# 6: Continued Fractions - Mathematics

6: Continued Fractions - Mathematics

## Continued Fractions I

Did you find the easiest way to calculate these? For example, you should be able to see that the last one is $<1overdisplaystyle 1+ >quad = quad <5over 8>.$ In this sequence of continued fractions you can always calculate one quickly by using the previous answer. The next fraction in this sequence is $<1overdisplaystyle 1+ >quad = quad <8over 13>.$ The numbers we get in order are $1, 2, 3, 5, 8, 13$. What do you think the next number is? Yes, these are the Fibonacci numbers. What do you think the next continued fraction in the sequence is?

Now let us find out what happens if the continued fraction goes on for ever. We write this as $f = <1overdisplaystyle 1+ >>.$ Can you see why we have $f = <1over 1+f>quad ?$ This gives the quadratic equation $f^2 + f -1 = 0$. Because $f$ is positive we get the one solution $f = -1over 2>,$ the ratio of the shorter to the longer side of the Golden Rectangle!

## Jq [ edit ]

We take one of the points of interest here to be the task of representing the infinite series a0, a1, . and b0, b1, . compactly, preferably functionally. For the type of series typically encountered in continued fractions, this is most readily accomplished in jq 1.4 using a filter (a function), here called "next", which, given the triple [i, [a[i], b[i]], will produce the next triple [i+1, a[i+1], b[i+1]].

Another point of interest is avoiding having to specify the number of iterations. The approach adopted here allows one to specify the desired accuracy in some cases, it is feasible to specify that the computation should continue until the accuracy permitted by the underlying floating point representation is achieved. This is done by specifying delta as 0, as shown in the examples.

We therefore proceed in two steps: continued_fraction( first next count ) computes an approximation based on the first "count" terms and then continued_fraction_delta(first next delta) computes the continued fraction until the difference in approximations is less than or equal to delta, which may be 0, as previously noted.

## MANUSCRIPTS

### 02/2021

New characterizations of the summatory function of the Möbius function
Full Text: arXiv/2102.05842 (math.NT) Software Reference: Mertens Function Computations (GitHub) Keywords: Möbius function Mertens function Dirichlet inverse Liouville lambda function prime omega function prime counting function Dirichlet generating function Erdős-Kac theorem strongly additive function. MSC (2010): 11N37 11A25 11N60 11N64 and 11-04.

### 04/2020

A catalog of interesting and useful Lambert series identities
Full Text: https://arxiv.org/pdf/2004.02976.pdf (math.NT, math.HO) Keywords: Lambert series Lambert series generating function divisor sum Anderson-Apostol sum Dirichlet convolution Dirichlet inverse summatory function generating function transformation series identity arithmetic function. MSC (2010): 05A15, 11Y70, 11A25, and 11-00.

### 08/2017

Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions
Full Text: https://arxiv.org/abs/1712.00608 (math.NT) Keywords: Lambert series factorization theorem matrix factorization partition function Hadamard product. MSC (2010): 11A25 11P81 05A17 05A19.

### 07/2017

Pair Correlation and Gap Distributions for Substitution Tilings and Generalized Ulam Sets in the Plane
Worked closely with Jayadev Athreya on the project. Full Text: https://arxiv.org/abs/1707.05509 Keywords: substitution tiling Ammann chair Ulam set directional distribution gap distribution pair correlation. MSC (2010): 52C20 06A99 11B05 62H11 52C23.

### 06/2017

New Factor Pairs for Factorizations of Lambert Series Generating Functions
With Mircea Merca. Full Text: https://arxiv.org/abs/1706.02359 Keywords: Lambert series factorization theorem matrix factorization partition function. MSC (2010): 11A25 11P81 05A17 05A19.

## Expressions

You can use the following operators and parentheses for the expressions:

• - for subtraction
• * for multiplication
• / for integer division
• % for modulus (remainder of the integer division)
• ^ or ** for exponentiation (the exponent must be greater than or equal to zero).
• <, ==, > <=, >=, != for comparisons. The operators return zero for false and -1 for true.
• AND, OR, XOR, NOT for binary logic. The operations are done in binary (base 2). Positive (negative) numbers are prepended with an infinite number of bits set to zero (one).
• SHL or <<: When b &ge 0, a SHL b shifts a left the number of bits specified by b . This is equivalent to a × 2 b . Otherwise, a SHL b shifts a right the number of bits specified by &minus b . This is equivalent to floor( a / 2 &minus b ). Example: 5 SHL 3 = 40.
• SHR or >>: When b &ge 0, a SHR b shifts a right the number of bits specified by b . This is equivalent to floor( a / 2 b ). Otherwise, a SHR b shifts a left the number of bits specified by &minus b . This is equivalent to a × 2 &minus b . Example: -19 SHR 2 = -5.
• n!: factorial ( n must be greater than or equal to zero). Example: 6! = 6 × 5 × 4 × 3 × 2 = 720.
• n!! . !: multiple factorial ( n must be greater than or equal to zero). It is the product of n times n &minus k times n &minus 2k . (all numbers greater than zero) where k is the number of exclamation marks. Example: 7!! = 7 × 5 × 3 × 1 = 105.
• p#: primorial (product of all primes less or equal than p ). Example: 12# = 11 × 7 × 5 × 3 × 2 = 2310.
• B(n): Previous probable prime before n. Example: B(24) = 23.
• F(n): Fibonacci number Fn from the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. where each element equals the sum of the previous two members of the sequence. Example: F(7) = 13.
• L(n): Lucas number Ln = F n -1 + F n +1
• N(n): Next probable prime after n. Example: N(24) = 29.
• P(n): Unrestricted Partition Number (number of decompositions of n into sums of integers without regard to order). Example: P(4) = 5 because the number 4 can be partitioned in 5 different ways: 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1.
• Gcd(m,n): Greatest common divisor of these two integers. Example: GCD(12, 16) = 4.
• Modinv(m,n): inverse of m modulo n , only valid when m and n are coprime, meaning that they do not have common factors. Example: Modinv(3,7) = 5 because 3 × 5 &equiv 1 (mod 7)
• Modpow(m,n,r): finds m n modulo r . Example: Modpow(3, 4, 7) = 4, because 3 4 &equiv 4 (mod 7).
• Jacobi(m,n): obtains the Jacobi symbol of m and n . When the second argument is prime, the result is zero when m is multiple of n , it is one if there is a solution of x ² &equiv m (mod n ) and it is equal to &minus1 when the mentioned congruence has no solution.
• IsPrime(n): returns zero if n is not probable prime, -1 if it is. Example: IsPrime(5) = -1.
• Sqrt(n): Integer part of the square root of the argument.
• NumDigits(n,r): Number of digits of n in base r . Example: NumDigits(13, 2) = 4 because 13 in binary (base 2) is expressed as 1101.
• SumDigits(n,r): Sum of digits of n in base r . Example: SumDigits(213, 10) = 6 because the sum of the digits expressed in decimal is 2+1+3 = 6.
• RevDigits(n,r): finds the value obtained by writing backwards the digits of n in base r . Example: RevDigits(213, 10) = 312.

You can use the prefix 0x for hexadecimal numbers, for example 0x38 is equal to 56.

## Continued fraction associated to KdV solitons

Background (may be skipped by those interested only in the basic question and not important associations):

The CF for the reciprocal is

Euler shows that the value of $q$ is defined by the Riccati equation

Then from the formulas in my contribution (Sept 18, 2014) to OEIS 008292 on the Eulerian numbers with $hat = frac$ ,

is an e.g.f. for the bivariate Eulerian polynomials $E_n(a,b)$ , whose coefficients are those of the h-vectors for the permutohedra,

with $(u.)^n = u_n = h_(a,b)$ a complete homogeneous polynomial in two indeterminates with $h_n(1,x)$ the h-vector of the $<(n-1)>$ -dimensional hypertetrahedron, is a log generating function for the complete homogeneous polynomials,

an instance of the Riccati equation

which can be written in terms of a Weierstrass elliptic function (see Buchstaber & Bunkova in the OEIS entry)

more generally the bivariate Eulerian row polynomials $E_n(a,b)$ of $A(x,a,b)$ with $E_0(a,b) =0$ are generated by

(see OEIS A145271 for a generator of compositional inverses via the refined Eulerian numbers)

So, with $x = a/p$ the continued fraction of Euler evaluates analytically as

with a discontinuity--a jump from $-1$ to $1$ as the argument passes through the origin from negative to positive values of $x$ .

The reciprocal, of course, is

with the same discontinuity at the origin $x=0$ .

The more natural presentation is

with no discontinuity for finite real argument $frac$ .

is called the hyperbolic formal group law and related to a generalized cohomology theory proposed by Lenart and Zainoulline.

This is the addition, or composition, law for velocities in special relativity for $c=1$ and the formula for the hyperbolic tangent of sums

See my post "The Elliptic Lie Triad: KdV and Riccati Equations, Infinigens, and Elliptic Genera" for relationships to a soliton solution to the KdV equation and an associated Riccati equation or my contribution to the MO-Q "Is there an underlying explanation for the magical powers of the Schwarzian?" for a briefer note on some aspects of the relationships.

What are continued fraction reps for

and what references for any specific rep are available (via the usual free sources)?

I suspect some version of Equation 4 in "Introduction to Chapter 3 on continued fractions [version 5, 29 January 2013]" by Xavier Viennot interpreted in terms of Dyck lattice paths should apply since the associahedra partition polynomials of OEIS A133437 for compositional inversion can be applied to $B(x,a,b)$ to obtain $A(x,a,b)$ and these associahedra face polynomials are a refinement of those of A126216, which are related to marked Dyck paths (and Schroeder--see Drake therein).

Another potential lead is A134264 / A125181 for compositional inversion via noncrossing partitions / Dyck paths of even length. See "A note on 2-distant noncrossing partitions and weighted Motzkin paths" by Ira Gessel and Jang Soo Kim, related to CFs.

I've scanned over dozens of references on orthogonal polynomials and continued fractions over the last couple of weeks, but didn't come across until just now "Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes–Rogers and Thron–Rogers polynomials, with coefficientwise Hankel-total positivity" by Mathias Petreolle, Alan Sokal, and Bao-Xuan Zhu. The footnote on p.77 states:

The identity (12.6) — that is, the S-fraction for the Eulerian polynomials — was found by Stieltjes [160, section 79]. Stieltjes does not specifically mention the Eulerian polynomials, but he does state that the continued fraction is the formal Laplace transform of $(1 − y)/(e^ − y), which is well known to be the exponential generating function of the Eulerian polynomials. Stieltjes also refrains from showing the proof: “Pour abreger, je supprime toujours les artifices qu’il faut employer pour obtenir la transformation de l’int´egrale definie en fraction continue” (!). But a proof is sketched, albeit also without much explanation, in the book of Wall [165, pp. 207–208]. The J-fraction corresponding to the contraction of this S-fraction was proven, by combinatorial methods, by Flajolet [52, Theorem 3B(ii) with a slight typographical error]. Dumont [41, Propositions 2 and 7] gave a direct combinatorial proof of the S-fraction, based on an interpretation of the Eulerian polynomials in terms of “bipartite involutions of [2n]” and a bijection of these onto Dyck paths. (The paper also contains, on p. 83, the partition polynomials of A190015, which they call the Euler symmetric polynomials, claiming to have introduced them originally in their paper. As noted in the OEIS entry, they are a scaled version of A145271, which I call the refined Eulerian partition polynomials mentioned above.) ## Maths IA – Exploration Topics Scroll down this page to find over 300 examples of maths IA exploration topics and ideas for IB mathematics students doing their internal assessment (IA) coursework. Topics include Algebra and Number (proof), Geometry, Calculus, Statistics and Probability, Physics, and links with other subjects. Suitable for Applications and Interpretations students (SL and HL) and also Analysis and Approaches students (SL and HL). Essential resources for IB students: Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications. There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful! The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s. Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories. I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here. Maths IA – Maths Exploration Topics A list with over 300 examples of maths IA exploration topics and ideas for IB mathematics students doing their internal assessment (IA) coursework. Suitable for Applications and Interpretations students (SL and HL) and also Analysis and Approaches students (SL and HL). Algebra and number 1) Modular arithmetic – This technique is used throughout Number Theory. For example, Mod 3 means the remainder when dividing by 3. 2) Goldbach’s conjecture: “Every even number greater than 2 can be expressed as the sum of two primes.” One of the great unsolved problems in mathematics. 3) Probabilistic number theory 4) Applications of complex numbers: The stunning graphics of Mandelbrot and Julia Sets are generated by complex numbers. 5) Diophantine equations: These are polynomials which have integer solutions. Fermat’s Last Theorem is one of the most famous such equations. 6) Continued fractions: These are fractions which continue to infinity. The great Indian mathematician Ramanujan discovered some amazing examples of these. 7) Patterns in Pascal’s triangle: There are a large number of patterns to discover – including the Fibonacci sequence. 8) Finding prime numbers: The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics. There is a$1 million prize for solving the Riemann Hypothesis and $250,000 available for anyone who discovers a new, really big prime number. 10) Pythagorean triples: A great introduction into number theory – investigating the solutions of Pythagoras’ Theorem which are integers (eg. 3,4,5 triangle). 11) Mersenne primes: These are primes that can be written as 2^n -1. 12) Magic squares and cubes: Investigate magic tricks that use mathematics. Why do magic squares work? 13) Loci and complex numbers 14) Egyptian fractions: Egyptian fractions can only have a numerator of 1 – which leads to some interesting patterns. 2/3 could be written as 1/6 + 1/2. Can all fractions with a numerator of 2 be written as 2 Egyptian fractions? 15) Complex numbers and transformations 16) Euler’s identity: An equation that has been voted the most beautiful equation of all time, Euler’s identity links together 5 of the most important numbers in mathematics. 17) Chinese remainder theorem. This is a puzzle that was posed over 1500 years ago by a Chinese mathematician. It involves understanding the modulo operation. 18) Fermat’s last theorem: A problem that puzzled mathematicians for centuries – and one that has only recently been solved. 19) Natural logarithms of complex numbers 20) Twin primes problem: The question as to whether there are patterns in the primes has fascinated mathematicians for centuries. The twin prime conjecture states that there are infinitely many consecutive primes ( eg. 5 and 7 are consecutive primes). There has been a recent breakthrough in this problem. 22) Diophantine application: Cole numbers 23) Perfect Numbers: Perfect numbers are the sum of their factors (apart from the last factor). ie 6 is a perfect number because 1 + 2 + 3 = 6. 24) Euclidean algorithm for GCF 25) Palindrome numbers: Palindrome numbers are the same backwards as forwards. 26) Fermat’s little theorem: If p is a prime number then a^p – a is a multiple of p. 28) Recurrence expressions for phi (golden ratio): Phi appears with remarkable consistency in nature and appears to shape our understanding of beauty and symmetry. 29) The Riemann Hypothesis – one of the greatest unsolved problems in mathematics – worth$1million to anyone who solves it (not for the faint hearted!)

30) Time travel to the future: Investigate how traveling close to the speed of light allows people to travel “forward” in time relative to someone on Earth. Why does the twin paradox work?

31) Graham’s Number – a number so big that thinking about it could literally collapse your brain into a black hole.

32) RSA code – the most important code in the world? How all our digital communications are kept safe through the properties of primes.

33) The Chinese Remainder Theorem: This is a method developed by a Chinese mathematician Sun Zi over 1500 years ago to solve a numerical puzzle. An interesting insight into the mathematical field of Number Theory.

34) Cesaro Summation: Does 1 – 1 + 1 – 1 … = 1/2?. A post which looks at the maths behind this particularly troublesome series.

35) Fermat’s Theorem on the sum of 2 squares – An example of how to use mathematical proof to solve problems in number theory.

36) Can we prove that 1 + 2 + 3 + 4 …. = -1/12 ? How strange things happen when we start to manipulate divergent series.

37) Mathematical proof and paradox – a good opportunity to explore some methods of proof and to show how logical errors occur.

38) Friendly numbers, Solitary numbers, perfect numbers. Investigate what makes a number happy or sad, or sociable! Can you find the loop of infinite sadness?

39) Zeno’s Paradox – Achilles and the Tortoise – A look at the classic paradox from ancient Greece – the philosopher “proved” a runner could never catch a tortoise – no matter how fast he ran.

40) Stellar Numbers – This is an excellent example of a pattern sequence investigation. Choose your own pattern investigation for the exploration.

41) Arithmetic number puzzle – It could be interesting to do an exploration where you solve number problems – like this one.

42) Normal Numbers – and random number generators – what is a normal number – and how are they connected to random number generators?

43) Narcissistic Numbers – what makes a number narcissistic – and how can we find them all?

44) Modelling Chaos – how we can use grahical software to understand the behavior of sequences

45) The Mordell Equation. What is the Mordell equation and how does it help us solve mathematical problems in number theory?

46) Ramanujan’s Taxi Cab and the Sum of 2 Cubes. Explore this famous number theory puzzle.

47) Hollow cubes and hypercubes investigation. Explore number theory in higher dimensions!

48) When do 2 squares equal 2 cubes? A classic problem in number theory which can be solved through computational power.

49) Rational approximations to irrational numbers. How accurately can be approximate irrationals?

50) Square triangular numbers. When do we have a square number which is also a triangular number?

51) Complex numbers as matrices – Euler’s identity. We can use a matrix representation of complex numbers to test whether Euler’s identity still holds.

52) Have you got a Super Brain? How many different ways can we use to solve a number theory problem?

IB Revision Notes for Analysis and Applications The IB Analysis and Approaches SL notes are a 60 page pd, the HL notes are a 112 page pdf and the SL Applications notes are 53 pages. All fully updated for the new syllabus. I would really recommend these resources for all IB students – it takes a lot of skill to successfully condense a syllabus into the essential content – and these notes really are of the highest quality. You can download these notes on my site here .

1a) Non-Euclidean geometries: This allows us to “break” the rules of conventional geometry – for example, angles in a triangle no longer add up to 180 degrees. In some geometries triangles add up to more than 180 degrees, in others less than 180 degrees.

1b) The shape of the universe – non-Euclidean Geometry is at the heart of Einstein’s theories on General Relativity and essential to understanding the shape and behavior of the universe.

2) Hexaflexagons: These are origami style shapes that through folding can reveal extra faces.

3) Minimal surfaces and soap bubbles: Soap bubbles assume the minimum possible surface area to contain a given volume.

4) Tesseract – a 4D cube: How we can use maths to imagine higher dimensions.

5) Stacking cannon balls: An investigation into the patterns formed from stacking canon balls in different ways.

6) Mandelbrot set and fractal shapes: Explore the world of infinitely generated pictures and fractional dimensions.

7) Sierpinksi triangle: a fractal design that continues forever.

8) Squaring the circle: This is a puzzle from ancient times – which was to find out whether a square could be created that had the same area as a given circle. It is now used as a saying to represent something impossible.

9) Polyominoes: These are shapes made from squares. The challenge is to see how many different shapes can be made with a given number of squares – and how can they fit together?

10) Tangrams: Investigate how many different ways different size shapes can be fitted together.

11) Understanding the fourth dimension: How we can use mathematics to imagine (and test for) extra dimensions.

12) The Riemann Sphere – an exploration of some non-Euclidean geometry. Straight lines are not straight, parallel lines meet and angles in a triangle don’t add up to 180 degrees.

13) Graphically understanding complex roots – have you ever wondered what the complex root of a quadratic actually means graphically? Find out!

14) Circular inversion – what does it mean to reflect in a circle? A great introduction to some of the ideas behind non-euclidean geometry.

15) Julia Sets and Mandelbrot Sets – We can use complex numbers to create beautiful patterns of infinitely repeating fractals. Find out how!

16) Graphing polygons investigation. Can we find a function that plots a square? Are there functions which plot any polygons? Use computer graphing to investigate.

17) Graphing Stewie from Family Guy. How to use graphic software to make art from equations.

18) Hyperbolic geometry – how we can map the infinite hyperbolic plane onto the unit circle, and how this inspired the art of Escher.

19) Elliptical Curves– how this class of curves have importance in solving Fermat’s Last Theorem and in cryptography.

20) The Coastline Paradox – how we can measure the lengths of coastlines, and uses the idea of fractals to arrive at fractional dimensions.

21) Projective geometry – the development of geometric proofs based on points at infinity.

22) The Folium of Descartes. This is a nice way to link some maths history with studying an interesting function.

23) Measuring the Distance to the Stars. Maths is closely connected with astronomy – see how we can work out the distance to the stars.

24) A geometric proof for the arithmetic and geometric mean. Proof doesn’t always have to be algebraic. Here is a geometric proof.

25) Euler’s 9 Point Circle. This is a lovely construction using just compasses and a ruler.

26) Plotting the Mandelbrot Set – using Geogebra to graphically generate the Mandelbrot Set.

27) Volume optimization of a cuboid – how to use calculus and graphical solutions to optimize the volume of a cuboid.

28) Ford Circles– how to generate Ford circles and their links with fractions.

29) Classical Geometry Puzzle: Finding the Radius. This is a nice geometry puzzle solved using a variety of methods.

31) The Shoelace Algorithm to find areas of polygons. How can we find the area of any polygon?

32) Soap Bubbles, Wormholes and Catenoids. What is the geometric shape of soap bubbles?

33) Can you solve an Oxford entrance question? This problem asks you to explore a sliding ladder.

34) The Tusi circle – how to create a circle rolling inside another circle using parametric equations.

35) Sphere packing – how to fit spheres into a package to minimize waste.

36) Sierpinski triangle – an infinitely repeating fractal pattern generated by code.

37) Generating e through probability and hypercubes. This amazing result can generate e through considering hyper-dimensional shapes.

38) Find the average distance between 2 points on a square. If any points are chosen at random in a square what is the expected distance between them?

39) Finding the average distance between 2 points on a hypercube. Can we extend our investigation above to a multi-dimensional cube?

40) Finding focus with Archimedes. The Greeks used a very different approach to understanding quadratics – and as a result had a deeper understanding of their physical properties linked to light and reflection.

41) Chaos and strange Attractors: Henon’s map. Gain a deeper understanding of chaos theory with this investigation.

Exploration Guides and Paper 3 Resources I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

Calculus/analysis and functions

1) The harmonic series: Investigate the relationship between fractions and music, or investigate whether this series converges.

2) Torus – solid of revolution: A torus is a donut shape which introduces some interesting topological ideas.

3) Projectile motion: Studying the motion of projectiles like cannon balls is an essential part of the mathematics of war. You can also model everything from Angry Birds to stunt bike jumping. A good use of your calculus skills.

4) Why e is base of natural logarithm function: A chance to investigate the amazing number e.

5) Fourier Transforms – the most important tool in mathematics? Fourier transforms have an essential part to play in modern life – and are one of the keys to understanding the world around us. This mathematical equation has been described as the most important in all of physics. Find out more! (This topic is only suitable for IB HL students).

6) Batman and Superman maths – how to use Wolfram Alpha to plot graphs of the Batman and Superman logo

7) Explore the Si(x) function – a special function in calculus that can’t be integrated into an elementary function.

8) The Remarkable Dirac Delta Function. This is a function which is used in Quantum mechanics – it describes a peak of zero width but with area 1.

9) Optimization of area – an investigation. This is an nice example of how you can investigation optimization of the area of different polygons.

10) Envelope of projectile motion. This investigates a generalized version of projectile motion – discover what shape is created.

11) Projectile Motion Investigation II. This takes the usual projectile motion ideas and generalises them to investigate equations of ellipses formed.

12) Projectile Motion III: Varying gravity. What would projectile motion look like on different planets?

13) The Tusi couple – A circle rolling inside a circle. This is a lovely result which uses parametric functions to create a beautiful example of mathematical art.

14) Galileo’s Inclined Planes. How did Galileo achieve his breakthrough understanding of gravity? Follow in the footsteps of a genius!

Statistics and modelling 1 [topics could be studied in-depth]

1) Traffic flow: How maths can model traffic on the roads.

2) Logistic function and constrained growth

3) Benford’s Law – using statistics to catch criminals by making use of a surprising distribution.

4) Bad maths in court – how a misuse of statistics in the courtroom can lead to devastating miscarriages of justice.

5) The mathematics of cons – how con artists use pyramid schemes to get rich quick.

6) Impact Earth – what would happen if an asteroid or meteorite hit the Earth?

7) Black Swan events – how usefully can mathematics predict small probability high impact events?

8) Modelling happiness – how understanding utility value can make you happier.

9) Does finger length predict mathematical ability? Investigate the surprising correlation between finger ratios and all sorts of abilities and traits.

10) Modelling epidemics/spread of a virus

11) The Monty Hall problem – this video will show why statistics often lead you to unintuitive results.

12) Monte Carlo simulations

14) Bayes’ theorem: How understanding probability is essential to our legal system.

15) Birthday paradox: The birthday paradox shows how intuitive ideas on probability can often be wrong. How many people need to be in a room for it to be at least 50% likely that two people will share the same birthday? Find out!

16) Are we living in a computer simulation? Look at the Bayesian logic behind the argument that we are living in a computer simulation.

17) Does sacking a football manager affect results? A chance to look at some statistics with surprising results.

18) Which times tables do students find most difficult? A good example of how to conduct a statistical investigation in mathematics.

19) Introduction to Modelling. This is a fantastic 70 page booklet explaining different modelling methods from Moody’s Mega Maths Challenge.

20) Modelling infectious diseases – how we can use mathematics to predict how diseases like measles will spread through a population

21) Using Chi Squared to crack codes – Chi squared can be used to crack Vigenere codes which for hundreds of years were thought to be unbreakable. Unleash your inner spy!

22) Modelling Zombies – How do zombies spread? What is your best way of surviving the zombie apocalypse? Surprisingly maths can help!

23) Modelling music with sine waves – how we can understand different notes by sine waves of different frequencies. Listen to the sounds that different sine waves make.

24) Are you psychic? Use the binomial distribution to test your ESP abilities.

25) Reaction times – are you above or below average? Model your data using a normal distribution.

26) Modelling volcanoes – look at how the Poisson distribution can predict volcanic eruptions, and perhaps explore some more advanced statistical tests.

27) Could Trump win the next election? How the normal distribution is used to predict elections.

28) How to avoid a Troll – an example of a problem solving based investigation

29) The Gini Coefficient – How to model economic inequality

30) Maths of Global Warming – Modeling Climate Change – Using Desmos to model the change in atmospheric Carbon Dioxide.

31) Modelling radioactive decay – the mathematics behind radioactivity decay, used extensively in science.

32) Circular Motion: Modelling a Ferris wheel. Use Tracker software to create a Sine wave.

33) Spotting Asset Bubbles. How to use modeling to predict booms and busts.

34) The Rise of Bitcoin. Is Bitcoin going to keep rising or crash?

35) Fun with Functions!. Some nice examples of using polar coordinates to create interesting designs.

36) Predicting the UK election using linear regression. The use of regression in polling predictions.

37) Modelling a Nuclear War. What would happen to the climate in the event of a nuclear war?

38) Modelling a football season. We can use a Poisson model and some Excel expertise to predict the outcome of sports matches – a technique used by gambling firms.

39)Modeling hours of daylight – using Desmos to plot the changing hours of daylight in different countries.

40) Modelling the spread of Coronavirus (COVID-19). Using the SIR model to understand epidemics.

42) The Martingale system paradox. Explore a curious betting system still used in currency trading today.

Statistics and modelling 2 [more simplistic topics: correlation, normal, Chi squared]

1) Is there a correlation between hours of sleep and exam grades?Studies have shown that a good night’s sleep raises academic attainment.

2) Is there a correlation between height and weight? (pdf). The NHS use a chart to decide what someone should weigh depending on their height. Does this mean that height is a good indicator of weight?

3) Is there a correlation between arm span and foot height? This is also a potential opportunity to discuss the Golden Ratio in nature.

4) Is there a correlation between smoking and lung capacity?

5) Is there a correlation between GDP and life expectancy? Run the Gapminder graph to show the changing relationship between GDP and life expectancy over the past few decades.

7) Is there a correlation between numbers of yellow cards a game and league position?
Use the Guardian Stats data to find out if teams which commit the most fouls also do the best in the league.

8) Is there a correlation between Olympic 100m sprint times and Olympic 15000m times?
Use the Olympic database to find out if the 1500m times have got faster in the same way the 100m times have got quicker over the past few decades.

9) Is there a correlation between time taken getting to school and the distance a student lives from school?

11) Is there a correlation between stock prices of different companies? Use Google Finance to collect data on company share prices.

13) Is there a correlation between height and basketball ability? Look at some stats for NBA players to find out.

14) Is there a correlation between stress and blood pressure?

16) Are a sample of student heights normally distributed? We know that adult population heights are normally distributed – what about student heights?

17) Are a sample of flower heights normally distributed?

18) Are a sample of student weights normally distributed?

19) Are the IB maths test scores normally distributed? (pdf). IB test scores are designed to fit a bell curve. Investigate how the scores from different IB subjects compare.

20) Are the weights of “1kg” bags of sugar normally distributed?

21) Does gender affect hours playing sport? A UK study showed that primary school girls play much less sport than boys.

22) Investigation into the distribution of word lengths in different languages. The English language has an average word length of 5.1 words. How does that compare with other languages?

23) Do bilingual students have a greater memory recall than non-bilingual students?
Studies have shown that bilingual students have better “working memory” – does this include memory recall?

Games and game theory

1) The prisoner’s dilemma: The use of game theory in psychology and economics.

3) Gambler’s fallacy: A good chance to investigate misconceptions in probability and probabilities in gambling. Why does the house always win?

4) Bluffing in Poker: How probability and game theory can be used to explore the the best strategies for bluffing in poker.

5) Knight’s tour in chess: This chess puzzle asks how many moves a knight must make to visit all squares on a chess board.

8) How to “Solve” Noughts and Crossess (Tic Tac Toe) – using game theory. This topics provides a fascinating introduction to both combinatorial Game Theory and Group Theory.

9) Maths and football – Do managerial sackings really lead to an improvement in results? We can analyse the data to find out. Also look at the finances behind Premier league teams

10) Is there a correlation between Premier League wages and league position? Also look at how the Championship compares to the Premier League.

11) The One Time Pad – an uncrackable code? Explore the maths behind code making and breaking.

12) How to win at Rock Paper Scissors. Look at some of the maths (and psychology behind winning this game.

13) The Watson Selection Task – a puzzle which tests logical reasoning. Are maths students better than history students?

Topology and networks

3) Chinese postman problem – This is a problem from graph theory – how can a postman deliver letters to every house on his streets in the shortest time possible?

4) Travelling salesman problem

5) Königsberg bridge problem: The use of networks to solve problems. This particular problem was solved by Euler.

6) Handshake problem: With n people in a room, how many handshakes are required so that everyone shakes hands with everyone else?

7) Möbius strip: An amazing shape which is a loop with only 1 side and 1 edge.

10) Codes and ciphers: ISBN codes and credit card codes are just some examples of how codes are essential to modern life. Maths can be used to both make these codes and break them.

11) Zeno’s paradox of Achilles and the tortoise: How can a running Achilles ever catch the tortoise if in the time taken to halve the distance, the tortoise has moved yet further away?

12) Four colour map theorem – a puzzle that requires that a map can be coloured in so that every neighbouring country is in a different colour. What is the minimum number of colours needed for any map?

13) Telephone Numbers – these are numbers with special properties which grow very large very quickly. This topic links to graph theory.

14)The Poincare Conjecture and Grigori Perelman – Learn about the reclusive Russian mathematician who turned down $1 million for solving one of the world’s most difficult maths problems. Mathematics and Physics 1) The Monkey and the Hunter – How to Shoot a Monkey – Using Newtonian mathematics to decide where to aim when shooting a monkey in a tree. 2) How to Design a Parachute – looking at the physics behind parachute design to ensure a safe landing! 3) Galileo: Throwing cannonballs off The Leaning Tower of Pisa – Recreating Galileo’s classic experiment, and using maths to understand the surprising result. 4) Rocket Science and Lagrange Points – how clever mathematics is used to keep satellites in just the right place. 5) Fourier Transforms – the most important tool in mathematics? – An essential component of JPEG, DNA analysis, WIFI signals, MRI scans, guitar amps – find out about the maths behind these essential technologies. 6) Bullet projectile motion experiment – using Tracker software to model the motion of a bullet. 7) Quantum Mechanics – a statistical universe? Look at the inherent probabilistic nature of the universe with some quantum mechanics. 8) Log Graphs to Plot Planetary Patterns. The planets follow a surprising pattern when measuring their distances. 9) Modeling with springs and weights. Some classic physics – which generates some nice mathematical graphs. 10) Is Intergalactic space travel possible? Using the physics of travel near the speed of light to see how we could travel to other stars. Maths and computing 1) The Van Eck Sequence – The Van Eck Sequence is a sequence that we still don’t fully understand – we can use programing to help! 2) Solving maths problems using computers – computers are really useful in solving mathematical problems. Here are some examples solved using Python. 3) Stacking cannonballs – solving maths with code – how to stack cannonballs in different configurations. 4) What’s so special about 277777788888899? – Playing around with multiplicative persistence – can you break the world record? 5) Project Euler: Coding to Solve Maths Problems. A nice starting point for students good at coding – who want to put these skills to the test mathematically. 6) Square Triangular Numbers. Can we use a mixture of pure maths and computing to solve this problem? 7) When do 2 squares equal 2 cubes? Can we use a mixture of pure maths and computing to solve this problem? 9) Coding Hailstone Numbers. How can we use computers to gain a deeper understanding of sequences? Further ideas: 1) Radiocarbon dating – understanding radioactive decay allows scientists and historians to accurately work out something’s age – whether it be from thousands or even millions of years ago. 2) Gravity, orbits and escape velocity – Escape velocity is the speed required to break free from a body’s gravitational pull. Essential knowledge for future astronauts. 3) Mathematical methods in economics – maths is essential in both business and economics – explore some economics based maths problems. 4) Genetics – Look at the mathematics behind genetic inheritance and natural selection. 5) Elliptical orbits – Planets and comets have elliptical orbits as they are influenced by the gravitational pull of other bodies in space. Investigate some rocket science! 6) Logarithmic scales – Decibel, Richter, etc. are examples of log scales – investigate how these scales are used and what they mean. 7) Fibonacci sequence and spirals in nature – There are lots of examples of the Fibonacci sequence in real life – from pine cones to petals to modelling populations and the stock market. 8) Change in a person’s BMI over time – There are lots of examples of BMI stats investigations online – see if you can think of an interesting twist. 9) Designing bridges – Mathematics is essential for engineers such as bridge builders – investigate how to design structures that carry weight without collapse. 10) Mathematical card tricks – investigate some maths magic. 11) Flatland by Edwin Abbott – This famous book helps understand how to imagine extra dimension. You can watch a short video on it here 12) Towers of Hanoi puzzle – This famous puzzle requires logic and patience. Can you find the pattern behind it? 13) Different number systems – Learn how to add, subtract, multiply and divide in Binary. Investigate how binary is used – link to codes and computing. 14) Methods for solving differential equations – Differential equations are amazingly powerful at modelling real life – from population growth to to pendulum motion. Investigate how to solve them. 15) Modelling epidemics/spread of a virus – what is the mathematics behind understanding how epidemics occur? Look at how infectious Ebola really is. 16) Hyperbolic functions – These are linked to the normal trigonometric functions but with notable differences. They are useful for modelling more complex shapes. 17) Medical data mining – Explore the use and misuse of statistics in medicine and science. 18)Waging war with maths: Hollow squares. How mathematical formations were used to fight wars. 19) The Barnsley Fern: Mathematical Art – how can we use iterative processes to create mathematical art? ## The ordinary and matrix continued fractions in the theoretical analysis of Hermitian and relaxation operators We consider the theory of the resolvent for Hermitian or relaxation operators, and we address the problem of the explicit evaluation of the Green's function. The continued fractions are shown to be an efficient and natural calculational tool. With respect to the literature, we provide here for the first time a systematic theory, which develops directly from the general Dyson equation. Our novel treatment allows us to extend the theory of ordinary continued fractions from scalar to matrix parameters it provides a unified formal treatment of both Hermitian and relaxation operators it makes transparent the natural link, overlooked in the literature, between continued fraction approach and renormalization group techniques finally it allows to establish the relationship with the moment method. A few examples are also reported to illustrate some relevant numerical or applicative aspects. ## Continued Fractions This is a thesis/capstone paper. The paper should have an abstract, an introduction and different sections.(i will attach the format) Talk about euclidean algorithm and continued fractions, finite and infinite cont. fract.,convergent continued fractions and linear algebra, periodic continued fractions and quadratic irrationals. CONTINUED FRACTIONS Abstract. In this paper, we will talk about continued fractions. To get started, we will discuss the development of the subject throughout history, we will give some definitions, theorems, proofs and some examples. We will show the expansion properties of continuous fractions and its the convergents. Lastly, we will touch based on Diophantine equations. We use only the theory of simple continued fractions to show how one may find fundamental solutions of these equations. 1. Introduction Continued fractions have a long history behind them. They started with the Euclidean algorithm. Continued fractions provide much insight into mathematical problems, particularly into the nature of numbers. In the computer field, continued fractions are used to give approximations to various complicated functions, rapid numerical results valuable to scientists and more. The purpose of this paper is to analyze the structures or patterns of irrational numbers expansions. First, let us start with the description of the Euclidean algorithm. 2. Euclidean Algorithm and continued fractions Though the Euclidean algorithm first appeared in Euclid?s The Elements, written around 300 BC, the algorithm itself is thought to have been in existence since around 500 BC. It is one of the oldest mathematical algorithms. Euclid?s method however, was applied geometrically as a method to find a common measure between two lines segments and numbers. Basically, the algorithm leads us to perform successive division. First of the smaller of the two numbers into the larger, followed by the resulting remainder divided into the divisor of each division until the remainder is equal to zero. This leads us to our first theorem. Theorem 2.1. The Euclidean algorithm always terminates after finitely many steps. Proof. We have: a = bq1 + r1, b = r1q2 + r2, r1 = r2q3 + r3, r2 = r3q4 + r4, rn-2 = rn-1qn + rn, rn-1 = rnqn+1 + 0, We know that: b > r1 > r2 > … > rn-2 > rr-1 > rn > 0. This only works for two numbers because they are integers. So the algorithm has to terminate. Theorem 2.2. Let a and b be positive integers. Then the Euclidean algorithm gives the gcd of (a,b). Proof. The key idea of the proof is to prove that rn is the greatest common divisor for a and b. We observe the following: gcd(a, b) = gcd(a – bq1,b) = gcd(r1, b) = gcd(r1, b – r1q2)=gcd(r1, r2) =gcd(r1 – r2q3, r2)= gcd(r3, r2). By mathematical induction, we end up seeing that gcd(a, b) = gcd(rn-1, rn)=gcd(rn, 0) = rn. Concluding the proof that rn is the gcd(a,b). This following example will be used to provide an idea as to why the Euclidean algorithm works. Example 2.1. Find the GCD of 11 and 6. By the Euclidean Algorithm we have: 11 = 1 x 6 +5 6 = 1 x 5 + 1 1 2 CONTINUED FRACTIONS 5 = 5 x 1 + 0 As we now have a remainder of 0, we stop the process. The new larger number is the GCD of the two original numbers. In our case, it is 1. By using the Euclidean algorithm, we can express rational numbers in a very special way. For instance, the Euclidean algorithm produces the following sequence of equations: Example 2.2. From our previous example, we divide both sides of each equation by the divisor of that equation, we obtain: 11 6 =1+5/6, 6 5 =1+1/5. By combining these equations, we find that 11 6 =1 + 1 1 + 1 5 These are what we will call Continued Fractions. 3. continued fractions and rational numbers Definition 3.1. An expression of the form a0 + 1 a1 + 1 a2 + 1 a3 + … where [a0 a1, a2, a3…] are positive real numbers is called a simple continued fraction. The real numbers a0 a1, a2, a3… are called partial quotients of the continued fraction. We will use this notation [a0, a1, a2, a3…] to represent the simple continued fraction in the above definition. There are different categories of continued fractions. In this paper, we really are referring to simple continued fractions, the only form we consider. Theorem 3.1. A number can be represented as a finite simple continued fraction if and only if it is a rational number. Proof. Let p/q, where p and q are integers with q > 0. Let r0 = p and r1 = q. Then the Euclidean algorithm produces the following sequence of equations: r0 = r1a1 + r2, 0 Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers Template:Mvar and Template:Mvar, p 2 − 2q 2 = ±1 if and only if << safesubst:#invoke:Unsubst||$B= p / q >> is a convergent of Template:Sqrt.

Continued fractions also play a role in the study of dynamical systems, where they tie together the Farey fractions which are seen in the Mandelbrot set with Minkowski's question mark function and the modular group Gamma.

The backwards shift operator for continued fractions is the map h(x) = 1/Template:Mvar − ⌊1/Template:Mvar⌋ called the Gauss map, which lops off digits of a continued fraction expansion: h([0 a1, a2, a3, …]) = [0 a2, a3, …] . The transfer operator of this map is called the Gauss–Kuzmin–Wirsing operator. The distribution of the digits in continued fractions is given by the zero'th eigenvector of this operator, and is called the Gauss–Kuzmin distribution.