Goldsmiths’ 2014 Mathematics

http://www.thegoldsmiths.co.uk/charity-education/education/science-for-society-courses/mathematics/

Monday 21st July

Introduction

This course took place in the mathematical sciences building of Queen Mary University of London. It looked at the mathematics behind applications as diverse as cryptography, drug testing and the internet. The course included a mix of lectures and workshops, with an introduction to mathematical computing using the Maple package and there was a visit to the National Codes Centre at Bletchley Park. The course was aimed at A level mathematics teachers but I went as I was interested in what mathematicians get up to and of course because mathematics is a necessary tool in physics.

http://www.qmul.ac.uk/

http://www.maths.qmul.ac.uk/

http://en.wikipedia.org/wiki/Queen_Mary_University_of_London

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The above left image shows the Queen’s building and the above right building shows the mathematical sciences building. The tiling pattern on the mathematical sciences building is Penrose tiling.

http://en.wikipedia.org/wiki/Penrose_tiling

A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry.

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I didn’t realise that maths could be so pretty

The course was coordinated by Dr Francis Wright, Reader in Mathematics and Director of Undergraduate Studies.

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I would like to thank Dr Wright and his team for their patience with me on this course. I would like to reassure all my past, present and future students (and their parents/carers) that my mathematical knowledge is more than adequate to teach GCSE maths and the maths components of A level physics but in this course I either met stuff for the first time or stuff I hadn’t seen for over thirty years so I asked a lot of “silly” questions (I always tell my students there is no such thing as silly questions).

This is also a lesson to my students that if you don’t use something you will lose it. So they need to continually go over their work in physics to remember and understand it (as Lord Rutherford once said “everything else is just stamp collecting”).

http://www.maths.qmul.ac.uk/~fjw/goldsmiths/2014/

Mathematical computing

Dr Frances Wright

http://www.maplesoft.com/

The program we used was Maple and they do have a free version which you can download. I was rather surprised that they use programs like this. I had this vision of mad maths professors writing things on blackboards. Pure mathematicians do prefer this method but they use white boards now, oh and they aren’t mad.

Maple is a computer algebra system with exact symbolic data, graphics and numbers with lots of digits.

Examples of exact symbolic algebra where you input the formula using palettes, templates and assignment (angles are always written radians and as fractions of p) can be seen below

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The radian is the standard unit of angular measure, used in many areas of mathematics. An angle’s measurement in radians is numerically equal to the length of a corresponding arc of a unit circle, so one radian is just less than 57.3 degrees (when the arc length is equal to the radius).

http://en.wikipedia.org/wiki/Radian

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http://www.mathsisfun.com/geometry/radians.html

An example of an exact number with lots of digits is 100! And an example of an approximate number with lots of digits is p (Maple uses p as 3.142 etc. is not exact enough).

An example of approximate to exact is

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Why is?

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Pythagoras’ Theorem

Why is

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At GCSE and at A level students are expected to know Pythagoras theorem and do calculations with their calculator.

http://en.wikipedia.org/wiki/Pythagorean_theorem

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The Pythagorean Theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

An example of graphics

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….. and you will get an image of a Mobius strip

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The Möbius strip or Möbius band, also Mobius or Moebius, is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.

http://en.wikipedia.org/wiki/M%C3%B6bius_strip

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August Ferdinand Möbius (above left) (17 November 1790 – 26 September 1868; German pronunciation was a German mathematician and theoretical astronomer. Johann Benedict Listing (above right) (25 July 1808 – 24 December 1882) was a German mathematician.

http://en.wikipedia.org/wiki/August_Ferdinand_M%C3%B6bius

http://en.wikipedia.org/wiki/Johann_Benedict_Listing

Working with Polynomials

Keyboard entry

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Factorisation: What is the factorisation of

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Answer

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The factorisation of big integers is the basis of internet security

http://en.wikipedia.org/wiki/Integer_factorization

Algorithmic factorisation – the first step is square-free factorisation. A polynomial is square-free if it has no repeated factors.

In mathematics, a polynomial is an expression consisting of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

http://en.wikipedia.org/wiki/Polynomial

Let’s aim to factorise

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The tools used can be polynomial differentiation, finding the polynomial’s greatest common divisor (GCD), using polynomial remainders, computing the square-free factor of the polynomial and using full factorisation.

http://en.wikipedia.org/wiki/Greatest_common_divisor

The answer is

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Polynomial graphics

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Can polynomials have corners?

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The above centre image makes it look like you can, but it isn’t a proper corner as you can see in the above right image.

Can you turn a circle into a square?

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The computer says no!

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. In 1882, the task was proven to be impossible.

http://en.wikipedia.org/wiki/Squaring_the_circle

Maths Apps and the Mobius Projects

Built-in Maths Apps,

http://www.maplesoft.com/support/help/Maple/view.aspx?path=MathApps/Guide

e.g Superellipse

http://www.maplesoft.com/support/help/Maple/view.aspx?path=MathApps/Superellipse

http://www.maplesoft.com/support/help/Maple/view.aspx?path=MathApps/CellularAutomata

Geometric Series (Maple 17/Maple 18)

http://www.maplesoft.com/support/help/Maple/view.aspx?path=MathApps/Series

http://www.maplesoft.com/support/help/Maple/view.aspx?path=MathApps/GeometricSeries

Other Computer Algebra Systems

Maple and Mathematica – widely used in universities (Maple costs £940 for academic institutions or £95 for a student)

TI-Nspire for calculators, ipad, Windows/Mac computers (software for computers costs £80 for a single user)

REDUCE – http://reduce-algebra.sourceforge.net/

Maxima – http://maxima.sourceforge.net/

Axiom – http://open-axiom.sourceforge.net/

MathPiper – http://www.mathpiper.org/

Microsoft Mathematics 4.0 –

http://www.microsoft.com/en-us/download/details.aspx?id=15702

Microsoft Mathematics Add-in 2013 for Word and OneNote

http://www.microsoft.com/en-us/download/details.aspx?id=36777

I was a bit concerned that using computers would stop mathematics students focusing on what the mathematics means but I was assured that it allows them to focus on the mathematics.

In the afternoon I had a go at using some of the applications to work out some problems. By clicking on the following links you may be able to have a go too.

http://www.maths.qmul.ac.uk/~fjw/goldsmiths/2014/FrancisWright/Mathematical_Computing_Workshop.mw

http://www.maths.qmul.ac.uk/~fjw/goldsmiths/2014/FrancisWright/Mathematical_Computing_Workshop.pdf

http://www.maths.qmul.ac.uk/~fjw/goldsmiths/2014/FrancisWright/The%20M%c3%b6bius%20Project.html

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