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Euclid - Mathematician

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Published in: Mathematics
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All details about the Mathematician - Euclid

Sumayya H / Dubai

2 years of teaching experience

Qualification: B.Ed (pursuing : 2020 - 2022), MSc Mathematics

Teaches: Maths, Mathematics

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  1. EUCLID OF ALEXANDRIA
  2. Euclid (350-300 BC)
  3. EucCids Early Life The documentation of Euclid's life is scant, at best. NV hile we have a great deal of his work in extant, facts about Euclid as a person come down to us mainly through little snippets by Proclus and Apollonius (among others). Proclus, a Greek philosopher living in the 5th century CAE, writes retrospectively and so his information must be taken with a grain of salt. He writes that Euclid taught in Alexandria, Egypt during the time of Ptolemy I Soter (4th-3rd century BCE). This places him as younger than Plato, but slightly older than Archimedes. Euclid was likely born around 300 BCE and resided in Alexandria, Egypt for most, if not all, of his life. Some historians think that he might have studied for a bit at Plato's Academy in Athens, but this is conjecture based on his style of teaching. Past this, nothing is known of Euclid
  4. ( EUCLID'SCAREE@) Often called the "Father of Geometry," Euclid was a teacher of mathematics, cultivating a school of pupils not unlike the style of the Academy. Proclus writes that Ptolemy once asked Euclid if there was a "shortened way to study geometry than the Elements ", to which Euclid replied "there was no royal road to geometry." This would suggest that not only was Euclid noteworthy among mathematicians and scientists in Alexandria, but was prominent enough to have an audience with the ruler of Egypt. As with details of his early life, we don't know specifics regarding his career, save for his extant works and the fact that he was a prominent teacher in Alexandria.
  5. EUCLD AS OF GEONffTRY
  6. EUCLI 1) S ELEMENTS Geometry. Eucliås: DATA:
  7. "The laws of nature are but the mathematical calculations of god " he wrote the most important and successful mathematical textbook of all time, the "Stoicheion" or 'Elements" Second most printed book in the world. ' probably true 'to ' proof that it is true ' compilation and explanation of all the known mathematics of his time, including the work of Pythagoras, Hippocrates, Theudius etc 0 13 volumes Gave definitions first - and then to get a starting point he made some assumptions called axioms or postulates 465 theorems and proofs, described in a clear, logical and elegant style, and using only a compass and a straight edge.
  8. Book 2 : Geometric algebra All theorems are constructions, Book 3 : general Pythagoras theorem-law of Book I : Triangles, parallelograms 5 axioms, 5 postulates Book 4 : Construction of regular polygons cosines PLANE GEOMETRY Book 5 : Ratios and proportions , geometric number theory Circles without areas and circumferences, Book 6 : Application of ratios to plane geometry, areas to solve quadratic equations
  9. Book 7 Number theory, antanerlsis Book 8 : Numbers in continued proportions Book 9 : There are infinite numbers, definition of one. Book 10 : Theory of irrational numbers, commesurable and incommesurable Book : Intersection of planes, lines and parallelepipeds Book 12 : Euxodus method- Ratio of areas and volumes of circles Book 13: Construction of 5 regular platonic solids from given sphere
  10. WHAT IS EUCLIDEAN GEOMETRY Euclidean Geometry is considered as an axiomatic system, where all the theorems are derived from the small number of simple axioms. Since the term "Geometry" deals with things like points, line, angles, square, triangle, and other shapes, the Euclidean Geometry is also known as the "plane geometry"
  11. A pointis that which has no part. A lineis breadthless length. A surfaceis that which has length and breadth only. A plane angleis the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. • And when the lines containing the angle are straight, the angle is called rectilinear A boundaryis that which is an extremity of anything A circleis a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. And the point is called the centerof the circle. Rectilinear figureas•e those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. Parallel straight line&e straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
  12. AXIOMS Things which are equal to the same thing are equal to one another. 0 If equals are added to equals, the wholes are equal. ( subtraction ) The whole is greater than the part Things which are double of same things are equal to one another. Things which are halves of same things are equal to one another
  13. POSTULATES A straight line may be drawn from any point to another A terminated line can be produced indefinitely A circle can be drawn with any centre and any radius All right angles are equal to one another 0 If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles
  14. c Euclid's method for constructing of an equilateral triangle from a given straight line segment AB using only a compass and straight edge was Proposition 1 in Book 1 of the "Elements" Euclid's proof of Pythagoras' Theorem revolved around the fact that. if a line is draw from the right angle. perpendicular to the hypotenuse. and down through the square on the hypotenuse, that square is split into two rectangles each having the same area as the squares on the otier two sides: Area 1 Area 2 Area 3 c Area a Area 1 Area 2 Area 3 • Area 4
  15. EUCLID'S CONSTRUCTION OF PLATONIC SOLIDS 5 regular so ids — tetrahedron, cube, octahedron, dodecahedron, icosahedron are called the platonic solds. He constructed these solids in a given sphere and cube.
  16. N/A
  17. EUCLID'S DIVISION LEMMA For any two postitve integers a and b. there exists a unique integer q and r such that a = bq + r. where O Sr S b Euclidean algorithm to find gcd of given 2 numbers " If a number be the least that is measured by prime numbers. it will not be measured by any other prime number except those originally rneasuring it." Lead to form-flation of fundamental theorem arithmetic - Otherwise known as the "unique factorization theorem." asserts that any integer greater than I can be represented as a product Of primes. and that the product is unique apart from the order in which the factors appear. For integers a and b. of which both are not zero, there exist integers x and y such that gcd (a: b) ax + by.
  18. Euclid identified the first four "perfect numbers", numbers that are the sum of all their divisors (excluding the number itself): 496 I $1 62 124 -+- 248; and 8,128 2 16 -+- 32 + 64 -k -+- 254 -+- 508 + + 2,032 + 4,064. Called them as triangular numbers — the sum of all the consecutive numbers up to their largest prime factor: 496 I + 5 + . + + . -+- 126 + Their largest prime factor is a power of 2 less one The number is always a product of this number and the previous power of two: 6 = 21(22— 1); 28 = 22(23— 1); 496 = 24(25 — 1); 8128 = 26(27— 1).
  19. Euclid — Euler theorem that relates perfect numbers to Mersenne primes — an even number is perfect if and only if it has the form 21' - t ( 2P— 1 ) where 2p— I is a prime number. Proof for is irrational Proof that there is an infinite number Of primes. Sum of angles of a triangle is 180 ( using parallel postulate ) Construction of regular polygons Application Of areas to solve quadratic equations.
  20. DATA — the importance of being given 94 geometric propositions given some item or property, then other items or properties are also "given"—that is, they can be determined. Points are given in position. Lines, ratios, angles are given in magnitude. Rectilinear figures are given in form. 0 Tool for problem solving -Data gives the mechanism on how to extend and apply what has been 'given' to perform a equations in 2 variables — existance of unique solution and solving in geometric terms E f KAEIAOT A E AOMEN A. MAPINor EVCLIDIS DATA. or VS A D GEOMETRIÆ Aro•rs 'ED as T. r 't. p HILO sopH1
  21. Phaenomena — A work on Spherical geometry — observing objects in space and using geometry to create measurements Divisions of Figures- It concerns the division of geometrical figures into two or more equal parts or into various ratios. Catoptrics- It examines the mathematical theory of mirrors, especially images formed by plane and spherical concave mirrors. Euclid's Optics-Euclid's Optics was an immensely influential book on light and vision
  22. Lost works.. ... Conics — work on conic sections — Appollonius completed Euclid's four books of conics and added 4 others. Porisms — little is known, believed to be extension of conics Pseudoria or book of fallacies — about errors in reasoning Surface loci — on surfaces and quadric surfaces Works on mechanics ( Arabic )
  23. Euclid's Death and Legacy Nv•Ve can assume that Euclid died in the mid- 3rd century BCE in Alexandria, but that is all we know. However, he left behind a legacy that has survived almost two and a half millennia. His work on geometry and theory is still in use today, governing even advanced models of dimension mathematics He is considered one of the greatest mathematicians to have ever lived, and a European Space Agency's spacecraft was even named in his honor, the Euclid Spacecraft.

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