# Maths at Bondi Beach

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Mathematics without the mystery. Try it for yourself!

•  Some years ago I picked up a standard high school textbook and randomly opened a page on trigonometry. There were many problems about ladders against walls. Nowhere could I find the one thing that for me is the hallmark of mathematical thinking - imagination.
• As part of an on-going experiment in maths education I am releasing extensive theoretical material coupled with problems and their solutions for interested teachers and students. The material is pitched at students who are curious about mathematics, yet perhaps fearful thanks to some teacher in an ill-fitting cardigan who humiliated them about quadratic equations. However, after a taste of some of its wonder, I hope that you might become passionate about one of the most sublime pleasures available.
• I was galvanised into doing this because of the parlous state of mathematics education at secondary and tertiary levels. Deeper laments can be found at http://www.maa.org/devlin/, in particular Lockhart’s Lament. The full story behind my concerns can be found here: http://www.review.ms.unimelb.edu.au/Forum.html
• I am not alone in my concerns.  The maths community in the United States spends an enormous amount of its time addressing the quality of content and new ways of delivering material.  Anyone seriously interested in maths should join the Mathematical Associaton of America http://www.maa.org
• The theoretical material is meant to extend a student's range of knowledge and excite interest.  Some of the material may appear too difficult and that may well be the case, but there is a benefit for students in going with the flow and seeing how the bits fit together even if some of the detail eludes them. I have gone to great effort to put all the reasoning in so there aren’t any major gaps.
• The problems are designed to establish a number of basic competencies and there is a range of difficulty.
• By way of overview I have taken the most "boring" mathematical proof technique in the universe (ie mathematical induction) and shown how it figures in a wide range of significant problems. Feynman's path integral approach which he used to investigate quantum mechanical behaviour actually involved induction (admittedly in a minor way) and I show how it was used in the context of explaining why light travels in a straight line.  Along the way the basic physics are also explained.  Other applications in logic, probability theory and much more are also explained.  If you want to know whether you can prove by induction if it is possible to escape the Gulag of chartered accountancy, there is a problem on the very subject. It may come in handy depending on your career choice!
• Even if you aren't interested in the theoretical material and you just want to do problems I will not be offended.  There are some interesting problems which bring together a range of different mathematical techniques which might assist in developing mathematical insights.  My proof of the volume of an n-dimensional ball using only elementary calculus and induction is an example of what can be done.
• This experiment is designed to get students interested in doing maths even though they may not have any intention of pursuing a highly mathematical career option. Because maths is an enabling discipline, the intellectual reach of other disciplines can be limited by lack of mathematical knowledge. Watson and Crick (and Ms Franklin whose role in the discovery also deserves recognition) used Fourier Theory to determine the structure of DNA.  They were molecular biologists/biochemists.
• I have made every attempt to ensure that the material is error free but in something of this scale errors will arise.  If you identify errors please email me with details via the contact page.
• To pique your interest in probability let’s start with Laplace’s famous problem concerning the rising of the sun.  If the sun has risen on n days in the past what is the probability it will rise tomorrow? Laplace’s perhaps counterintuitive answer was 2/3 notwithstanding that he assumed that the probability of  the sun rising on any particular day was ½.  The young Kai Lai Chung was puzzled by this and developed a combinatorial proof for this miraculous result.  Probability is full of such conundrums.  Kai Lai Chung went on to be a famous probability theorist.
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1.    An individual may download the material on this site for personal educational or research use.  You are not authorised to place the item on your own website or the website of any other entity.  If you want to provide access to an article, provide a link to this website.

2.    If you are an educational institution you  may use the content on this website for free PROVIDED THAT:

(a)  The number of students in any one course using the content does not exceed 50 (fifty).  You MUST acknowledge the source of the material by providing the content’s URL on this site.

(b)  If the number of students referred to in (a) exceeds 50, you MUST email me at mathsatbondibeach@gmail.com and seek specific permission.

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