Mathematica 2014

Mathematica 2014

MATHEMATICA 2014

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DCM ATHEMATICA 2014

E DITORS Xiaofeng Xu Leslie Leung

S UPERVISOR Dr Purchase

C OVER D ESIGNED B Y Kenza Wilks

A RTICLE C ONTRIBUTORS

Kaichun Liu

Charlie Sparkes

Mitchell Simmonds

Xiaofeng Xu

Leslie Leung

Ayman D’Souza

Faraz Taheri

Mr Ottewill

Harry Goodhew

Dr Purchase

Kit George

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Table of contents

What is e?

By Kaichun Liu

A-Level Statistics in Investment Banking and Portfolio Theory

By Charlie Sparkes

Mathemagics

By Mitchell Simmonds

How to trap the prisoners?

By Xiaofeng Xu

An Introduction to Modular Arithmetic and its Applications

By Leslie Leung

Proof by Induction or Contradiction

By Ayman D’Souza

Cavalieri’s Principle

By Faraz Taheri

Where’s the policeman?

By Mr Ottewill

Hilbert's 6 th Problem

By Harry Goodhew

BMO question – two equations, three unknowns

By Kit George

How calculators calculate

By Dr Purchase

New Mathematics for an Old Problem

By Mr Ottewill

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What is e? By Kaichun Liu

Before writing something really related to the topic, I would like to draw your attention to π first. What is π? If I ask my little sister who is 12 years old what π is she can definitely tell me that π is a constant and has the value of 3.1415926…… I have no idea how many decimal places she can count to but I am pretty sure it will be enough for calculations. However, what is the definition of π,or what does π actually represent? The question is also easy and I will answer for my little sister – it is a mathematical constant that is the ratio of a circle's circumference to its diameter. All right, what if I replace π with e and ask you the same question? If you can recall its value to the nineth decimal place immediately please text Dr. Purchase and see if he is impressed. However, it is not enough, so far, you have only got 1 mark. WHAT IS THE DEFINATION? WHAT DOES e STAND FOR?!!! To be honest, I myself actually had no idea. Thus I asked help from the mighty Wikipedia. Wikipedia generously provided me with a beautifully organized definition -- The mathematical constant e is the base of the natural logarithm. “ What is natural logarithm then?” The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.

Did you really tell me anything?

So what is e? Let us take a look at this example first.

Let’s assume that a cell will split into two in a day, so mathematically, after a day, we will get two new cells. If we need to write a formulae of how many cells we ݕ ൌ 2 ௫ However, according to cell biology, each 12 hours, which the split is halfway through, there are a number of new cells which equals the total number of the original cells have been produced and have started splitting into next generation. In fact, the growth speeds up as time goes—although the rate of splitting (the ratio) remains at 100%, the base number is continuously growing, which makes will get after x days, that will be

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the 100% arithmetically bigger that the 100% a moment ago.

Thus a day can be divided into two parts, within each part the total number of cells grows by 50%. ݃ ݎ ݋ ݐݓ ݄ ൌ ሺ1 ൅ 100%2 ሻ ଶ ൌ 2.25 In fact, if there are 100 cells at first, after 24 hours, we can get 225 cells. If we can chage the rule of splitting and make it “producing 1/3 of the cells after 8 hours ” The formulae would be ݃ ݎ ݋ ݐݓ ݄ ൌ ሺ1 ൅ 100%3 ሻ ଷ ൌ 2.37037 … If we make the split carry on forever and continuesly, the formulae would be ݃ ݎ ݋ ݐݓ ݄ ൌ lim ௡՜ஶ ൬1 ൅ 100%݊ ൰ ௡ ൌ 2.718281828 “Coincidently”,the number we get here just equals the “e” When the rate of increasing stays at 100%, within a unit of time the cells can only increase to 271.8% In fact e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and moreover it represents the idea that all continually growing systems are scaled versions of a common rate. e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more.

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A-Level Statistics in Investment Banking and Portfolio Theory By Charlie Sparkes Investment banking is often made out to be like an impossible, unpredictable science that only those audacious enough to wear pin striped suits can ever fathom. However, as you may know, a crucial part of an investment banker’s day to day regime requires a lot of pondering over the magical relationship between risk and return and how to optimize the performance of a portfolio based on past and predicted >Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51

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