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Mathematics is nearly as old as humanityitself:evidence of a sense of geometry andinterest in geometric pattern has beenfound in the designs of prehistoric potteryand textiles and in cave paintings.
Primitive counting systems were almostcertainly based on using the fingers of oneor both hands, as evidenced by thepredominance of the numbers 5 and 10 asthe bases for most number systems today.
ANCIENT MATHEMATICS MEDIEVAL AND RENAISSANCE MATHEMATICS WESTERN RENAISSANCE MATHEMATICS MATHEMATICS SINCE THE 16TH CENTURY
ANCIENT MATHEMATICS Greek Mathematics Mesopotamian(Babylonian ) Egyptian Mathematics
Babylonian MathematicsThere mathematics was dominatedby arithmetic, with an emphasis onmeasurement and calculation ingeometry and with no trace oflater mathematical concepts suchas axioms or proofs.
Babylonian MathematicsIn the Babylonian system, using clay tablets consisting ofvarious wedge-shaped marks, a single wedge indicated 1and an arrow-like wedge stood for 10 Numbers up through59 .The number 60, however, was represented by the samesymbol as 1, and from this point on a positional symbol wasused. For example, a numeral consisting of a symbol for 2followed by one for 27 and ending in one for 10 stood for 2 ×602 + 27 × 60 + 10.
Babylonian Mathematics The Babylonians in time developed a sophisticated mathematics by which they could find the positive roots of any quadratic equation. The Babylonians had a variety of tables, including tables for multiplication and division, tables of squares, and tables of compound interest. They could solve complicated problems using Pythagoras theorem; one of their tables contains integer solutions to the Pythagorean equation, a2 + b2 = c2, The Babylonians were also able to sum not only arithmetic and some geometric series.. In geometry, they calculated the area of rectangles, triangles, and trapezoids, the volumes of simple shapes such as bricks and cylinders. No copy right –Mrs.P.Nayak,K.V.Fort william
Egyptian MathematicsThe earliest Egyptian texts,composed about 1800 BC, reveal adecimal numeration system withseparate symbols for thesuccessive powers of 10 (1, 10, 100,and so forth) , just as in the systemused by the Romans. Numbers wererepresented by writing down thesymbol for 1, 10, 100, and so on, asmany times as the unit was in a
Egyptian MathematicsThe Egyptians were able to solve all problems ofarithmetic that involved fractions, as well as someelementary problems in algebra. In geometry, the Egyptians arrived at correct rulesfor finding areas of triangles, rectangles, andtrapezoids, and for finding volumes of figures suchas bricks, cylinders, and, of course, pyramids. To find the area of a circle, the Egyptians used thesquare on of the diameter of the circle, a valueclose to the value of the ratio known as pi, butactually about 3.16 rather than pis value of about3.14
Greek Mathematics The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians. The new element in Greek mathematics was theinvention of an abstract mathematics founded on a logical structureof definitions, axioms, and proofs. According to later Greek accounts, this developmentbegan in the 6th century BC with Thales of Miletus and
Culver PicturesPythagoras is Considered thefirst true mathematician 6th-century BC. The followers of thismovement, Pythagoreans, werethe first to teach that the Earthis a sphere revolving aroundthe Sun.
•Studies of odd and even numbers and of prime andsquare numbers, essential in number theory..• In geometry the great discovery of the school wasthe hypotenuse theorem, or Pythagoras theorem,which states that the square of the hypotenuse of aright-angled triangle is equal to the sum of thesquares of the other two sidesHe gave the idea about Perfect Numbers
The Greek mathematician Euclid , wholived around 300 bc, wrote Elements, a13-volume work on the principles ofgeometry and properties of numbers. Hiswork was rediscovered in the 15thcentury, when it was translated fromArabic, and until recent years has beenthe principal source for the study ofgeometry.
Aryabhata, also spelt Aryabhatta (476-c. 550),Hindu astronomer and mathematician, born inPataliputra (modern Patna), India.He was known to the Arabs as Arjehir, and hiswritings had considerable influence on Arabicscience. Aryabhata held that the Earth rotates onits axis, and he gave the correct explanation ofeclipses of the Sun and the Moon. In mathematicshe could solve quadratic equations, althoughmany of his geometric formulas were incorrect.His only extant work is the Aryabhatiya, a seriesof astronomical and mathematical rules and
Culver PicturesArchimedes made extensivecontributions to theoreticalmathematics, in particular geometry.Through his study of conic sections hederived formulas for the areas of circlesand parabolas, and his work became thebasis for the development of calculus inthe 17th century.
MEDIEVAL AND RENAISSANCE MATHEMATICS Al-Karaji completed Muhammad al- Khwarizmis algebra of polynomials to include even polynomials with an infinite number of terms. Ibrahim ibn Sinan continued Archimedes investigations of areas and volumes, Kamal al-Din and others applied the theory of conic sections to solve optical problems.
MEDIEVAL AND RENAISSANCE MATHEMATICS Thus mathematicians extended the Hindu decimal positional system of arithmetic from whole numbers to include decimal fractions. In 12th-century Persian mathematician Omar Khayyam generalized Hindu methods for extracting square and cube roots to include fourth, fifth, and higher roots.
MEDIEVAL AND RENAISSANCE MATHEMATICS Finally, a number of Muslim mathematicians made important discoveries in the theory of numbers, while others explained a variety of numerical methods for solving equations. Together with translations of the Greek classics, these Muslim works were responsible for the growth of mathematics in the West during the late Middle Ages. Italian mathematicians such as Leonardo Fibonacci and Luca Pacioli depended heavily on Arabic sources
The Italian mathematician LeonardoFibonacci was largely responsible forintroducing the advances made by Arabicand Indian mathematicians to Europe. HisLiber Abaci, published in 1202, helpedspread this knowledge and promoted theArabic numerals that we use today.
Bhaskara (1114-c. 1160), one of the mostoutstanding of Indian mathematicians. Hismajor works were :Lilavati, Bijaganita, Siddanta Siromani.The Bijaganita analyses algebraicexpressions and explores solutions toquadratic equations..
WESTERN RENAISSANCE MATHEMATICSThe discovery, an algebraic formula for thesolution of both the cubic and quarticequations, was published in 1545 by theItalian mathematician Gerolamo Cardano inhis Ars Magna. The discovery drew theattention of mathematicians to complexnumbers and stimulated a search forsolutions to equations of degree higher than
WESTERN RENAISSANCE MATHEMATICS The 16th century also saw the beginnings of modern algebraic and mathematical symbols, as well as the remarkable work on the solution of equations by the French mathematician François Viète.
During the 17th century, the greatestadvances were made in mathematicssince the time of Archimedes andApollonius. The century opened with thediscovery by the Scottish mathematicianJohn Napier of logarithms,
n v t o r l i u b ti u o ti nThe science of number theory, which had lain o i n zdormant since the medieval period, illustrates the s e17th-century advances built on ancient learning. t d o mIt was Diophantus Arithmetica that stimulated m a a tFermat to advance the theory of numbers greatly. t h h e e m m a a ti
Isaac Newton, one of the greatestscientists of all time, revolutionizedmathematics in the 17th century. Hewas responsible for the invention ofcalculus and for advances inalgebra, analytic geometry, and thetheory of equations.
n v t o r l i u Two important developments in pure b ti geometry occurred during the century. u o ti n The first was the publication, in Discourse on o i Method (1637) by Descartes, of his discovery n z of analytic geometry, which showed how to s e use the algebra that had developed since the t d Renaissance to investigate the geometry of o m curves. m a. a t The second development in geometry was the t h publication by the French engineer Gérard h e Desargues in 1639 of his discovery of e m projective geometry. m a a ti
René Descartes founded analyticgeometry, which uses algebra torepresent geometric lines andcurves in terms of axes andcoordinates. He also contributedto the theory of equations .
The 17th-century thinkerGottfried Leibniz made manycontributions to mathematics.He formulated the theory ofcalculus
French mathematician Gaspard Monge invented differential geometry. Also in France, Joseph Louis Lagrange gave a purely analytic treatment of mechanics in his great Analytical Mechanics (1788), in which he stated the famous Lagrange equations for a dynamical system His contemporary, Laplace, wrote The Analytic Theory of Probabilities (1812) and the classic Celestial Mechanics (1799-1825), which earned him the title of the “French Newton”.
This illustration shows the Swiss mathematicianbrothers Jakob ( left ) and Johann ( right )Bernoulli discussing a geometrical problem. Thebrothers both made important contributions tothe early development of calculus.
French astronomer and mathematician Pierre SimonLaplace was best known for applying the theory ofgravitation. The mathematical procedures Laplace developed tomake his calculations laid the foundation for laterscientific investigation of heat, magnetism, andelectricity.
The greatest mathematician of the 18th century was Leonhard Euler, a Swiss, who made basic contributions to calculus and to all other branches of mathematics. He wrote textbooks on calculus, mechanics, and algebra that became models of style for writing in the areas of Newtons ideas based on kinematics and velocities, Leibnizs explanation , based on infinitesimals, and Lagranges the idea of infinite series.No copy right –Mrs.P.Nayak,K.V.Fort william
Although hindered by loss of sight,Leonhard Euler was an importantcontributor to both pure and appliedmathematics. Euler is best known for hisanalytical treatment of mathematics andhis discussion of concepts in calculus,but he is also noted for his work inacoustics, mechanics, astronomy, andoptics.
i e o a r p n s e C , d a I i r nIn a l 1821 a French mathematician, Augustin cLouis Cauchy, succeeded in giving a F ./ r Slogically satisfactory approach to calculus. i cHe based his approach only on finite e iquantities and the idea of a limit. d e r n i c c e h S G o a u u r
Augustin Louis Cauchy wasone of the most brilliantmathematicians of the 19thcentury, making importantcontributions to the fields offunctions, calculus, andanalysis.
In the 18th century the Swedishastronomer Anders Celsius invented thecentigrade or Celsius scale of 100 degreesbetween the freezing and boiling points ofwater for the measurement oftemperature. The Celsius scale is one ofthe most commonly used measurementscales in the world.
. Early in the century, Carl Friedrich Gauss gave asatisfactory explanation of complex numbers, and thesenumbers then formed a whole new field for analysis,Another important advance in analysis was Fouriersstudy of infinite sums whose terms are trigonometricfunctions. Known today as Fourier series, they are stillpowerful tools in pure and applied mathematics. In addition, the investigation of which functions could beequal to Fourier series led Cantor to the study of infinitesets and to an arithmetic of infinite numbers.
Gauss was one of the greatest mathematicians who everlived. Diaries from his youth show that this infantprodigy had already made important discoveries innumber theory, an area in which his book DisquisitionesArithmeticae (1801) marks the beginning of the modernera. While only 18, Gauss discovered that a regularpolygon with m sides can be constructed by straight-edgeand compass when m is a power of two times distinctprimes of the form 2n + 1.
German mathematician CarlFriedrich Gauss contributedto many areas ofmathematics, includingprime numbers, probabilitytheory, algebra, andgeometry.Gauss also applied hismathematical work totheories of electricity andmagnetism. The magneticunit of intensity is named inhis honour.
Klein applied it to the classification of geometriesin terms of their groups of transformationsLie applied it to a geometric theory of differentialequations by means of continuous groups oftransformations known as Lie groups. In the 20thcentury, algebra was also applied to a general formof geometry known as topology.
Ramanujan, Srinivasa (1887-1920),Indian mathematician known for hiswork on number theory, whose geniusbrought him from obscurity to a briefbut remarkable collaboration with G. H.Hardy at Cambridge.
Ramanujan was born at Erode, in TamilNadu state, south India December 22,1887, into a poor Brahmin family. Hisfather was an accountant with a clothmerchant; his mother earned a fewrupees singing bhajans at the temple.The young Ramanujan quickly showeda single-minded love for mathematics.
However, his neglect of other subjects incollege led him to fail and lose hisscholarship. With no money, he gave up hisstudies and eventually found a small job atthe Madras Port Trust in 1911. Before leavingschool, Ramanujan had bought himself acopy of G. S. Carrs A Synopsis ofElementary Results in Pure and AppliedMathematics.
He worked his way through thissystematically and began his own research,publishing articles in Indian mathematicaljournals and soon becoming recognized as aremarkable mathematician.
It was his letter to the English mathematician G. H.Hardy, at the University of Cambridge, discussingand questioning some of Hardys published work,that brought him, after considerable efforts by Hardyand his colleagues, to Cambridge in 1914, on aresearch scholarship. Here he was able to study andresearch freely. His health was frail, however, and in1917 he became very ill, probably from tuberculosis.He was elected Fellow of Trinity College, and aFellow of the Royal Society, in 1918, at the age of 31 .
Ramanujans main interest had been thestudy of numbers, and his most remarkableresults were in the partitioning of numbers.He also worked on identities, modularequations, and mock-theta functions. Thenotebooks he left, full of the fevered work ofhis last days, are still being studied. Hisextraordinary intuition, and unorthodoxmethods, led to some of the strangest andmost beautiful formulae in mathematics.
He had proved that zero dividedby zero was neither zero nor one,but infinity.
Another subject that was transformedin the 19th century, notably byEnglish mathematician GeorgeBooles Laws of Thought (1854) andCantors set theory, was thefoundations of mathematics. Towardsthe end of the century, however, aseries of paradoxes was discoveredin Cantors theory. One such paradox,found by English mathematicianBertrand Russell, aimed at the
The German mathematician GeorgCantor was renowned for hisdevelopments in the field of settheory. His line of inquiry led inthe 20th century to thefundamental investigation of thenature of mathematical logic.
In the 19th century the Britishmathematician George Booledeveloped a form of algebra,known as Boolean algebra, whichtoday is very important tocomputer operations, such as inthe use of Internet searchengines.
Current Mathematicsharles Babbage in 19th-century England who designedmachine that could automatically perform portant calculations.ilbert could not have foreseen seems destined to play even greater role in the future development mputations based on a programme of instructionsored on cards or tape.abbages imagination outran the technology of his day, d it was not until the invention of the relay, then of thecuum tube, and then of the transistor, that large-scale,ogrammed computation became feasible with
The inventor of the Difference Engine, asophisticated calculator, themathematician Charles Babbage is alsocredited with conceiving the first truecomputer. With the help of Augusta AdaByron, Babbage created a design for theAnalytical Engine, a machine remarkablylike the modern computer, evenincluding a memory.