: x = 0.\bar{9}\]

    \[10x=9.\bar{9}\]

    \[10x = 9 + x\]

    \[10x-x = 9\]

    \[9x = 9\]

    \[x=1=0.\bar{9}\]

Simple, and surprising. Although also somewhat disturbing.

The equals sign tends to show that the object on one side is identical to the object on the other. In this case, they look like two completely different objects! We all know what "1" is, and 0.\bar{9} looks distinctly different.

Of course, based on Lara Alcock's advice, my first thought was "how on earth do I adjust my intuition to make this feel completely logical?!"

After some thinking, the following helped me a little bit: Before, I considered 0.\bar{9} to be a number that was infinitely close to 1, due to the infinity of "9"s after the decimal point. I now, however, consider 0.\bar{9} to be an infinitely accurate representation of 1.

For me, if I consider an approximation to be "infinitely accurate", then that approximation is the object it's trying to approximate. ie: An infinitely accurate approximation of 1 is, essentially, 1.

My opinion of this may change, but this is what's helping me make sense of this for the moment....

Analysis, Books, Maths, Text Books

The Maths Club

After my first year, I had a look at the clubs and societies you could join as a member of the OU. I thought it would help me feel a little less like I was learning so remotely. Sure enough, there was a mathematics club, M500.

A nice little bonus that you get for being a member is a discount for the bi-annual revision sessions that are held by the mathematics faculty at the OU. So when I started my last module, I signed up for a revision weekend that was arranged to be held about a month before my exam.

It was great meeting other students. With all that remote learning, it was great to be reminded that you weren't alone. We even had nightly trips to the pub! Just like a physical university!!!

Towards the end of the weekend, one of my fellow students, studying the same module as me, mentioned a book that he'd recently ordered.  It was written by David Brannan, a man apparently partially responsible for the syllabus of our first double-credit module in our second stage, and wrote the book to reflect the content of that module. My classmate thought he buy it to give him a bit of an edge when he finally started the monster-module. In our class he gave everyone the details. The book is called A First Course in Mathematical Analysis, and at over 450 pages, it looks like one serious introduction!

aFirstCourseInMathematicalAnalysis

Finding out about this book couldn't have been any more perfect. It means I can start my easy introductory module this year, and read through my new book on analysis so I'm ahead of the game for when my double-credit module starts in 2016! Perfecto.