MODULE 8 - GAUSS (1777 - 1855)

GAUSS (1777 - 1855)

“Mathematics is the queen of sciences and

Arithmetic is the queen of mathematics”

By Gauss

Born	23  April 1777,Brunswick, Germany (1707-04-15)
Died	23 February 1855 (age 77)
Residence	Germany,
Nationality	German
Fields	Mathematician, telegraphic, astronomy
Institutions	university of Gottingen
Alma mater	Helmstadt.

Gauss the greatest mathematician of the world was born in Brunswick (Germany) on 23^rd April, 1777.He was the son of a day laborer and small contractor. His father name was Jerold doyatric gauss. From the early childhood, gauss was having great interest in mathematics.

In 1799 he wrote a book on mathematics titled “Disquisitions Arithmaeticae.” In the same year he received his Doctor’s degree at Helmstadt. In 1807 he was appointed as director of the Gottingen observatory cum professor of mathematics at the University of Gottingen. In 1809 he published his another book titled “Thoria Motus” (a work on the application of mathematics to celestial mechanics.).  Again in 1827 he published another book titled “Supervision Curves.”

His contributions

v  17 sides Polygon: In the year of 1796, Gauss invented the process of constructing a polygon of seventeen sides with the help of rural and compass.

v  Trigonometry: He discovered four formulae in spherical trigonometry, which is also known as “Gauss Analogies.”

v  Astronomy: he calculated the orbits of two new planets (shares and phallus) and satellites. In 1809 he published his book on astronomy.

v  Differential geometry: theory of surfaces he did many researches

v  Statistics (Theory of Errors): he discovered the famous law of “Gaussian law of Normal Distribution of Errors” in theory of probability.

v  Mathematical analysis: he invented mathematical of motion and growth as had been developed by Newton and leibnitz.

v  Hyper complex number: he solved the hyper complex numbers (a+bi+cj+dk).

v  Non-Euclidean geometry: he also did a good work in that side. He gave lot of research conclusion in geometry.

v  Magnetism: Gauss is also famous for his scientific work in the field of magnetism and electricity in which he tried the possibility of sending telegraphic signals from Gottingen to a neighbouring town.

v  Telegraph: He was initiated to found the telegraphic signal in an intercom mode. He invented some of the telegraphic theorem and practice.

v  Book: In 1799 he wrote first book on mathematics. I.e. “Disquisition Arithmeticae” in which he showed that every integral rational equation in a single variable has atleast one root.

v  Others: He spent most of the time to problems of astronomy, theory of surface complex numbers, least squares and hyperbolic geometry. He invented new algorithms by introducing the theory of congruencies of figures. 

Gauss has touched many topics not only in the field of astronomy, but pure and applied sciences and that is why he is known as the real founder of modern German mathematics. He died on February 23, 1855.

MODULE 7 - LEONHARD PAUL EULER (1707 – 1783)

LEONHARD PAUL EULER (1707 – 1783)

FATHER OF TOPOLOGY

Born	15 April 1707, Basel, Switzerland (1707-04-15)
Died	18 September 1783 (aged 76), St. Petersburg, Russia
Residence	Prussia, Russia, Switzerland
Nationality	Swiss
Fields	Mathematician and Physicist
Institutions	Imperial Russian Academy of Sciences ,Berlin Academy
Alma mater	University of Basel
Doctoral advisor	Johann Bernoulli

Leonhard Paul Euler (15 April, 1707 – 18 September, 1783) was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. He wrote 886 books in mathematics. Euler is the father of topology. 

Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for his work in mechanics, optics, and astronomy

EULER CONTRIBUTIONS:

Euler worked in almost all areas of mathematics: geometry, calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. Euler's name is associated with a large number of topics.

MATHEMATICAL NOTATION

Ø   f(x): he introduced the concept of a function and was the first to write f(x) to denote the function f applied to the argument x.

Ø   Trigonometric Functions: He also introduced the modern notation for the trigonometric functions,

Ø   e,i,Σ,π : The letter “e”  for the base of the natural logarithm (now also known as Euler's number), the Greek letter “Σ” for summations and the letter “i” to denote the imaginary unit. The use of the Greek letter “π” to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it did not originate with him.

ANALYSIS

Ø  Exponential  Expansion: Euler is well-known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms,

Ø  Power Series Expansions: Euler discovered the power series expansions for e and the inverse tangent function.

Ø  Complex Exponential Function: He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. For any real number 

Ø   Euler created the theory of hyper geometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions.

NUMBER THEORY

Ø  Prime Number: He found a formula for prime number. Ie. X²-X-41=one prime number. If we substitute the value X=1 to 40 then we get the prime number.

Ø  Mersenne Prime: By 1772 Euler had proved that 2³¹ − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.

Ø  Totient Function: He also invented the totient function φ(n) which is the number of positive integers less than the integer n that are coprime to n. Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem.

Ø  Euler's interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy.  He proved that the sum of the reciprocals of the primes diverges. Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares,

GEOMETRY

 Ü   Euler line , Euler's circle

GRAPH THEORY

             Seven Bridges of Königsberg: In 1736, Euler solved the problem known as the Seven Bridges of Königsberg. The city of Königsberg, Prussia was set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not: there is no Eulerian circuit. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory.

V − E + F = 2 : Euler also discovered the formula V − E + F = 2 relating the number of vertices, edges, and faces of a convex polyhedron and hence of a planer graph.

PHYSICS AND ASTRONOMY

• Euler-Bernoulli Beam Equation: Euler helped develop the Euler-Bernoulli beam equation, which became a cornerstone of engineering.
• His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career.  His calculations also contributed to the development of accurate longitude tables.  In addition, Euler made important contributions in optics.

APPLIED MATHEMATICS:

 ·            F₅:  He found the symbol of F₅   ie.  F₅   =4,294,967,297 = 641 X 6 700 417

·            Music: One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. 

MODULE 6 - EUCLIDE (325BC – 265BC)

EUCLIDE (325BC – 265BC)

FATHER OF GEOMETRY

Born	325 BC (1707-04-15)
Died	265 BC
Residence	Egypt, Athens, Alexandria,
Nationality	Greek
Fields	Mathematics (geometry),
Institutions	Royal school at Alexandria in Egypt
Alma mater	Plato academy in Athens
Teacher	Archimedes

Eulcid, best known for his treatise on mathematics, ‘Elements’ was born about 325 BC. Little is known about his life except that he taught mathematics about 300 BC in Royal School at Alexandria in Eqypt that has been founded by Ptolemy, the successor of Alexander the great. He must have studies in Plato’s academy in Athens and learnt the geometry of Eudoxus and theaetetus of which he was familiar. He is considered a one of the great Greek mathematicians.

Euclid Elements / BOOKS:

Books are given the content of subject matter of the thirteen books of the elements.

Ü   Book-1 : Proves elementary theorems about triangles and parallelograms and ends with  Pythagorean theorem.

Ü   Book –2 : has been known as “geometric algebra” because it states algebraic identities  as theorems about equivalent geometric  figures. This book contains a construction of “the section”, the division of a line into two parts such that the ratio of the original line to the larger segment. It also generalizes the Pythagrorean theorem to arbitrary triangles, a result that is equivalent to the law of cosines.

Ü   Book –3 : gives details of properties of circles.

Ü   Book –4 : deals with the construction of regular polygons, in particular the pentagon.

Ü   Book –5 : develops a general theory of ratios and proportions. It formed the foundation for a geometric theory of numbers the foundation for a geometric theory of numbers until an analytic theory developed in the late 19^th century.

Ü   Book-6 : applies the theory of ratios to plane geometry, mainly triangles and parallelograms, culminating in the “application of areas” a procedure for solving Quadratic problems by geometric means.

Ü   Book-7 : describes a method, antanaresis (now known as Euclidean algorithms), for finding the greatest common divisor of two or more numbers.

Ü   Book-8 : examines numbers in continued proportions, now known as geometric sequences (such as ax, ax2, ax3, ax4).

Ü   Book-9 : proves that there are infinite numbers.

Ü   Book-10 : which comprises roughly one-fourth of the Elements seems disproportionate  to the importance of its classification of incommensurable lines and areas.

Ü   Book-11 : deals with the intersections of planes, lines and parallelepipeds (solids with parallel parallelograms as opposite faces).

Ü   Book-12 : provides Eudoxus’ method of exhaustion to prove that areas of a circle are to one another as the squares of their diameters and the  volumes of spheres are to one another as the cubes of their diameters.

Ü   Book-13 : culminates with the construction of the five regular Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given spheres.

Book 1.       Triangles, parallelogram, Pythagorean theorem

Book 2.       Geometric algebra : Algebraic identities.

Book 3.       Properties of Circles

Book 4.       Construction of regular polygons

Book 5.       Geometrical Theory : Theory of ratios and Proportion

Book 6.       Idea of proportion applied to similar figure

Book 7.       Even, odd numbers, numerical theory of proportion

Book 8.       Continued proportion

Book 9.       Number theory

Book 10.     Irrational

Book 11.     Solid geometry

Book 12.     Method of exhaustion used to show that circles are

Proportional to their diameters etc.

Book 13.     Regular solids.

Euclid also wrote other books in mathematics and a few in physics. The important of them are ‘data’, ‘on Division of Figures’, ‘Phenomena’, ‘optics’, etc.

@    Data:  Data deals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of the Elements. It comprising supplementary geometrical material concerned with algebraic problems is important in the development of algebra.

@    Division of figures: consists of 36 propositions concerning the division of various figures into two or more equal parts or parts in given ratios.

@    Phenomena: Phenomena, a treatise on spherical astronomy survives in Greek; it deals with the celestial sphere and contains 25 geometrical propositions.

@    Optics: apparent shapes of cylinder and cones with viewed from different angles.

Euclid Contributions:

Ø  Euclid proved what is generally known as Euclid’s second theorem that prime numbers are infinite.

Ø  He thought about the three current problems of time namely

-Dividing an angle into three equal parts

-Making double of a cube

-Obtaining square from a cube

Ø  He solved unresolved problems related to irrational numbers

Ø  He also discussed the so called Euclidean algorithm for finding the greatest common divisor of two numbers

Ø  He is created with the well-known proof of the Pythagorean Theorem.

Ø  He also formulated few mathematical riddles.

Ø  Euclid axioms:

Ø   We connect two points in a straight line

Ø   We extend the line in both sides

Ø   Right angle triangle consists of 90 degree

Ø   We can draw the circle from the center point.