#### Research on Pythagoras and his work.

Beyond Pythagoras Introduction: Research on Pythagoras and his work Pythagoras of Samos is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure. We do have details of Pythagoras's life from early biographies which use important original sources yet are written by authors who attribute divine powers to him, and whose aim was to present him as a god-like figure. What we present below is an attempt to collect together the most reliable sources to reconstruct an account of Pythagoras's life. There is fairly good agreement on the main events of his life but most of the dates are disputed with different scholars giving dates which differ by 20 years. Some historians treat all this information as merely legends but, even if the reader treats it in this way, being such an early record it is of historical importance. Pythagoras's father was Mnesarchus, while his mother was Pythais and she was a native of Samos. Mnesarchus was a merchant who came from Tyre, and

#### Investigate the probability of someone rolling a die and the probability of it landing on particular number for a player to win the game

We have been asked to investigate the probability of someone rolling a die and the probability of it landing on particular number for a player to win the game. For A to win he/she must roll a 1 and if he/she does this they have won the game. For B to win, first of all A must lose and they must roll 2 or a 3 and then they have won the game. For C to win they must roll a 4,5 or 6 and of course B must have lost. I have to investigate these tasks: . The probability of A, B or C winning. 2. Who will be the most likely winner? 3. Most likely length of the game. I have first of all drawn a tree diagram so it is easier to interpret and it is easier to see things visually: From this I tried to find the probability that no one wins in Round 1 and this is how I did it: P (LLL) = 1- (5 x 2 x 1) 6 3 2 P (LLL) = 1 - 5 18 P (LLL) = 13 18 I also found the probability of A, B and C winning in Round 1: P (A) wins = 1 6 P (B) wins = 5 x 1 = 5 6 2 18 P (C) wins = 5 x 2 x 1 = 5 6 3 2 18 In the second round the probabilities of winning will be different, as you must say that no one won in the last round. This is how I found out the probability of A, B and C winning in the second round: P (A) wins = 5 x 2 x 1 x 1 = 5 6 3 2 6 108 P (B) wins = 5 x 2 x 1 x 5 x 1 = 25 6 3 2 6 2 324 P (C) wins = 5 x 2 x 1 x 5 x 2 x 1 =

#### Maths Dice Investigation

Rules 10 counters are placed in the centre of the dodecahedron. Two dice are then rolled the amount of the two numbers on the dice are then recorded. A counter from the centre of the dodecahedron is then placed into "A" wins or "B" wins depending on the sum of the two numbers. E.g. 3 + 2 = 5 therefore it goes into A wins 6 + 4 = 10 so obviously B wins And so on. this continues until there are no counters remaining inside the dodecahedron. Prediction I predict that B will win because it is more likely to get the numbers that when added together make the sum of the numbers found in B. Results Game Counters in A Counters in B Who won ? 5 5 draw 2 4 6 b won 3 3 7 b won 4 5 5 draw 5 5 5 draw 6 4 6 b won 7 3 7 b won 8 6 4 a won 9 2 8 b won 0 4 6 b The results table clearly see that after rolling 2 dice 10 times the number of counters placed in either "A" wins or "B" wins add up to 10. This is obviously because we used 10 counters. After carefully looking at the results table it becomes apparent that "A Wins" "B Wins" Win = 1 Win = 6 Draw = 3 Draw = 3 Lose = 6 Lose = 1 We can write this as a probability equation: P(A Wins) =

#### Beyond Pythagoras P.1 Pythagoras Theorem is a2+b2= c2 'a' is being the shortest side, 'b' being the middle side and 'c' being

Mahmoud Elsherif Beyond Pythagoras P.1 Pythagoras Theorem is a2+b2= c2 'a' is being the shortest side, 'b' being the middle side and 'c' being the longest side (hypotenuse) of a right angled triangle. The numbers 3,4,5 satisfy this condition and so 32+ 42=52 Because 32= 3*3=9 42=4*4=16 52=5*5=25 32+ 42=52 9+16=25 25=25 This proves Pythagoras Theorem goes with the right angled triangle with the numbers 3,4,5. Next I shall prove that Pythagoras's Theorem applies to 5,12,13 right angled triangle. 52+122=132 Because 52= 5*5=25 22= 12*12=144 32= 13*13=169 Mahmoud Elsherif Beyond Pythagoras P.2 This satisfies the Theorem of Pythagoras's goes with these numbers 5,12,13. Finally I shall prove that Pythagoras's Theorem applies to 7,24,25 right angled triangle. 72+ 242=252 Because 72= 7*7=49 242= 24*24= 576 252=25*25=625. So a2+b2=c2 72+242=252 49+576=625 This proves Pythagoras Theorem goes with the right angle triangle with the sides 7,24,25 Shortest Side Middle Side Longest Side 3 4 5 5 2 3 7 24 25 9 40 41 1 60 61 3 84 85 Mahmoud Elsherif Beyond Pythagoras P.3 I shall find the prediction of the shortest side first. 3,5,7 It goes up in 2 so in my conclusion so it will become 3,5,7,9,11,13 Now I will find the difference between them. The difference is 2 Next I shall find the prediction

#### Pythagoras [Samos, 582 - 500 BC].

Pythagoras [Samos, 582 - 500 BC] Like Thales, Pythagoras is rather known for mathematics than for philosophy. Anyone who can recall math classes will remember the first lessons of geometry that usually start with Pythagoras famous proposition about right-angled triangles: a²+b²=c². Pythagoras found this principle two and a half millennia ago -around 532 BC- and with it his name and philosophy have survived the turbulences of history. His immediate followers were strongly influenced by him, and even until today Pythagoras shines through the mist of ages as one of the brightest figures of early Greek antiquity. What he found out about triangles has been the beginning of mathematics in Western culture, and ever since mathematics -the art of demonstrative and deductive reasoning- has had a profound influence on Western philosophy, which can be observed down to Russel and Wittgenstein. Pythagoras' influence found an expression in visual art and music as well, particularly in the renaissance and baroque epoch. The far-reaching imprint of his ideas is yet more impressive if we consider that he did not leave any original writings. Instead, all what is known about Pythagoras was handed down by generations of philosophers and historiographers, some of whom, like Heraclitus, opposed his views. In this light it is remarkable that Pythagoras' teachings have survived relatively

#### Beyond Pythagoras

Beyond Pythagoras Maths Pure Coursework 1 By: Ben Ingram 0R Beyond Pythagoras Pythagoras Theorem: Pythagoras states that in any right angled triangle of sides 'a', 'b' and 'c' (a being the shortest side, c the hypotenuse): a2 + b2 = c2 E.g. 1. 32 + 42 = 52 9 + 16 = 25 52 = 25 2. 52 + 122 = 132 3. 72 + 242 = 252 25 + 144 = 169 49 + 576 = 625 132 = 169 252 = 625 All the above examples are using an odd number for 'a'. It can however, work with an even number. E.g. 1. 102 + 242 = 262 100 + 576 = 676 262 = 676 N.B. Neither 'a' nor 'b' can ever be 1. If either where then the difference between the two totals would only be 1. There are no 2 square numbers with a difference of 1. 32 9 42 16 52 25 62 36 72 49 82 64 92 81 102 100 112 121 As shown in the above table, there are no square numbers with a difference of anywhere near 1. Part 1: Aim: To investigate the family of Pythagorean Triplets where the shortest side (a) is an odd number and all three sides are positive integers. By putting the triplets I am provided with in a table, along with the next four sets, I can search for formulae or patterns connecting

#### Beyond Pythagoras

Mahmoud Elsherif Beyond Pythagoras P.1 Pythagoras Theorem is a2+b2= c2 'a' is being the shortest side, 'b' being the middle side and 'c' being the longest side (hypotenuse) of a right angled triangle. The numbers 3,4,5 satisfy this condition and so 32+ 42=52 Because 32= 3*3=9 42=4*4=16 52=5*5=25 32+ 42=52 9+16=25 25=25 This proves Pythagoras Theorem goes with the right angled triangle with the numbers 3,4,5. Next I shall prove that Pythagoras's Theorem applies to 5,12,13 right angled triangle. 52+122=132 Because 52= 5*5=25 22= 12*12=144 32= 13*13=169 Mahmoud Elsherif Beyond Pythagoras P.2 This satisfies the Theorem of Pythagoras's goes with these numbers 5,12,13. Finally I shall prove that Pythagoras's Theorem applies to 7,24,25 right angled triangle. 72+ 242=252 Because 72= 7*7=49 242= 24*24= 576 252=25*25=625. So a2+b2=c2 72+242=252 49+576=625 This proves Pythagoras Theorem goes with the right angle triangle with the sides 7,24,25 Shortest Side Middle Side Longest Side 3 4 5 5 2 3 7 24 25 9 40 41 1 60 61 3 84 85 Mahmoud Elsherif Beyond Pythagoras P.3 I shall find the prediction of the shortest side first. 3,5,7 It goes up in 2 so in my conclusion so it will become 3,5,7,9,11,13 Now I will find the difference between them. The difference is 2 Next I shall find the prediction of

#### Beyond Pythagoras

BEYOND PYTHAGORAS MATHS COURSEWORK INTRODUCTION Pythagoras was a Greek mathematician and philosopher. He lived in 400 BC and was one of the first great mathematical thinkers. He spent most of his life in Sicily and southern Italy. He had a group of follows who went around and thought other people what he had taught them who were called the Pythagoreans. Pythagoras himself is best known for proving that the Pythagorean Theorem was true. The Sumerians, two thousand years earlier, already knew that it was generally true, and they used it in their measurements, but Pythagoras proved that it would always be true. The Pythagorean Theorem says that in a right triangle, the sum of the squares of the two right-angle sides will always be the same as the square of the hypotenuse (the long side). A2 + B2 = C2 Pythagoras theorem can also help in real life. Here is an example: Say you were walking though a park and wanted to take a short cut. With Pythagoras's theorem you could work out exactly how long you would have to walk though the grass, rather then talking the long route by walking on the paths. PLAN I am going to investigate the three triangles I have been given. They are all right-angled triangles, with 3 sides, all different lengths. The three triangles satisfy the Pythagoras theorem. The theorem states that the hypotenuse side (longest side) must equal the 2 shorter

#### Beyond Pythagoras

Beyond Pythagoras In this investigation, I shall be studying the relationship between the lengths of the three sides of right angled triangles, their perimeters and their areas. I aim to be able to: -Make predictions about Pythagorean triples -Make generalizations about the lengths of side -Make generalizations about the perimeter and area of corresponding triangles My Table of Results for the Triangles n Length of Shortest Side Length of Middle Side Length of Longest Side Perimeter Area 3 4 5 2 6 2 5 2 3 30 30 3 7 24 25 56 84 4 9 40 41 90 80 5 1 60 61 32 330 6 3 84 85 85 546 7 5 12 13 240 840 8 7 44 45 300 224 9 9 80 81 380 710 0 21 220 221 460 2310 1 23 264 265 552 3036 2 25 312 313 650 3900 3 27 364 365 756 4914 4 29 420 421 870 6090 5 31 480 481 992 7440 6 33 544 545 122 8976 The reason I used certain triples in my table (for example there is also another triple for 9 as the shortest side which has 12 as the middle side length and 15 as the hypotenuse) is because they followed the pattern I was looking at. Even though some of the other triple combinations might have worked, I chose the ones out of them which went best with the other combinations. The 9,40,41 triple has a difference of 1 between its middle and longest side lengths, which is what the other triples

#### Beyond Pythagoras

Daniel Cook 10R Beyond Pythagoras: Year 10 GCSE Coursework I am going to study Pythagoras' theorem. Pythagoras Theorem is a2 + b2 = c2. 'a' being the shortest side, 'b' being the middle side and 'c' being the longest side (hypotenuse) of a right angled triangle. For example, I will use 32 x 42 = 52 . This is because: 32 = 3 x 3 = 9 42 = 4 x 4 = 16 52 = 5 x 5 = 25 So.. 9 +16 = 25 For this table, I am using the term a, b, b + 1 Triangle Number (n) Length of shortest side Length of middle side Length of longest side Perimeter Area 3 4 5 2 6 2 5 2 3 30 30 3 7 24 25 56 84 4 9 40 41 90 80 5 1 60 61 32 330 6 3 84 85 83 546 7 5 12 13 240 840 8 7 44 45 296 224 Formulas Shortest side = 2n + 1, n being the triangle number Middle side = 2n2 + 2n. This is because: Triangle Number 1 = 2 x 2 2 = 3 x 4 3 = 4 x 6 4 = 5 x 8 = 2n2 + 2n 5 = 6 x 10 6 = 7 x 12 7 = 8 x 14 8 = 9 x 16 Longest side = 2n2 + 2n + 1 Perimeter = A + B + C Area = A x B x 0.5 Box Methods Shortest² = (2n + 1) ² 2n +1 2n 4n² 2n +1 2n = 4n² + 4n + 1 Middle² = (2n² + 2n)² 2n² +2n 2n² 4n4 4n³ +2n 4n³ 4n² = 4n4 + 8n³ + 4n² Longest² = (2n² + 2n + 1)² 2n² +2n +1 2n² 4n4 4n³