DC Mathematica 2016

DC Mathematica 2016

DC M ATHEMATICA 2016

E DITORS

Minghao Zhang

Jack Kurtulus

Charles Cheung

Arthur Cheung

Johnnie Baggs

S UPERVISOR

Dr Purchase

C OVER D ESIGNED B Y

Tate Sun

A RTICLE C ONTRIBUTORS

Arthur Cheung

Jay Connor

Toby Evans

Harry Goodwin

Thomas Kuijlaars

Joseph Lane

Theo Macklin

Park Jun Sang

Shawn Shen

Stanley Traynor

Minghao Zhang

Zhengyuan Zhu

Table of Contents

The Sum of All Natural Numbers? ................................................................ 1 By Theo Macklin (Yr 11)

Globalization and Cointegration – Study of Stock Markets in China and USA ................................................................................................................. 5 By Zhengyuan Zhu (Yr 13)

The Mathematics inside the optical fibre .................................................... 14 By Shawn Shen (Yr 13)

The Fourth Dimension and Those Above It................................................ 23 By Stanley Traynor (Yr 7)

Marble on a turntable ................................................................................... 25 By Minghao Zhang (Yr 12)

Mr Ottewill - How he won a hundred pounds with ease............................. 27 By Arthur Cheung (Yr 12)

Pythagoras and His Cult .............................................................................. 28 By Park Jun San (Yr 9)

Sir Roger Penrose, mathematician and physicist ........................................ 31 By Joseph Lane (Yr 8)

Maths Problems from History...................................................................... 33 By Harry Godwin (Yr 10)

Much Ado about Nothing: the History of Zero ........................................... 35 By Jay Connor (Yr 8)

Phi - not to be confused with pie!................................................................. 37 By Thomas Kuijlaars (Yr 9)

Square Numbers – I bet you didn’t know this! ............................................ 38 By Toby Evans (Yr 8)

The Sum of All Natural Numbers?

1 + 2 + 3 + 4 + 5 … = −1

12⁄

By Theo Macklin (Y11)

This succinct, self-confident statement is a reason to hate Euler. Unfortunately, the maths that leads us to this arrogant but strangely compelling summation as simple as it comes. The idea that the infinite sum of positive whole numbers should give a negative fraction is, frankly, unimaginable; however, once again, the sly, mischievous talons of infinity delight in ruining the fundamental bedrock of nursery addition. Despite this initital mental impasse, there are in fact two ways to prove that the sum of positive integers does reach that ridiculous -1/12. There is the scary sounding proof by Leonhard Euler that utilises zeta function regularisation and the more accessible proof by Srinivasa Ramanujan. It should be obvious which one I intend to start with.

Ramanujan’s Proof:

Ramanujan’s proof requires two realisations: these require us to utilise algebra to represent the constants of series, let’s call them C, C 1 and C 2 . The first is such:

 1 = 1 − 1 + 1 − 1 + 1 − 1 + 1 … = 1 2⁄

This innocent equation is the premise of this proof and while it seems initially unintuitive, on further inspection it makes logical sense. Where n is odd in this series the value of the sequence up to that point is 1. Conversely, where n is even the value is 0. Since ∞ ∉ 𝑂 ∪ 
the logical value for the sequence at the infinite point is the average of the two: 1 2⁄ . The next stage tackles another new sequence with the knowledge gained from the last. Here the sequence C 2 is used:

 2

= 1 − 2 + 3 − 4 + 5 …

2 2 = (1 − 2 + 3 − 4 + 5 … ) + (1 − 2 + 3 − 4 + 5 … ) = 1 − 1 + 1 − 1 + 1

2 2

=  1

= 1

2⁄

1

 2 = 1 4⁄ This sequence also causes unease with a strange blend of positive and negative integers culminating in a fraction. Fortunately, this brings us to a position from which we can prove our titular summation:

 = 1 + 2 + 3 + 4 + 5 …

 −  2

= (1 + 2 + 3 + 4 + 5 … ) − (1 − 2 + 3 − 4 + 5 … ) = 4 + 8 + 12 +

16 + 20 …

 −  2

= 4 =  − 1

4⁄

3 = −1

4⁄

∴  = −1

12⁄

And there we have it: a concise and intuitive proof of what is a deeply unintuitive sum. For all that work, however, something feels wrong: incomplete. This foundation-rocking piece of mathematics was all so simple! Surely there must be more to it that that; I, like that part of you deep inside, crave intelligible squiggles littering the page. Fear not, however, for I mentioned Euler’s proof: a forest of differentiation, sigmas and Riemann-Zeta functions. Despite this, the mathematics itself is quite rational and very satisfying.

Euler’s Proof:

Euler’s proof begins with a seemingly unrelated series and the proof of its constant, K :

 = 1 +  +  2 +  3 …

 + 1 = 1 +  +  2 +  3 … = 

 −  = 1 = (1 − )

∴  = 1 +  +  2 +  3 … = 1

;  < 1

(1 − ) ⁄

Following this, we differentiate K:

𝑥 = 1 + 2 + 3 2 + 4 3 … = 1

(1 − ) 2 ⁄

And let  = −1 :

1 − 2 + 3 − 4 + 5 … = 1

2 2 ⁄ = 1

4⁄ =  2

2

Straight away we can see a similarity to Ramanujan’s method in Euler’s except he has reached his equivalent of C 2 via stronger mathematical logic. Reaching this point I feel a quick pause is in order while we quickly go over the Riemann-Zeta function. This is a function that was first introduced by Euler and then generalised by Bernhard Riemann. It behaves as follows:

∞

3
⁄ … = 1 −
+ 2 −
+ 3 −
…

⁄

() = ∑ 1

= 1

1
⁄ + 1

2
⁄ + 1

=1

In practical terms, the Riemann-Zeta function is the infinite sum of 1, divided by integers beginning at 1, which have been placed to the power of the argument, s . Returning to the problem at hand, Euler’s proof continues:

(2 −
)() = 2 −
+ 4 −
+ 6 −
…

(1 − 2(2 −
))() = (1 + 2 −
+ 3 −
… ) − 2(2 −
+ 4 −
+ 6 −
… ) = 1 − 2 −
+ 3 −
…

Set  = −1

1 − 2(2 −−1 ) = −3

(−1) = 1 −−1 + 2 −−1 + 3 −−1 … = 1 + 2 + 3 …

1 − 2 −−1 + 3 −−1 − 4 −−1 … = 1 − 2 + 3 − 4 …

∴ −3(1 + 2 + 3 + 4 … ) = 1 − 2 + 3 − 4 … = 1 4⁄

1 + 2 + 3 + 4 + 5 … = −1

12⁄

It is notable that along the path of Euler’s proof there were close parallel at points to Ramanujan’s proof despite the fact that they were separated by around 250 years and Ramanujan worked, for much of his career, in isolation from the mathematical world, having taught himself and developed his own system of maths. Other than for the pleasure in maths, however, the sceptics are asking: “But what use is it?” and leaning back with a smug look smeared across their faces confident that a series reliant on infinity has no application in the real world. And these neigh-sayers would be correct if it weren’t for the great mathematical receptacle of quantum field theory that applies the -1/12 result in computing the Casimir effect. This result also has applications in bosonic string theory as it allows calculation of the number of physical dimensions and

3

rectifies the failure of string theory to be consistent in dimensions other than 26.

Overall, despite these pertinent usages, it is unlikely that this summation will be used in day to day life, but I feel that this concise reality of mathematics is truly under-appreciated. It was for this reason that I felt I should put it under spotlight so that the idea of the innate logic of numbers can be dispelled in favour of their actual mind-bending reality. However, I think that this brief line of numbers truly sums up the majesty in pure maths

Bibliography:

‘1 + 2 + 3 + 4 +…’ (7/04/16) Wikipedia . Available at: https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF [Accessed: 10/04/16]

‘ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12’ (9/01/14) YouTube: Numberphile. Available at: https://www.youtube.com/watch?v=w-I6XTVZXww

‘Sum of Natural Numbers (second proof and extra footage)’ (11/01/15) YouTube: Numberphile. Available at: https://www.youtube.com/watch?v=E- d9mgo8FGk&feature=youtu.be

Cargal, J.M. (1988) Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff.

Maths Puzzle 1 Each of the Four Musketeers made a statement about the four of them, as follows. a. Artagnan: “Exactly one is lying.” b. Athos: “Exactly two of us are lying.” c. Porthos: “An odd number of us is lying.” d. Aramis: “An even number of us is lying.” How many of them were lying (with the others telling the truth)?

4

Globalization and Cointegration – Study of Stock Markets in China and USA

By Zhengyuan Zhu (Yr 13)

Abstract This paper investigates the integrating and long term casual underlying relationship of share prices in Shanghai Stock Exchange (China) and S&P 500(USA) 1 . On the basis of the Dickey Fuller unit root and the Durbin Watson test, my results suggest that both shares have an integration of order one (unit root). More importantly, a long term relationship measured by cointegration does not exist between these two indices, they are statistically weakly cointegrated.

1. Introduction

a. Impact of globalization

By the end of last century, the international economy started moving towards a single market, due to the liberalization of trade and production, which allowed better and more efficient allocation of resources. 2 The growth and integration of the world capital markets is in fact one of the engines of globalization. As foreign exchange and bond markets become more integrated, the law of one price begins to apply throughout the world. The national stock exchanges are moving towards increasing linkages to other international stock exchanges. The linkages are extended from developed stock markets to other emerging stock markets, as well as from stock markets to other financial and banking systems. 3 The Shanghai Stock Exchange (SSE) was founded on Nov. 26th, 1990 and in operation on December 19th the same year. “Shares in mainland China have recorded their biggest one-day fall for more than eight years following a sell-off towards the end of the trading day . The Shanghai Composite closed down 8.5% at 3,725.56 after more weak economic >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

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