Euclid and his contribution in development of math

This is a presentation on Euclid's life.

1. Euclid and his contribution in development of math matics Presented By – Akshay Kumar Kushawaha Class – 10 th , B Roll No. 03 Radiant Academy Summer Vacation Holiday Homework

2. Euclid was a Greek mathematician, often referred to as the "Father of Geometry". Euclid was born in 365 B.C. He went to school at Plato's academy in Athens, Greece. He founded the university in Alexandria, Egypt. He taught there for the rest of his life. One of his students was Archimedes. Euclid was kind, fair, and patient. Once, when a boy asked what the point of learning math was, Euclid gave him a coin and said, "He must make gain out of what he learns." Another time, he was teaching a king. When the king asked if there was an easier way to learn geometry Euclid said, "There is no royal road to geometry." Then he sent the king to study. In his time he was thought of as being too thorough. Now, in our time, we think he wasn't thorough enough. Euclid died in 275 B.C. Euclid's most famous work was the Elements. This series of books was used as a center for teaching geometry for 2,000 years. It has been translated into Latin and Arabic. The Elements were divided into thirteen books, which subjects are as follows: Books 1-6= plane geometry, books 7-9= number theory, book 10= Eudoxus's theory of irrational numbers, and books 11-13= solid geometry. More than 1,000 editions of Elements have been published since 1482. Elements were popular until the 20th century. Summer Vacation Holiday Homework Euclid’s Biography

3. Summer Vacation Holiday Homework Continued…………. Born:   c. 365 BC Birthplace:  Alexandria, Egypt Died:   c. 275 BC Location of death:   Alexandria, Egypt Cause of death:   unspecified Occupation:   Mathematician, Educator Nationality:   Ancient Greece Executive summary:   Father of geometry University: Plato's Academy, Athens, Greece Teacher: Library of Alexandria, Alexandria, Egypt Asteroid Namesake 4354 Euclides Lunar Crater Euclid (7.4S, 29.5W, 11km dia, 700m height) Eponyms Euclidean geometry Slave-owners Author of books: Elements (13 volumes)  Data (plane geometry)  On Divisions (geometry)  Optics (applied mathematics)  Phenomena (astronomy)

4. Summer Vacation Holiday Homework EUCLID’S DEFINITONS Some of the definitions made by Euclid in volume I of ‘The Elements’ that we take for granted today are as follows :- A point is that which has no part. A line is breadth less length. The ends of a line are points. A straight line is that which has length only. Euclid's construction of a regular dodecahedron.

5. Summer Vacation Holiday Homework Continued…………. A surface is that which has length and breadth only. The edges of a surface are lines A plane surface is a surface which lies evenly with the straight lines on itself Axioms or postulates are the assumptions which are obvious universal truths. They are not proved. Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning .

6. EUCLID’S AXIOMs SOME OF EUCLID’S AXIOMS WERE :- Things which are equal to the same thing are equal to one another. i.e. if a=c and b=c then a=b. Here a, b and c are same kind of things. If equals are added to equals, the wholes are equal. i.e. if a=b and c=d, then a+c = b+d Also a=b then this implies that a+c = b+c .

7. Continued….. If equals are subtracted, the remainders are equal. Things which coincide with one another are equal to one another. Things which are double of the same things are equal to one another The whole is greater than the part. That is if a > b then there exists c such that a =b + c. Here, b is a part of a and therefore, a is greater than b. Things which are halves of the same things are equal to one another.

8. EUCLID’S FIVE POSTULATES EUCLID’S POSTULATES WERE :- POSTULATE 1 :- A straight line may be drawn from any one point to any other point Axiom :- Given two distinct points, there is a unique line that passes through them POSTULATE 2 :- A terminated line can be produced infinitely

9. Continued….. POSTULATE 3 :- A circle can be drawn with any centre and any radius POSTULATE 4 :- All right angles are equal to one another POSTULATE 5 :- If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

10. THEOREMS WITH PROOF THEOREM :- Two distinct lines cannot have more than one point in common PROOF :- Two lines ‘l’ and ‘m’ are given. We need to prove that they have only one point in common Let us suppose that the two lines intersects in two distinct points, say P and Q That is two line passes through two distinct points P and Q But this assumptions clashes with the axiom that only one line can pass through two distinct points Therefore the assumption that two lines intersect in two distinct points is wrong Therefore we conclude that two distinct lines cannot have more than one point in common

11. Euclid Division lemma THEOREM :- Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, ≤ r < b.

12. Thank You