#### Greek Maths

Mas3039 Mathematics: History and Culture Topic 2: The Greek Legacy Essay (2): Discuss Archimedes' double reductio ad absurdum proof for the quadrature of the parabola. Compare and contrast this to a modern calculus proof of the same result. Archimedes of Syracuse (287 - 212 BC) is known as the greatest mathematician of his time and is considered to be one of the greatest of all time. He dominated Greek maths in the third century BC despite not being a native of the city of Alexandria, the centre of mathematical activity.1 The son of an astronomer, Archimedes is credited with many great discoveries in mathematics, mechanics and engineering. During the second Punic war Syracuse was besieged by Romans and we are told that Archimedes invented war machines such as catapults, ropes and pulleys, and devices to set fire to the ships to keep the enemy at bay.1 Archimedes did not think much of these inventions but it meant that mathematics and science were brought "more within the appreciation of the people in general".2 Archimedes' work was both productive and thoroughly detailed and he was never reluctant to share his methods of discovery. What was different about Archimedes compared to other mathematicians of his time was the fact that his work illustrated his method of discovery of a theorem prior to presenting a rigorous proof. This was to stop people claiming his work to be

#### Hange of sign, Newton-Raphson and the rearrangement method and are going to use them to find roots of different equations, and hence

INTRODUCTION For my investigation, I will analyse the use three methods which are called the: change of sign, Newton-Raphson and the rearrangement method and are going to use them to find roots of different equations, and hence compare the merits and flaws of the methods with each other. I will analyse which is the best in terms of factors such as a speed of convergence and ease of use with available software and hardware.cocb 1. For Change of sign method (Decimal search) I use these equations: 2. For Newton-Raphson method (Fixed point iteration) I use these equations: -3ex +2x3+6=0 x3-2x2-2x+3=0 3. For Rearranging f(x) = 0 in the form x = g(x), I use these equations: In my course work, I will use computer (hardware) and Autograph (plot the graph), and Microsoft Excel (spreadsheet), Microsoft Word. Change of sign method: This method finds the roots of an equation by looking at the points where the graph crosses x axis. In these case the value of f(x) change sign from positive to negative or vice versa. And the root must be some where between two values that change. For this equation: Function . I am going to use the decimal search. Check the shape of the graph This is quadratic equation a = 6, b = 0, c = -4 D = b2-4ac = 0)2- 4x6x(-4) = 96>0 the graph has 2 turning points. As the graph shows there are three roots. I calculated each of the f(x) values in

#### Lesson Plan for Physical Education Year GroupKS1 AreaSports hall/gym with apparatus ThemeTravelling in different directions

Lesson Plan for Physical Education Year Group KS1 Area Sports hall/gym with apparatus Theme Travelling in different directions Learning Objectives They should improve on their knowledge of travelling in different directions and successfully put these different ways together in a sequence on both the floor and the apparatus. N.C Reference/ELGs PE KS1 1a, 1b, 2a, 2b, 3a, 3b, 5, 8a, 8b, 8c, 8d. Resources Apparatus including 5 planks, 6 mats, 4 trestle tables and 2 benches. Vocabulary Travel, sequence, link, combine, direction, sideways, backwards, forwards. Activities In the warm up ask the children to move using only their feet. Ask them to come up with different ways that they could do this. For example running, walking, skipping, hopping etc. Then ask them to move using both their hands and their feet - see how many different ways they can come up with. Encourage them to keep moving, move into space and to change direction. (8 minutes) Ask the children to come up with three or four ways of travelling using a combination of hands and feet remembering to think about direction. Using mats they should practice these three or four travelling movements. They should also think of a starting and finishing position. (15 minutes) Supervise the children putting out their designated apparatus. In groups of 6 ask the children to practice their sequence on the particular

#### Determination of moment of inertia of a uniform rectangular bar and a connecting rod using the trifilar suspension, and by swinging the connecting rod as a compound pendulum.

The Trifilar Suspension Tutor : Dr. Yan Name : Manilka Abeysuriya Course : Aeronautical Engineering Group : A - 1 Date : 27 / 01 / 2003 Title : Determination of moment of inertia of a uniform rectangular bar and a connecting rod using the trifilar suspension, and by swinging the connecting rod as a compound pendulum. Introduction : Moment of Inertia can be described as a measure of "unwillingness to change the current motion" of a certain body of mass. In the experiment, the main objective is to find the moment of inertia of a uniform body and an irregular shaped body. Toward achieving this First the center of mass of the bar and the connecting rod was found by balancing them on a knife-edge. Then the bar was placed on the trifilar suspension which is a circular platform suspended by three equally spaced wires of equal length, such that the center of mass of the rod is over the center of the circular platform. Then the whole system is given a small angular displacement, and the periodic time for the oscillations is determined by measuring the time taken for 20 oscillations. By using the equation 1 the moment of inertia of the bar about the axis through its center of mass can be calculated. The same procedure is followed for the connecting rod and its moment of inertia about the axis passing through its center of mass was found.

#### Banach-Tarski Paradox Talk

The Banach-Tarski paradox is a result of research in to set theory. Stefan Banach and Alfred Tarski, after which the paradox is named, were Polish mathematicians. It states that it is possible to take a solid sphere (a "ball"), cut it up into a finite number of pieces, rearrange them using only rotations and translations, and re-assemble them into two identical copies of the original sphere, effectively doubling the volume of the sphere. It's important to be clear what we're talking about. The most important point is that when we mention a sphere, we're talking about a mathematical sphere. Physical spheres and mathematical spheres are not entirely congruent, the most important distinction being that a mathematical sphere is infinitely divisible. As a physical sphere contains a finite (albeit large) number of atoms, it is not so. For simplicity, we shall assume that our mathematical sphere, S, has radius 1. As such, it can be defined as follows: S = {(x,y,z) | x2+y2+z2 ? 1 } That is, S is defined as the set of points that lie within a 3-dimensional spherical area in R3, where R is the set of all real numbers. The Definition: Ai n Aj = Ø for each i and j between 1 and n such that i=j (no two pieces overlap each other) A1 ? A2 ? ... ? An = S (assembling all the pieces yields the original sphere S) There exist T1, T2, ... Tn, where each Ti represents some finite

#### The trifilar suspension is used to determine the moments of inertia of a body about an axis passing through its mass center.

Introduction The trifilar suspension is used to determine the moments of inertia of a body about an axis passing through its mass center. The apparatus consists of a circular platform suspended by three equi-spaced wires of equal length. The body under consideration is placed with its mass centre exactly in the middle of the circular platform. The platform is given a small circular displacement about the vertical axis through its center, and is released. The periodic time of the subsequent motion is obtained by measuring the time taken to complete a definite number of oscillations. Then from the formula the moment of inertia of the body can be calculated. In Addition the moment of inertia of a connecting rod was found. The apparatus was a knife edge to suspend the rod and record the oscillations. This was the second part of the experiment. T=2? L I 0+I r2g M0+M Where L=length of suspension wires. r=radius from center to attachment points of the platform. I0=Moment of inertia of the platform. M0=Mass of platform I=Moment of inertia of body M=Mass of body g=Acceleration due to gravity Results Mass of bar .95 kg Mass of rod .85 kg period (s) Platform .73 Platform with the bar 2.14 Platform with the rod 2.99 Knife edge wide side down length (m) 0.238 period (s) .09 wide side up length (m) 0.098 period (s) 0.97 Theoretical moment of

#### In this investigation I have been asked to find out how many squares would be needed to make up a certain pattern accorting to its sequence.

In this experiment I am going to require the following: A calculator A pencil A pen Variety of sources of information Paper Ruler In this investigation I have been asked to find out how many squares would be needed to make up a certain pattern according to its sequence. The pattern is shown on the front page. In this investigation I hope to find a formula which could be used to find out the number of squares needed to build the pattern at any sequential position. Firstly I will break the problem down into simple steps to begin with and go into more detail to explain my solutions. I will illustrate fully any methods I should use and explain how I applied them to this certain problem. I will firstly carry out this experiment on a 2D pattern and then extend my investigation to 3D. The Number of Squares in Each Sequence I have achieved the following information by drawing out the pattern and extending upon it. Seq. no. 1 2 3 4 5 6 7 8 No. Of cubes 1 5 13 25 41 61 85 113 I am going to use this next method to see if I can work out some sort of pattern: Sequence Calculations Answer =1 1 2 2(1)+3 5 3 2(1+3)+5 13 4 2(1+3+5)+7 25 5 2(1+3+5+7)+9 41 6 2(1+3+5+7+9)+11 61 7 2(1+3+5+7+9+11)+13 85 8 2(1+3+5+7+9+11+13)+15 113 9 2(1+3+5+7+9+11+13+15) +17 145 What I am doing above is shown with the aid of a diagram below; If we take sequence 3:

#### How many squares are there on a chessboard?

How many squares are there on a chessboard? The aim of this investigation is to find out how many squares and rectangles of specific sizes can be found on a chessboard, and too see if there is a common sequence and algebraic formulae for each example. I have started with the simplest example that is to count the different combinations of squares on a range of boards from 2x2 board up to an 8x8 board. Results from the count 2x2 board x1=4 2x2=1 3x3 board x1=9 2x2=4 3x3=1 4x4 board x1=16 2x2=9 3x3=4 4x4=1 5x5 board x1=25 2x2=16 3x3=9 4x4=4 5x5=1 6x6 board x1=36 2x2=25 3x3=16 4x4=9 5x5=4 6x6=1 7x7 board x1=49 2x2=36 3x3=25 4x4=16 5x5=9 6x6=4 7x7=1 8x8 board x1=64 2x2=49 3x3=36 4x4=25 5x5=16 6x6=9 7x7=4 8x8=1 When the 8x8 boards results are analyzed a quadratic sequence can be identified i.e the second difference is a constant. term No of squares st diff 2nd diff 64 2 49 5 3 36 3 2 4 25 1 2 5 6 9 2 6 9 7 2 7 4 5 2 8 3 2 Using the general term for a quadratic sequence where A= 1/2 the constant 2nd difference, which is 2x 1/2 = 1 Yn=An + Bn + C So y1 = 1x1 + B x 1 + C = 64 So B + C = 64 - 1 B + C = 63 Y2 = 1x2 + B x 2 + C = 49 4 + 2B + C = 49 so 2B + C = 45 we now solve the simultaneous equation B + C = 63 2 B + C = 45 ?- B = 18 OR B = - 18 if B = - 18 then C = 63 - (-18)=81 so the

#### 2D and 3D Sequences.

2D and 3D Sequences Project Plan of Investigation In this experiment I am going to require the following: A calculator A pencil A pen Variety of sources of information Paper Ruler In this investigation I have been asked to find out how many squares would be needed to make up a certain pattern according to its sequence. The pattern is shown on the front page. In this investigation I hope to find a formula which could be used to find out the number of squares needed to build the pattern at any sequencial position. Firstly I will break the problem down into simple steps to begin with and go into more detail to explain my solutions. I will illustrate fully any methods I should use and explain how I applied them to this certain problem. I will firstly carry out this experiment on a 2D pattern and then extend my investigation to 3D. The Number of Squares in Each Sequence I have achieved the following information by drawing out the pattern and extending upon it. Seq. no. 1 2 3 4 5 6 7 8 No. Of cubes 1 5 13 25 41 61 85 113 I am going to use this next method to see if I can work out some sort of pattern: Sequence Calculations Answer =1 1 2 2(1)+3 5 3 2(1+3)+5 13 4 2(1+3+5)+7 25 5 2(1+3+5+7)+9 41 6 2(1+3+5+7+9)+11 61 7 2(1+3+5+7+9+11)+13 85 8 2(1+3+5+7+9+11+13)+15 113 9 2(1+3+5+7+9+11+13+15) +17 145 What I am doing above is shown with the aid of a

#### Show how a Stratigraphical Sequence can be Deduced. How can Fossils be Used to Tell the Relative Age

Show how a Stratigraphical Sequence can be Deduced. How can Fossils be Used to Tell the Relative Age "Stratigraphy is the key to understanding the Earths crust and it's materials, structure and past life." "Within geology the study of time is the study of Stratigraphy." The earth's crust consists of bodies of rocks that can be divided into two groups: layered and unlayered. Layered rock bodies are described as stratified and unlayered are described as massive. The most common example of stratified rocks are sedimentary rocks. These have been built up by layer upon layer of sediments, some of which will be vastly similar and in some cases will have changed in character rapidly. Three basic principles must be recognised before a stratified rock sequence can be analysed. Firstly we must accept superposition that states that when a layer of rock was forming the layer beneath it was older. Secondly we must assume originalhorizontality,the idea that layers of rock were originally deposited horizontally and finally original lateralcontinuity. This means the layers of rock extend laterally until physically constrained in some way; this may be a shoreline or an upstanding relief feature. These principles allow a rock sequence to be seen as a record of geological events over time, with the oldest rocks representing the most ancient events at the bottom of the sequence.