mathematics in engineering lecture notes-3: famous...

MATHEMATICS IN ENGINEERING

City University London 2012-13

By:A. Sayyad

Lecture Notes-3: Famous Mathematicians

Famous Mathematicians

� Mathematics differs from natural science in that scientists subject truth claims to tests by experiments, while mathematical propositions are conclusions of mathematical proofs.

� According to Alfréd Rény (and/or Paul

Erdos, “A mathematician is a device for

turning coffee into theorems.”


� Just the Facts:� A mathematician is someone who studies and adds

to the intellectual body of mathematics.

� Mathematicians are smart.

� If you read the first fact above and sniggered at the word "body," then you're probably not smart enough to be a mathematician!!


� These are the mathematicians that known as the best.These are the mathematicians who should have their owntrading cards. These are the ones who should becentrefolds in the math textbooks.

� In this notes, the biography of some of the best knownmathematicians of all time and their achievements will bereviewed.

� Different references used within these notes areavailable for more studies.


� Joseph Louis Lagrange (1736-1813) Italy, France

� Joseph-Louis Lagrange (born Giuseppe LodovicoLagrangia) was a brilliant man who advanced to become a teen-age Professor shortly after first studying mathematics.

� Lagrange is usually considered to be a

French mathematician, but he has been

referred as an Italian mathematician.


� He excelled in all fields of analysis and number theory,and made key contributions to the theories ofdeterminants, continued fractions, and many other fields.

� He developed partial differential equations, developedthe calculus of variations and developed terminologyand notation (e.g. the use of f'(x) and f''(x) for afunction’s 1st and 2nd derivatives).

� He also proved a fundamental Theorem of GroupTheory.


� He laid the foundations for the theory of polynomialequations which later completed by some othermathematicians .

� Number theory was almost just a diversion for Lagrange,whose focus was analysis.

� Nevertheless he was the master of that field as well,proving difficult and historic theorems including Wilson'stheorem :n is prime if and only if (n -1)! + 1 is divisibleby n.


� The other one is the Lagrange's Four-Square Theorem (everypositive integer is the sum of four squares), and that

n·x2 + 1 = y2 has solutions for every positive non-square

integer n.

� Lagrange had so many contributions to physics includeunderstanding of vibrations.

� He found an error in Newton's work and published thedefinitive treatise on sound, celestial mechanics (e.g.explanation of why the Moon keeps the same face pointedtowards the Earth), and the discovery of the Lagrangianpoints (e.g., in Jupiter's orbit).


� Lagrange's textbooks were noted for clarity and inspiredmost of the 19th-century mathematicians on this list.

� Unlike Newton, who used calculus to derive his results butthen worked backwards to create geometric proofs forpublication, Lagrange relied only on analysis. "Nodiagrams will be found in this work" he wrote in thepreface to his masterpiece Mécanique analytique.

� largely he was self taught and did not have the benefitof studying with leading mathematicians. Howevere, hewrote his first paper , when he was just 18 years old.


� Although he was still only 19 years old, Lagrange wasappointed professor of mathematics at the RoyalArtillery School in Turin on 28 September 1755.

� He did an enormous work on calculus of variations andapplied it to mechanics.

� In papers which were published from him, Lagrangestudied the integration of differential equations andmade various applications to topics such as fluidmechanics (where he introduced the Lagrangian function).


� Also contained are methods to solve systems of linear differential equations which used the characteristic value of a linear substitution for the first time.

� Another problem to which he applied his methods was the study the orbits of Jupiter and Saturn.

� Napoleon named Lagrange to the Legion of Honour and Count of the Empire in 1808.


� John von Neumann (1903-1957) Hungary, U.S.A.

� He is one of the most prolific geniuses in history,making major contributions in many

branches of pure mathematics, and

applied mathematics.

� He was an essential pioneer of both quantumphysics and computer science.


� Von Neumann pioneered the use of models in set theory,thus improving the axiomatic basis of mathematics.

� He proved a generalized spectral theorem sometimescalled the most important result in operator theory andby that, he developed von Neumann Algebras.

� He was first to state and prove the minimax theorem andthus invented game theory.

� He invented cellular automata, famously constructing aself-reproducing automaton.


� He invented elegant definitions for the counting numbers(0 = {}, n+1 = n ∪∪∪∪ {n}).

� He also worked in analysis, matrix theory, measuretheory, numerical analysis, ergodic theory, grouprepresentations, continuous geometry, statistics andtopology.

� He also contributed with lots of great discoveries in thefield of physics.


� By treating the universe as a very high-dimensionalphase space, he constructed an elegant mathematicalbasis (now called von Neumann algebras) for theprinciples of quantum physics.

� He played a key role in the design of conventional,nuclear and thermonuclear bombs, and he alsoadvanced the theory of hydrodynamics.


� One of his greatest achievements was to apply gametheory and to economics, becoming a major figure in thatfield.

� His contributions to computer science are many: inaddition to co-inventing the stored-program computer, hewas first to use pseudo-random number generation, finiteelement analysis, the merge-sort algorithm, and a"biased coin" algorithm.


� By implementing wide-number software he joined several other great mathematicians (Archimedes, Apollonius, Liu Hui, Hipparchus, Madhava, Ramanujan) in producing the best approximation to πof his time.

� At the time of his death, von Neumann was working on a theory of the human brain.


� Johann P. G. L. Dirichlet (1805-1859) Germany

� Dirichlet was preeminent in algebraic and analytic

number theory, but did advanced work in several

other fields as well.

� He discovered the modern definition of function, theVoronoi diagram of geometry, and important concepts indifferential equations, topology, and statistics.


� His proofs were noted both for great ingenuity andunprecedented rigor.

� As an example of his careful rigor, he found afundamental flaw in Steiner's Isoperimetric Theoremproof which no one else had noticed.

� Although the impossibility of xk + yk = zk for k = 4 andk = 3, had been proved, but Dirichlet became famous byproving impossibility for k = 5 at the age of 20 andlater he proved the case k = 14.


� His Unit Theorem, considered one of the most importanttheorems of algebraic number theory.

� The Unit Theorem is unusually difficult to prove. it is saidthat Dirichlet discovered the proof while listening to musicin the Sistine Chapel.

� A key step in the proof uses "Dirichlet's PigeonholePrinciple", a trivial idea but which Dirichlet applied withgreat ingenuity.


� He invented a method of L-series to prove the importanttheorem (Gauss' conjecture) that any arithmetic series (withouta common factor) has an infinity of primes.

� It was Dirichlet who proved the fundamental Theorem ofFourier series: that periodic analytic functions can always berepresented as a simple trigonometric series.

� In mechanics he investigated the equilibrium of systemsand potential theory. These investigations began in 1839 withpapers which gave methods to evaluate multiple integralsand he applied this to the problem of the gravitationalattraction of an ellipsoid on points both inside and outside.


� He turned to Laplace's problem of proving the stability ofthe solar system and produced an analysis whichavoided the problem of using series expansion withquadratic and higher terms disregarded.

� This work led him to the Dirichlet problem concerningharmonic functions with given boundary conditions.

� And Finally, In 1852 he studied the problem of a sphereplaced in an incompressible fluid, in the course of thisinvestigation becoming the first person to integrate thehydrodynamic equations exactly.


� M. al-Khowârizmi (ca 780-850) Persia, Iraq

� Al-Khowârizmi worked as a mathematician, astronomer andgeographer early in the Golden Age of Islamic science.

� He introduced the Hindu decimal system to the Islamicworld and Europe; invented the horary quadrant, improvedthe sundial, developed trigonometry tables and improvedon Ptolemy's astronomy and geography.

� He wrote the book Al-Jabr, which demonstrated simplealgebra and geometry, and several other influential books.


� He made major contributions to the fields of algebra,trigonometry, and geography.

� He corrected Ptolemy’s geographical maps as regards toAfrica and the Middle East and published the bookKitab surat al-ard (“The image of the earth”).

� With his publication about the calculation with HinduNumerals he promotes the use of the Indian system ofnumeration in the Middle-East followed by Europe.


� His book was translated into Latin in the 12th century withthe name “Algoritmi de numero Indorum”, because hisname was rendered in Latin as “Algoritmi” he is indirectresponsible for the term algorithm.

� His books made a significant contribution to theadvancement of mathematics (solving linear andquadratic equations including geometric principles forcompleting the square) in Europe.


� John Brehaut Wallis (1616-1703) England

� Wallis began his life as a savant at arithmetic and amedical student (he may have contributed to the concept ofblood circulation).

� It is said he once calculated the square root of a 53-digitnumber to help him sleep and remembered the result in themorning.


� He may be named to become perhaps the most brilliantand influential English mathematician before Newton.

� He made major advances in analytic geometry, but alsocontributions to algebra, geometry and trigonometry.

� He is especially famous for using negative and fractionalexponents and taking the areas of curves, and treatinginelastic collisions.

� he and Huygens were first to develop the law ofmomentum conservation.


� He was the first European to solve Pell's Equation.

� He was the first great mathematician to consider complexnumbers legitimate; and first to use the symbol ∞. Walliscoined several terms including "continued fraction,""induction," "interpolation," "mantissa," and"hypergeometric series.“

� Wallis created an infinite product formula for pi, whichmight be (but isn't) written today as:

π = 2 ∏k=1,∞ 1+(4k2-1)-1


� At the end, we will briefly go through the achievementsof some of other great mathematicians of all time:

� Madhava of Sangamagramma (1340-1425) India

� Madhava is most famous for his work with Taylor series, discoveringidentities like sin q = q - q3/3! + q5/5! - ..., formulae for π, includingthe one attributed to Leibniz, and the then-best knownapproximation π ≈ 104348 / 33215.


� Girolamo Cardano (1501-1576) Italy

� He was first to publish general solutions to cubic andquartic equations, and first to publish the use of complexnumbers in calculations.

� Cardano introduced binomial coefficients and the binomialtheorem, and introduced and solved the geometrichypocyloid problem, as well as other geometric theorems.

� Cardano is credited with Cardano's Ring Puzzle, stillmanufactured today and related to the Tower of

Hanoi puzzle.


� René Déscartes (1596-1650) France

� His famous mathematical theorems include the Rule of Signs(for determining the signs of polynomial roots), the elegantformula relating the radii of Soddy kissing circles, histheorem on total angular defect, and an improved solutionto the Delian problem (cube-doubling).

� He improved mathematical notation (e.g. the use ofsuperscripts to denote exponents). He also discoveredEuler's Polyhedral Theorem, F+V = E+2.


The End

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