DC Mathematica 2017

DC Mathematica 2017

0 2017

DULWICH COLLEGE | MATHEMATICA 2017

Editors:

Theo Macklin

Phillip Cloud

Supervisor:

Dr Purchase

Cover Art By:

Theo Podger

Contributors:

Simon Xu

Jakub Dranczewski

Andrew Ng

Hin Chi Lee

Timothy Moulding

Toby Evans

Jay Connor

Lunzhi Shi

Joshua du Parc Braham

Joseph Lazzaro | Ayman D’Souza

Jack Kurtulus

Theo Macklin

Mr Ottewill

1

CONTENTS

Interesting Integer Sequences and Their Stories

3

Lunzhi Shi

Y10

Can You Trisect an Angle?

8

Simon Xu

Y11

Measuring Infinities: Or a short journey into the number theory

11

Jakub Dranczewski

Y13

Mathematical History- Zeno’s Paradoxes

14

Toby Evans

Y9

Parity

17

Hin Chi Lee

Y11

Taylor Series and Euler’s identity

21

Andrew Ng

Y12

What is the Probability of Two Random Numbers Being Coprime?

24

Theo Macklin Y12

Are the Renewed Statistics and Probability Behind the "Draft Lottery" Used in the NBA Reliable and an Accurate Method of Levelling a Team's Ability? 28 Jack Kurtulus Y13

A Golden Opportunity: Mathematical Patterns in Nature

33

Jay Connor

Y9

Sequences and Architecture

35

Timothy Moulding

Y10

Modelling Infectious Diseases

42

Joseph Lazzaro & Ayman D’Souza

Y12

Fractals: Are They Just Mathematical Curiosities?

46

Joshua du Parc Braham Y13

From a Question by Dr Purchase

55

Mr Ottewill

Puzzles

60

2

Interesting Integer Sequences and Their Stories

Lunzhi Shi

I was first introduced to the Pascal’s Triangle back in year 5 when I was asked to produce a program that loops and gives a constant output of the triangle. As I began to learn more mathematics, I realized that the triangle was far more than an array as you see it.

If you aren’t sure about what Pascal’s Triangle is, it’s like the food triangle: the further down the more there is. One number in the array is simply the sum of the two above it, taking empty spaces as zero and the initial as 1: it keeps on going non-stop. Now over with the definition, let’s first take a look at the applications of the triangle. I will start off with this easy one related to everyday algebra: Try expanding this: ( +
) . Now this: ( +
) 2 . That will be 2 + 2
+
2 . How about ( +
) 3 ? It will get more and more complicated.

But why am I mentioning binomial expansion? Is that somehow related to the triangle? Have a look at the rearranged table of the expansions (up to the power of 5):

Row Expansion 0

1 +

1 2 3 4 5

1 x

1 y

1 x 2

1 y 2

+

2 xy

+

1 x 3

3 x 2 y

3 xy 2

1 y 3

+

+

+

1 x 4

4 x 3 y

6 x 2 y 2

4 xy 3

1 y 4

+

+

+

+

1 x 5

5 x 4 y

10 x 3 y 2

10 x 2 y 3

5 xy 4

1 y 5

+

+

+

+

+

As you can see, the coefficients draw out the first five rows of the Pascal’s Triangle. Say we have ( +
)
, you then need to find row
to get the coefficient. The row and the expansion are magically related. The powers also come in with a very clear pattern of adding up and subtracting down to zero. This fun fact leads to so called binomial theorem. To further explain this, you possibly know that we need to use (
) or ( choose
). This is combinatorics; representing the number of ways of picking

3

elements from a set of elements. We can deduce the numeric value of ( choose
) using the formula:

!
! ( −
)!

(

) =

Now have a look at the pattern to the right – we have formed Pascal’s Triangle. Using this method, we can create a bridge to better understand the relationship between Binomial Theorem and Pascal’s Triangle.

Therefore, according to what’s shown above, for (x + y) n , we can write down an expansion of:

0

1

2 )
−2
2 + ⋯+ (

− 1

( +
)
= (

)

0 + (

)
−1
1 + (

) 1

−1 + (

) 0

Or:

( +
)
= ∑(
𝑘 )
=0

−


hence binomial theorem.

If we expand the binomial theorem further into multinomial theorem it’s going to become more interesting. We can first have a look at the example ( +
+ )
. As it is three- dimensional, we can visualize this with a matching Pascal’s Pyramid. And if we go into factorizing, it can be represented as:

( +
+ )
= ∑ 


−−

,

What if we have, say, subjects in the brackets and visualizing polygons can no longer support us? Since everything follows a pattern, there is a formula for multinomial theorem:



)
= ∑ (

( 1

+ 2

+ ⋯+

)∏ 

𝑘 1

, 𝑘 2

, … , 𝑘

 1 + 2 +⋯+ =

=1

Going through the full proof would take a very long time. There are articles you can find online that explain the theorem step by step – it only adds

another loop of getting a product on top of the coefficient calculation.

Now, on to some other sequences, there are a lot to find in this triangle and I will go through some briefly. Say we take all the odd numbers in a Pascal’s Triangle and leave the rest

4

alone. Zoom out, we discover this pattern known as the Sierpinski Triangle. Same goes for the pyramid.

We also have the Fibonacci sequence which is a bit hard to spot.

You have possibly worked with probabilities with this sequence in school. A famous example is flipping coins. You will either get heads or tails. Suppose you throw it
times. The possible number of results can be found in row
in Pascal’s triangle. The total possibilities of either heads or tails appearing times, ( ≤
) , is shown in the array with the ℎ representing the total possibilities. Let’s draw a table:

Throws

Possible Results

Pascal’s Triangle

Throws

Possible Results

Pascal’s Pyramid

H T S

1 1 1 1 2 1 2 1 2

H T

1 1 1 2 1 1 3 3 1

1

1

HH HT TH TT

HH HT TH TT TS ST SS SH HS

2

2

HHH HHT HTH THH HTT THT TTH TTT

3

…

…

…

…

…

…

The same theory applies to the pyramid model. Suppose a very skilled coin flipper has 1 3 the probability of landing a coin standing straight up, the table would look like the one above on the right. So where does this perfect creation come from? We need to look back to France in 1654. In 1654 Blaise Pascal completed his Traité du Triangle Arithmétique and later the titular array was given the name Pascal’s Triangle. However, the original discovery of this array, which later led to the more famous Pascal’s Triangle, actually took place in China. In the 13 th century, mathematician Yang Hui officially presented the triangle. Hence it is still called the Yang Hui Triangle, 杨辉 三角 , in China. The earliest official mathematics in China dates back to the Shang Dynasty (1600-1050 BC). Back then, calculations were done using rods of bamboo. This first number system could even handle complex calculations pretty well. It was, however, just a counting method- the written numbers were not that efficient. Jumping 1500 years ahead, the numeric system had improved a lot to make mathematics more accessible. In the 13 th century, China’s mathematics reached its peak as many new features were discovered by mathematicians and the government decided to make education of mathematics to everyone possible.

5

Yang Hui was one of the mathematicians that stood out during this time. To the right is the “Yang Hui’s Magic Concentric Circle”, another very interesting integer array. Counting the four circles and the eight semi-circles, the sum of the numbers in each semi- circle is exactly 69. There are also eight radii which, if you add the numbers up, also give 69. The second sum

is 147 – which is the sum of the numbers on four diameters and the sum of the numbers in each circle plus 9 in the center.

Talking about magic circles, the Ding Yi Dong ( 丁易 东 ) magic circle is also worth mentioning. In this model, the sum of each circle is 200. Also, taking a number on one circle and adding it to the number which is exactly on the opposite side of that circle, you always get 50 (for example, 14+36=50 and 45+5=50). Knowing some of these properties, we can deduce other properties like the sum of each diameter would always be 325 (25+6x50).

Magic squares, in turn, are constructed with similar properties. Again, the square is always related to a ‘magic’ constant which is the sum of the integers in each row, column and diagonal. In this 3 x 3 model, it is 15. This model is called the Lo Shu Square, which is the earliest magic square discovered in China. The more advanced magic squares were officially recorded in Greece in the 14 th century. These models were combined with a much

deeper meaning- linked to planets and gods. The magic squares were said to be written into medieval magic books. Ranging from the 3 x 3 model to the 9 x 9 one, they were given names of: Saturn, Jupiter, Mars, Sol (Sun), Venus, Mercury, Luna (Moon). Astronomy is very different though, but still very well connected with maths. Later people constructed the magic cube that has similar properties. Well, people never stop exploring, do they? Soon the term ‘magic hypercubes’ were introduced by mathematician John R. Hendricks.

The cubes follow the exact same rule, that is, the sum on any axis is always the same. Let’s have a look at the previously innocent-looking 3x3 model in 4d. People have been creating crazy things all the time, including the ones they can’t really see, or understand very well.

6

You might wonder the formula for the magic number, since no matter how you construct this magic cube, the stem of the magic number will be related to it. Suppose a n -dimensional magic cube arranged in an
×
× …×
structure, the formula for the magic number would be:

(

+ 1) 2

(
) =

Looking into the higher dimensions has always been a hard thing to do, and the seemingly convincing results we get could sometimes be more confusing than the question itself. Imagine throwing a paper ball to a whiteboard with 2-dimensional creatures on it. What they would see would only be a cross-section of the

paper ball and its changing shapes will not follow their laws of physics. But just now, we put ourselves as ‘flatlanders’. The 4d model above is technically correct, but if a 4d visitor decides to take you into his dimension to show you this magic cube, you would be seeing blobs and spheres and other things that make absolutely no sense - could this explain why movie makers decide to put a blurring and confusing background every time there’s a scene of time travel? There are so many integer sequences out there and you can just pick one from the pool and spend the whole afternoon looking into it. There is even an encyclopaedia just for integer sequences – oeis.org , or The Online Encyclopaedia of Integer Sequences. By 2015, there were at least 250,000 sequences on their list – go have a look and see if you find something new. The earliest, pre-historic mathematics that could be 20,000 years old started as human began to quantify things around them – time, matter, etc. The concept of numbers is always the essence of our world and writing this article has been a good experience; reminding me of a page I haven’t turned over for a long time. And most importantly of all, I hope you, fellow mathematicians, have enjoyed reading this.

7

Can You Trisect an Angle in a Straight Edge Compass Construction?

Simon Xu

Using a straight edge and a compass we can construct many lengths if we are given a unit length. If we take the unit length as ‘one’, we can copy its length using the compass and add it to itself to construct ‘two’; add another one to construct ‘three’; take one away from ‘four’ to make ‘three’. When we can do addition and subtraction, multiplication and division a length can be made as well to represent the answer. Here we can see how some geometrical operations are equivalent to axiomatic algebra.

So what are the rules for straightedge-compass construction? What is the limit for what number we can construct?

Rules for straight/edge compass construction:

Basic operations for straight- edge/ compass construction: Creating the length through two existing points Creating a circle through one point with centre another point Creating the point which is the intersection of two existing, non-parallel lines Creating the one or two points in the intersection of a line and a circle (if they intersect) Creating the one or two points in the intersection of two circles (if they intersect)     

In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. All rational numbers are constructible, and all constructible numbers are algebraic numbers. A field of rational numbers ( ℚ ) is a set containing operations including addition and multiplication. Also, if and
are constructible numbers with
≠ 0 , then ±
, ×
, , and √ are all constructible. A complex number is constructible if, and only if, the real and imaginary parts are both constructible. In a straight-edge/compass construction, we can also create numbers such as root 2 by using the method as shown on the right, where we can construct as any rational number, and we can construct a number = +
√ , where ,
,  are rational numbers, which forms a new field, F1, based on the field of rational numbers, 𝐹0 . We can construct another number from another field by constructing  = +
√ where ,
,  are from 𝐹1 , and so on we can construct many numbers and values. angle, and then prove that one third of that angle is not constructible, so we use that as an example to prove that we cannot trisect that angle. To do this, we need to prove that we cannot construct a number such as (40) , which means that an angle of 40˚ is not constructible, but as we can construct a 120˚ angle and therefore the operation of trisecting a 120˚ angle is not possible.
Therefore, to prove that a given angle cannot be trisected in straight-edge/compass construction, we need to construct an

8

Theorem: If a cubic has any constructible root then it has at least one real root.

Proof:

Suppose that the in cubic:

3 +  2 +  +  = 0

there are constructible roots but none in ℚ (the set of rational numbers) and there is some value such that the field 𝐹() is the closest field to the rational number field in which a root exists. I.e. that the root is = +
√𝑘 for ,
, 𝑘 from a field lower than 𝐹() , 𝐹( − 1) and √𝑘 is in 𝐹() .

Then we put = +
√𝑘 into the cubic:

We can create many fields from the field of rational numbers

( +
√𝑘) 3 + ( +
√𝑘) 2 + ( +
√𝑘) +  = 0

If we expand the brackets we get:

( + 3 +
2 𝑘 +  2 + 
2 𝑘 + ) + (3 2
+
3 𝑘 + 2
+ 
)√𝑘 = 0

which we can simplify as :

+ √𝑘 = 0

So either √𝑘 = −
/ or
=  = 0 . Because −
/ is in 𝐹( − 1) and √𝑘 is not, this case cannot be true; therefore S=T=0 must be the true case.

Therefore we ask: is = −
√𝑘 a root?

( −
√𝑘) 3 + ( −
√𝑘) 2 + ( −
√𝑘) +  = 0

( + 3 +
2 𝑘 +  2 + 
2 𝑘 + ) − (3 2
+
3 𝑘 + 2
+ 
)√𝑘 = 0

∴
− √𝑘 = 0

Because
=  = 0 , this is also true. However, a cubic cannot have just 2 real roots (unless the roots are repeated). Let +
√𝑘 = 1 , −
√𝑘 = 2 , and the third (either repeated or not repeated) be 3 . I.e.:

( − 1

)( − 2

)( − 3

) = 0

3 + ( 1

) 2 + ⋯ = 0

+ 2

+ 3

Comparing with the original cubic, 3 +  2 +  +  = 0 , we can say that:

) = −( + + 3

) = −2 − 3

 = ( 1

+ 2

+ 3

3

= −2 − 

9

Because a and p are in the field 𝐹( − 1) , 3 is in the field 𝐹( − 1) . This contradicts with what we supposed at the beginning in which 𝐹() is the lowest field which has a constructible root. So if a cubic has constructible roots, at least one root is rational; if a cubic has no rational roots, it has no constructible roots.

How can we use this theorem?

To prove that we cannot trisect angle with a straight-edge and a compass, we need to show that there exists an angle that cannot be trisected using a straightedge and a compass. For example, a 120˚ angle cannot be trisected, because we can prove that (40) is not constructible. To do this we need to put (40) into a cubic function.

Using the trigonometric identity:

(3) = 4 3 () − 3()

in which =40˚:

(120) = 0.5

0.5 = 4 3 (40) − 3(40)

If we let (40) = then the cubic will become:

8 3 − 6 − 1 = 0

Which we can prove has no rational roots by typing it into a calculator. So we can say that it has no constructible roots. Because (40) is a root of this function, (40) is not constructible, therefore an angle of 120˚ which we can construct cannot be trisected.

In conclusion, we cannot trisect an angle in a straightedge-compass construction.

10

Measuring Infinities: a Short Journey into Number Theory

Jakub Dranczewski

Could you say that one infinity is somehow bigger than another one? It seems completely senseless. How can one claim that one thing of infinite size is greater than another thing of infinite size if, by definition, they both have no proper end? Let’s just stop here for a second though. What does it actually mean that two sets of things have the same number of elements? Your first answer would probably be ‘if you count the elements in them, you end up with the same number’. But what if we cannot count the number of elements in the set? This definition fails in that case. We have to look for something more general, and that something is a simple action: matching the elements of the two compared sets into pairs. If we manage to find a method of matching that leaves no elements of either set without a pair, we have proven that the number of elements in both sets is the same! Is this in any way useful for you? Well, that surely depends on your definition of usefulness, but one can make some fairly decent proofs with the above method. For example, we can quickly prove that the number of even numbers is exactly the same as the number of positive natural numbers! ‘But wait!’, you scream, ‘shouldn’t there be twice as many natural numbers as even numbers? Surely there is an additional odd number for every even number’. Intuitively that is correct, but let’s use the pairing method and see what happens. Take every natural positive number and multiply it by two, then assign the result as a pair to that number. If you write that down you’ll notice that all the even numbers were assigned to a natural positive number. … This means that the number of positive integers and the number of even integers are exactly the same. We just compared two infinite sets! It can be proven in a similar way that the number of integers is the same as the number of negative integers (I for one was initially rather surprised by that fact) or that there are exactly as many points on a line segment as on the entire line. The proofs go on and on! But hey, one could argue that all infinities are of the same size – as they are all infinite this seems to be a fair assumption. So now the real fun begins: let us prove that some infinities are bigger than others! 1 2 3 4 5 6 7 … ▼ ▼ ▼ ▼ ▼ ▼ 10 ▼ 12 ▼ 14 2 4 6 8

Not all infinities are equal

The proof above involved the size of a set of all integers. Let’s call this amount ℵ 0 (aleph-zero, the term is actually used by mathematicians, and it originates from a Hebrew letter). Now to help us find an infinity bigger than that I’ll quickly go through the proof that the size of the set of all rational numbers is also ℵ 0 (bear with me, it’s simple). First, make a table as below:

0 1

1

2

3

4

… … … … … …

1/1

2/1

3/1

4/1

-1

-1/1

-2/1

-3/1

-4/1

2

1/2

2/2

3/2

4/2

-2

-1/2

-2/2

-3/2

-4/2

…

…

…

…

…

11

Now follow the blue line, omitting doubles, and assign subsequent positive integers to every cell you visit. You just numbered all the rational numbers! Since they are numbered, they all have a pair in the natural numbers set and therefore the amount of them is the same (another name for ℵ 0 -sized sets stems from that proof method: we call them countable since you can count their elements using natural numbers). Thus, the number of rational numbers can also be written as ℵ 0 . Now, onto the greater infinities. Imagine a number line and two rational numbers as close to each other as possible. Notice that there are still a lot of numbers between them that are not rational. As you get closer and closer you discover an infinite amount of real numbers between each pair of rational numbers, which there is, of course, an infinite amount of. It’s like infinity squared! This exclamation is, in fact, a rather close description of the number of real numbers. There are a lot of them, even more than rational numbers. The proof above relayed strongly on intuition, but there is, of course, a more rigorous one, which I will not write down here, but it involves imagining a list of all the rational numbers and creating new numbers out of them that are not on the list. The amount of real numbers is called continuum. This brings us to one of the most interesting problems of the set theory: the continuum hypothesis. The hypothesis simply asks whether there is an infinity that is bigger than ℵ 0 , but smaller than continuum. And yet it landed on David Hilbert’s list of the twenty-two important open questions in mathematics, and generally puzzled mathematicians for a very long time. The interesting part of the hypothesis is that it cannot be proven. It cannot be disproven either. It was mathematically proved to be unprovable, and then mathematically proved to be undisprovable, which is just weird. It just hangs somewhere in-between, neither true nor false, and mathematicians try to come up with new ways to crack it (since it is not impossible that with completely new tools the hypothesis could be one day settled). But we do not need to consider problems this complicated to have some fun with infinities. There are many problems that are easier and still fun. Imagine a hotel that has an infinite amount of numbered rooms, each housing a mathematician. Now, what shall the hotel’s management team do if a new mathematician arrives? They certainly should not just dismiss him or make him sleep on a random couch – mathematicians are sensitive beings; the poor guy could catch a cold or something. The hotel’s staff needs to find an empty room in a hotel in which every single room is already taken. The resolution is actually surprisingly simple – just make every mathematician move to the room next door (if he was in room one, he should go to room two, if he was in room two, he should go to room three, and so on). Now room one is free! A swift and mind-bending solution, involving merely the pain of infinite mathematicians having to change rooms all of a sudden, which they would surely be willing to endure for a colleague in need. … Now what happens if a coach with infinite mathematicians arrives and they all want a room? What if an infinite number of coaches arrive, each with an infinite number of mathematicians inside? The world of infinity problems seems to be truly endless and I do encourage you to look yourself for some of the wonderful and often weird things that happen in there. The Hilbert’s Hotel – the absolute classic of number theory 1 2 3 4 5 6 Is there something in between?

12

Sources:

 Most of this article is based on notes taken by me during a lecture by Krzysztof Ciesielski (Ph.D.) at the Jagiellonian University in Krakve. Nevertheless, below I cite some interesting webpages that can be a good start in your number theory research:  https://www.ias.edu/ideas/2011/kennedy-continuum-hypothesis - Can the Continuum Hypothesis Be Solved? by Juliette Kennedy  https://en.wikipedia.org/wiki/Hilbert’s_paradox_of_the_Grand_Hotel - a thorough description and analysis of the Hilbert’s Hotel problem.

13

Mathematical History: Zeno’s Paradoxes

Toby Evans

Zeno was an Italian Mathematician, he was born in about 490 BC and died around 425 BC but during his brief life he made some great discoveries. Since he lived so long ago, we know little about him; however we do know that he was a philosopher. Many of his paradoxes are known to us today, and in this article, I will explain the paradox of Achilles and the Tortoise. Achilles was a legendary Greek hero; he was a terrific fighter and could run very quickly, whereas a tortoise is an animal known for being slow.

Zeno

Zeno stated that if the tortoise was given a small head start, then despite Achilles being so much faster, he could never catch up and overtake. The logic behind this was that Achilles would have to cover the distance the tortoise had already moved to catch up with it, however by this time the tortoise would have moved on. Hence Achilles would then have to cover that additional distance too, and, whilst doing this, the tortoise would move on even further. This cycle would continue endlessly, with the tortoise always moving further away. Whilst the gap might narrow, the tortoise would always win this race.

The conclusion is that anything, given a head start and no matter how slow it is going, will inevitably beat a faster object, for example a formula one car would lose to a snail if the snail starts the race

14

first. This, however, seems unlikely. Therefore, I shall try to disprove Zeno’s statement with an example, using proof by contradiction.

If Achilles is travelling at twice the speed of the tortoise and the tortoise travels 1 metre a second and has a 1 second head start, then the following sequence occurs.

Time /seconds

Achilles /metres

Tortoise /metres

1 2 3 4 5

0 2 4 6 8

1 2 3 4 5

9

Tortoises are slow!

8

7

6

5

4

3

2

Slow and steady doesn’t win!

1

0

0

1

2

3

4

5

6

Achilles (metres)

Tortoise (metres)

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