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Mathematics is beautiful Mathematics is useful Of these two aspects, which one is more important? by V. Kannan University of Hyderabad on 18 th August 2012 at VBIT

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Part I : Beauty of Mathematics Part II : Usefulness of Mathematics Part III : Place of Mathematics among other subjects Part IV : Srinivasa Ramanujan Part V : Views of scientists Feynmann, Dirac, etc

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Beauty of Mathematics 1. Mathematics is needed to understand beauty. 2. Mathematics exhibits beauty. 3. Mathematics can be used to create beauty. 4. Mathematics is tested by its beauty. 5. Mathematics enhances the beauty of life.

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Beauty of Mathematics Mathematics is needed to understand beauty To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in - Richard Feynman

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Richard Phillips Feynman (1918-1988) American physicist Nobel prize in physics 1965

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Beauty of Mathematics Mathematics can show beauty Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. Bertrand Russell (1872-1970)

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Bertrand Arthur William Russell British philosopher Born in 1872 Ravenscroft,Wales Nobel Prize for Literature In 1950 Died in 1970 February 02 at Penrhyndeudraeth, Wales Bertrand Russell

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Beauty of Mathematics The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful. J.H.Poincare (1854-1912) (cited in H.E.Huntley, The Divine Proportion)

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French Mathematician 1854-1912 The Last Universalist Jules Henri Poincaré

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Beauty of Mathematics Mathematics can create beauty Mathematics can be used to create beautiful designs. Of the many ways to do so, here are three: 1. Iterated Function Systems to create fractals. 2. Julia Sets of quadratic polynomials. 3. Mandelbrot set.

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Barnsley Fern Michael Barnsley

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Sierpinski Gasket Sierpinski

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Julia set Julia

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Mandelbrot Set Mandelbrot

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Beauty of Mathematics Mathematics is tested by beauty The mathematician's patterns, like the painters or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. G. H. Hardy (1877 - 1947) A Mathematician's Apology, Cambridge University Press, 1994.

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An English Mathematician 1877-1947 Fellow of the Royal Society Godfrey Harold Hardy

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Beauty of Mathematics Mathematics enhances the beauty of life Russell's paradox : Call the set of all sets that are not members of themselves "R. If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell In 1910

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Beauty of Mathematics Russels paradox : A barber who shaves all those men who do not shave themselves. When one thinks about whether the barber should shave himself or not, the paradox begins to emerge.

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Usefulness of Mathematics The range of applications of matrix eigen values problems is incredibly large. - p.376, Kreyszig, Advanced Engineering Mathematics Stretching of an elastic membrane Markov Processes Population models Vibrating systems of two masses on two springs

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Usefulness of Mathematics Application of triple integrals to mass distribution. Application of triple integrals to moment of inertia. Application of divergence theorem to heat flow. Application of Fourier analysis to mechanics and electrostatics.

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Place of Mathematics among other subjects Antiquity: Some subjects are old but extinct Example : Archery Some subjects are new : Example : Computer Science Some subjects are both old and new : Example : Medicine and Mathematics

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Place of Mathematics among other subjects List of subjects studied 5000 years ago: Four upavedas : (a)Dhanurveda : Archery (b)Ayurveda : Medicine (c)Gandharvaveda : Music (d)Sthapatya veda : Archeology

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Place of Mathematics among other subjects Six ancilary subjects : a)Phonetics b)Grammar c)Prosody : Rules of poetry d)Etymology e)Astronomy f)Rituals

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Place of Mathematics among other subjects From other Vidyasthanas : a)Logic b)Meemamsa : Interpretation and Analysis c)Purana : Mythology d)Ethics

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Place of Mathematics among other subjects 64 Kalas : Many are extinct now.

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Place of Mathematics among other subjects Three categories : 1.Subjects that are for beauty, charm, elegance e.g. : Music, fine arts 2. Subjects that are for application, uses, welfare e.g. : Engineering, technology 3. Subjects that are for search of truth e.g. : Astronomy, Physics Where does Mathematics belong to ? In all the three.

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Srinivasa Ramanujan(1887-1920) Lived in Kumbakonam Contributed to Number theory and combinatorics

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Srinivasa Ramanujan, native to India, was an extraordinary mathematician. A child prodigy, he was self-educated. -G.D.Anderson, M.Vuorinen

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Every positive Integer was one of his (Ramanujans) personal friends -J.E.Littlewood.

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The greatest mathematician of the century -Prof.Julian Huxley.

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A gift from heaven -E.T.Bell.

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The Door – Number Problem There is a street in which the houses are numbered serially from 1 onwards. The total number of houses is roughly between 50 and 500. A mathematician lives in a house whose number has this peculiar property : The sum of all numbers in the left side = The sum of all numbers in the right side Find the number of this house. REFERENCE : A 2009 film God, zero and infinity.

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Ramanujans Answer :204 i.e., 1+2+…+203 = 205+206+…288. The next pair of numbers is 1189 &1681.

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I have never met his equal and I can compare him only with Euler and Jacobi. I learnt from him much more than he learnt from me. -G.H.Hardy

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A single look at them (Ramanujans formulae) is enough to show that they could only be written down by a mathematician of the highest class. -G.H.Hardy

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A single look at them (Ramanujans formulae) is enough to show that they could only be written down by a mathematician of the highest class. -G.H.Hardy

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Views of scientists As time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. - Paul A M Dirac

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Views of scientists It is more important to have beauty in one's equations than to have them fit experiment... It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. - Paul A M Dirac

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Views of scientists The mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which nature has chosen. - Paul A M Dirac

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Views of scientists Theoretical physicists accept the need for mathematical beauty as an act of faith... For example, the main reason why the theory of relativity is so universally accepted is its mathematical beauty. - Paul A M Dirac

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Views of scientists For many parts of Nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity, without the aid and intervening of the mathematics, of which sort are perspective, music, astronomy, cosmography, architecture, engineery, and divers others. -Sir Francis Bacon

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Views of scientists The burden of the lecture is just to emphasize the fact that it is impossible to explain honestly the beauty of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case. -Richard Feynman

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