By Harman Bikram Singh Riar X-B BHAVAN VIDYALAYA ,CHANDIGARH

1. Presentation In Mathematics By Harman Bikram Singh Riar X-B BHAVAN VIDYALAYA ,CHANDIGARH

2. Presentation On Pythagoras & His Theorem (Mathematician)

3. Pythagoras About his life Pythagoras is often referred to as the first pure mathematician. He was born on the island of Samos, Greece in 569 B.C. Various writings place his death between 500 BC and 475 BC in Metapontum , Lucania , Italy. His father, Mnesarchus, was a gem merchant. His mother's name was Pythais. Pythagoras had two or three brothers. Some historians say that Pythagoras was married to a woman named Theano and had a daughter Damo, and a son named Telauges, who succeeded Pythagoras as a teacher and possibly taught Empedocles. Others say that Theano was one of his students, not his wife, and say that Pythagoras never married and had no children

4. Pythagoras was well educated , and he played the lyre throughout his lifetime , knew poetry and recited Homer. He was interested in mathematics, philosophy, astronomy and music and was greatly in fluenced by Pherekydes (philosophy) , Thales (mathematics and astronomy) and Anaximander ( philosophy, geometry ). Pythagoras left Samos for Egypt in about 535 B.C. to study with the priests in the temples.Many of the practices of the society He created later in Italy can be traced to the beliefs of Egyptian priests, such as the codes of secrecy, striving for purity, and refusal to eat beans or to wear animal skins as clothing. Ten years later , when Persia invaded Egypt, Pythagoras was taken prisoner and sent to Babylon (in what is now Iraq), where he met the Magoi, priests who taught him sacred rites. Iamblichus (250-330 AD), a Syrian philosopher, wrote about Pythagoras, "He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians...“. In 520 BC,Pythagoras, now a free man, left Babylon and returned to Samos, and sometime later began a school.. His methods of teaching were not popular with the leaders of Samos, and their desire for him to become involved in politics did not appeal to him, so he left.

5. Establishes School of Philosophy and Science Pythagoras established his school in Croton around 529 B.C. Focusing primarily on religion, philosophy, and mathematics, the school was known as a "homakoeion," meaning a gathering place for people to learn. The school's success can be attributed to the charismatic personality of Pythagoras. In a relatively short period of time, he established a large following, broken up into two groups--the "akousmatikoi," who primarily studied the philosophical teachings of Pythagoras, and the “ mathematikoi , " who focused on theoretical mathematics. Students of Pythagoras' school, known as Pythagoreans , followed a number of strict rules. For instance, they were conscientious vegetarians, took a vow of silence for the first five years of their membership, and kept no written records. Liberal in nature, Pythagoras' school was open to everyone, including women, who it allowed to share in the instructing. Following Pythagoras' own belief in simplicity and disdain for worldly honors, Pythagoreans took no credit for their own work or discoveries, attributing all findings either to their master or the group.

6. Pythagoras Believed:- • All things are numbers. Mathematics is the basis for everything, and geometry is the highest form of mathematical studies. The physical world can understood through mathematics. • The soul resides in the brain, and is immortal. It moves from one being to another, sometimes from a human into an animal, through a series of reincarnations called transmigration until it becomes pure. Pythagoras believed that both mathematics and music could purify. • Numbers have personalities, characteristics, strengths and weaknesses. • The world depends upon the interaction of opposites, such as male and female, lightness and darkness, warm and cold, dry and moist, light and heavy, fast and slow. • Certain symbols have a mystical significance. • All members of the society should observe strict loyalty and secrecy.

7. Because of the strict secrecy among the members of Pythagoras' society, and the fact that they shared ideas and intellectual discoveries within the group and did not give individuals credit, it is difficult to be certain whether all the theorems attributed to Pythagoras were originally his, or whether they came from the communal society of the Pythagoreans. Some of the students of Pythagoras eventually wrote down the theories, teachings and discoveries of the group, but the Pythagoreans always gave credit to Pythagoras as the Master for: • The sum of the angles of a triangle is equal to two right angles. • The theorem of Pythagoras - for a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. The Babylonians understood this 1000 years earlier, but Pythagoras proved it. • Constructing figures of a given area and geometrical algebra. For example they solved various equations by geometrical means. • The discovery of irrational numbers is attributed to the Pythagoreans, but seems unlikely to have been the idea of Pythagoras because it does not align with his philosophy the all things are numbers, since number to him meant the ratio of two whole numbers. • The five regular solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron). It is believed that Pythagoras knew how to construct the first three but not last two. • Pythagoras taught that Earth was a sphere in the center of the Kosmos (Universe), that the planets, stars, and the universe were spherical because the sphere was the most perfect solid figure. He also taught that the paths of the planets were circular. Pythagoras recognized that the morning star was the same as the evening star, Venus.

8. Numbers were the very foundation of Pythagorean philosophy, which maintained that numbers were both mystical in nature and had a reality of their own outside of the human mind. Impressed by the Ratios that existed in musical harmonies, astronomy, and geometrical shapes, the Pythagoreans developed a theory that all things, in essence, were numbers and related through numbers. Out of this belief, they developed certain representations for numbers: 1 was a point, 2 a line, 3 a surface, and 4 a solid. Moral qualities were also numbers, with 4 representing justice and 10 (known as the tetractys) representing the sum of all nature due to its being the sum of 1+2+3+4 (the point, line, plane, and solid). While the Pythagoreans' mystical belief in numbers is largely ignored today, their belief that one could penetrate the secrets of the universe through numbers led them to conduct a careful study of mathematical theory, focusing on the principals of geometry.

9. In Later Years In his own time and the first few centuries following his death, Pythagoras was revered by many, with some accounts of his deeds reaching mythic proportions (such as being able to walk on water). His influence can be found in such famous Greek scientists and philosophers as Euclid and Plato. However, Pythagoras was also viewed with disdain by some, primarily because of his philosophies (such as transmigration of the soul, or reincarnation), which resembled the philosophy of the Orient rather than the Greek philosophy of his day. Although Pythagoras and his school prospered, their controversial philosophies and strong political influence created enemies. With the rise of the democratic party in southern Italy, Pythagoras and his students became the objects of persecution. The democrats considered the Pythagoreans as elitists who placed themselves above others and were also suspicious of their secret rites. This mistrust came to a head sometime between 500 B.C. and 510 B.C., resulting in the destruction of the Pythagorean school and campus. The exact fate of Pythagoras himself remains uncertain. According to some accounts, Pythagoras was killed during this attack. Other accounts indicate that he escaped and lived to be nearly 100 years old. In keeping with Pythagoras' mythic stature, he is also said to have ascended bodily into heaven. Regardless of the fate of Pythagoras, his followers continued his teachings in other lands, eventually returning to Italy to reestablish the Pythagorean school in Tarentum. In terms of influence, the Pythagoreans outlasted all other philosophies of ancient Greece. They established a strong presence and/or influence both in Egypt and ancient Rome, where the Senate erected a statue in honor of Pythagoras as "the wisest and bravest of Greeks."

10. Pythagoras Theorem In mathematics, the Pythagoras' theorem is a relation among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof,although knowledge of the theorem almost certainly predates him. The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

11. In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). This is usually summarized as follows: The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation: or, solved for c: If c is already given, and the length of one of the legs must be found, the following equations can be used (The following equations are simply the converse of the original equation): or This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of c osines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle it reduces to the Pythagorean theorem.

12. Proof of Pythagoras Theorem Proof using similar triangles :- Like most of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles. Like most of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles. Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC.

13. The similarities lead to the two ratios..: As so These can be written as Summing these two equalities, we obtain In other words, the Pythagorean theorem:

14. Consequences and uses of the theorem 1.Pythagorean triples A Pythagorean triple has 3 positive numbers a, b, and c, such that a2 + b2 = c2. In other words , a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Evidence from megalithic monuments on the Northern Europe shows that such triples were known before the discovery of writing.Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13). List of primitive Pythagorean triples up to 100 (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97).

15. 2. The existence of irrational numbers One of the consequences of the Pythagorean theorem is that irrational numbers, such as the square root of two, can be constructed. A right triangle with legs both equal to one unit has hypotenuse length square root of two. The Pythagoreans proved that the square root of two is irrational, and this proof has come down to us even though it flew in the face of their cherished belief that everything was rational. According to the legend, Hippasus, who first proved the irrationality of the square root of two, was drowned at sea as a consequence . 3. Distance in Cartesian coordinates The distance formula in Cartesian coordinates is derived from the Pythagorean theorem. If (x0, y0) and (x1, y1) are points in the plane, then the distance between them, also called the Euclidean distance, is given by More generally, in Euclidean n-space, the Euclidean distance between two points , and , is defined, using the Pythagorean theorem, as:

16. Cultural references to the Pythagorean theorem 1. In The Wizard of Oz, when the Scarecrow receives his diploma from the Wizard, he immediately exhibits his "knowledge" by reciting a mangled and incorrect version of the theorem: "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. Oh, joy, oh, rapture. I've got a brain!" The "knowledge" exhibited by the Scarecrow is incorrect. The accurate statement would have been "The sum of the squares of the legs of a right triangle is equal to the square of the remaining side." 2. In an episode of The Simpsons, Homer quotes the Oz Scarecrow's quote, thus turning the theorem into a cultural reference to a cultural reference. After finding a pair of Henry Kissinger's glasses in a toilet at the Springfield Nuclear Power Plant, Homer puts them on and quotes the scarecrow's mangled formula. A man in a nearby toilet stall then yells out "That's a right triangle, you idiot!" (The comment about square roots remained uncorrected.) 3. Similarly, the Speech software on an Apple MacBook references the Scarecrow's incorrect statement. It is the sample speech when the voice setting 'Ralph' is selected.

17. 4. In the English version of the seventeenth Asterix book "The Mansions of the Gods", Julius Caesar uses the services of an architect named "Squaronthehypotenus" to develop the estate near the village, through which he hopes to absorb the Gaulish village into the Roman culture. • In 2000, Uganda released a coin with the shape of a right triangle. The tail has an image of Pythagoras and the Pythagorean theorem, accompanied with the mention "Pythagoras Millennium". • In the Major-General's Song, "About binomial theorem I'm teeming with a lot o' news, With many cheerful facts about the square of the hypotenuse." • Greece, Japan, San Marino, Sierra Leone, and Surinam have issued postage stamps depicting Pythagoras and the Pythagorean theorem. • 7. In South Park, Eric Cartman shows the Pythagorean theorem during a slide show at show and tell. • 8. In Smart Guy, the teacher is asking the class what the formula for Pythagorean Theorem is. T.J. was supposed to be stupid but replied, "A squared plus B squared equals C squared...". ----------------------------------X---------------------------------