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Numbers

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It provides information about realnumbers

It provides information about realnumbers

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  • 1. Maths projectHistory of MathematicsDone byPavan 9EJ.H.P.S
  • 2. The area of study known asthe history of mathematics isprimarily an investigation intothe origin of discoveriesin mathematics and, to a lesserextent, an investigation intothe mathematical methods andnotation of the past.
  • 3. The first method of counting was counting on fingers. Thisevolved into sign language for the hand-to-eyecommunication of numbers. But this was not writing.Tallies by carving notches in wood, bone, and stone wereused for at least forty thousand years. Stone age cultures,including ancient Native American groups, used tallies forgambling with horses, slaves, personal services and trade-goods.Roman numerals evolved from this primitive system of cuttingnotches .It was once believed that they came from alphabeticsymbols, or from pictographs like the hand, but thesetheories have been disproved.
  • 4. Before the modern age and the worldwidespread of knowledge, written examples ofnew mathematical developments havecome to light only in a few locales. The mostancient mathematical texts availableare Plimpton 322 Babylonian mathematicsGreek mathematics greatly refined themethods (especially through theintroduction of deductive reasoningand mathematical rigor in proofs) andexpanded the subject matter ofmathematics.
  • 5. Egyptian mathematics c. 2000-1800 BC andthe Moscow Mathematical Papyrus Egyptianmathematics c. 1890 BC. All of these texts concernthe so called Pythagorean theorem which seems to bethe most ancient and widespread mathematicaldevelopment after basic arithmetic and geometry.The study of mathematics as a subject in its own rightbegins in the 6th century BC with the Pythagoreanswho coined the term "mathematics" from the ancientGreek word (mathema), meaning "subject ofinstruction.
  • 6. Chinese mathematics made earlycontributions, including a place valuesystem.The Hindu-Arabic numeralsystem and the rules for the use of itsoperations, in use throughout the worldtoday, likely evolved over the course ofthe first millennium AD in India and wastransmitted to the west via Islamicmathematics Many Greek and Arabic textson mathematics were then translated intolatin which led to further development ofmathematics in medieval Europe.
  • 7. From ancient times through the MiddleAges, bursts of mathematical creativitywere often followed by centuries ofstagnation. Beginningin Renaissance Italy in the 16th century,new mathematical developments,interacting with new scientificdiscoveries, were made at anincreasingpace that continues through thepresent day.
  • 8. Indian mathematicsThe earliest civilization on the Indian subcontinent isthe Indus Valley Civilization that flourished between2600 and 1900 BC in the Indus river basin. Their citieswere laid out with geometric regularity, but no knownmathematical documents survive from this civilizationThe oldest extant mathematical records from Indiaare the Sulba Sutras (dated variously between the 8thcentury BC and the 2nd century AD), appendices toreligious texts which give simple rules for constructingaltars of various shapes, such as squares, rectangles,parallelograms, and others
  • 9. zeroZero was invented independently by the Babylonians, Mayansand Indians (although some researchers say the Indian numbersystem was influenced by the Babylonians). The Babylonians gottheir number system from the Sumerians, the first people in theworld to develop a counting system. Developed 4,000 to 5,000years ago, the Sumerian system was positional — the value of asymbol depended on its position relative to other symbols.Robert Kaplan, author of "The Nothing That Is: A Natural Historyof Zero," suggests that an ancestor to the placeholder zero mayhave been a pair of angled wedges used to represent an emptynumber column. However, Charles Seife, author of "Zero: TheBiography of a Dangerous Idea," disagrees that the wedgesrepresented a placeholder.
  • 10. India: Where zero became a numberSome scholars assert that the Babylonian concept wove its way downto India, but others give the Indians credit for developing zeroindependently.The concept of zero first appeared in India around A.D. 458.Mathematical equations were spelled out or spoken in poetry or chantsrather than symbols. Different words symbolized zero, or nothing, suchas "void," "sky" or "space." In 628, a Hindu astronomer andmathematician named Brahmagupta developed a symbol for zero — adot underneath numbers. He also developed mathematical operationsusing zero, wrote rules for reaching zero through addition andsubtraction, and the results of using zero in equations. This was the firsttime in the world that zero was recognized as a number of its own, asboth an idea and a symbol. By the 1600s, zero was used fairly widelythroughout Europe. It was fundamental in Rene Descartes’ Cartesiancoordinate system and in Sir Isaac Newton’s and Gottfried WilhemLiebniz’s developments of calculus. Calculus paved the way for physics,engineering, computers, and much of financial and economic theory.
  • 11. A Persian mathematician, Mohammed ibn-Musa al-Khowarizmi, suggested that a little circle should beused in calculations if no number appeared in thetens place. The Arabs called this circle "sifr," or"empty." Zero was crucial to al-Khowarizmi, who usedit to invent algebrain the ninth century. Al-Khowarizmi also developed quick methods formultiplying and dividing numbers, which are knownas algorithms — a corruption of his name.Zero found its way to Europe through the Moorishconquest of Spain and was further developed byItalian mathematician Fibonacci, who used it to doequations without an abacus, then the mostprevalent tool for doing arithmetic. This developmentwas highly popular among merchants, who usedFibonacci’s equations involving zero to balance theirbooks.
  • 12. Maths in differrentcountries• The Ishango Bone, found in the area of the headwaters of theNile River (northeastern Congo), dates as early as 20,000 BC.One common interpretation is that the bone is the earliestknown demonstration[7] of sequences of prime numbers andAncient Egyptian multiplication. Predynastic Egyptians of the5th millennium BC pictorially represented geometric spatialdesigns. It has been claimed that Megalithic monuments inEngland and Scotland from the 3rd millennium BC, incorporategeometric ideas such as circles, ellipses, and Pythagoreantriples in their design1 Early mathematics2 Ancient Near East (c. 1800-500 BC)2.1 Mesopotamia2.2 Egypt3 Ancient Indian mathematics (c. 900BC—AD 200)4 Greek and Hellenistic mathematics(c. 550 BC—AD 300)5 Classical Chinese mathematics(before c. 4th century BC— AD 1300)6 Classical Indian mathematics (c.400—1600)7 Islamic mathematics (c. 800—1500)8 Medieval European mathematics (c.500—1400)8.1 The Early Middle Ages (c. 500—1100)8.2 The Rebirth of Mathematics inEurope (1100—1400)9 Early Modern European mathematics(c. 1400—1600)The earliest known mathematics in ancient Indiadates back to circa 3000-2600 BC in the Indus ValleyCivilization (Harappan civilization) of North India andPakistan, which developed a system of uniformweights and measures that used the decimal system,a surprisingly advanced brick technology whichutilised ratios, streets laid out in perfect right angles,and a number of geometrical shapes and designs,including cuboids, barrels, cones, cylinders, anddrawings of concentric and intersecting circles andtriangles. Mathematical instruments discoveredinclude an accurate decimal ruler with small andprecise subdivisions, a shell instrument that servedas a compass to measure angles on plane surfaces orin horizon in multiples of 40–360 degrees
  • 13. A shell instrument used to measure 8–12 whole sections of thehorizon and sky, and an instrument for measuring the positionsof stars for navigational purposes. The Indus script has not yetbeen deciphered; hence very little is known about the writtenforms of Harappan mathematics. Archeological evidence has ledsome historians to believe that this civilization used a base 8numeral system and possessed knowledge of the ratio of thelength of the circumference of the circle to its diameter, thus avalue of ?. Dating from the Shang period (1600—1046 BC), theearliest extant Chinese mathematics consists of numbersscratched on tortoise shell . These numbers use a decimalsystem, so that the number 123 is written (from top to bottom)as the symbol for 1 followed by the symbol for a hundred, thenthe symbol for 2 followed by the symbol for ten, then thesymbol for 3. This was the most advanced number system in theworld at the time and allowed calculations to be carried out onthe suan pan or Chinese abacus. The date of the invention of thesuan pan is not certain, but the earliest written reference was inAD 190 in the Supplementary Notes on the Art of Figureswritten by Xu Yue.
  • 14. Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (modern Iraq) from thedays of the early Sumerians until the beginning of the Hellenistic period. It is named Babylonianmathematics due to the central role of Babylon as a place of study, which ceased to exist during theHellenistic period. From this point, Babylonian mathematics merged with Greek and Egyptian mathematicsto give rise to Hellenistic mathematics. Later under the Arab Empire, Iraq/Mesopotamia, especially Baghdad,once again became an important center of study for Islamic mathematics.In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics isderived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tabletswere inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of theseappear to be graded homework.The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliestcivilization in Mesopotamia. They developed a complex system of metrology from 3000 BC. From around2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometricalexercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions,algebra, quadratic and cubic equations, and the calculation of Pythagorean triples (see Plimpton 322).[11]The tablets also include multiplication tables, trigonometry tables and methods for solving linear andquadratic equations. The Babylonian tablet YBC 7289 gives an approximation to ?2 accurate to five decimalplaces.Babylonian mathematics was written using a sexagesimal (base-60) numeral system. From this we derive themodern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle.Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors. Also, unlike theEgyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in theleft column represented larger values, much as in the decimal system. They lacked, however, an equivalentof the decimal point, and so the place value of a symbol often had to be inferred from the context.Egypt.
  • 15. Egyptian mathematicsThe Rhind papyrus (c. 1650 BC [3]) is another major Egyptianmathematical text, an instruction manual in arithmetic and geometry.In addition to giving area formulas and methods for multiplication,division and working with unit fractions, it also contains evidence ofother mathematical knowledge (see [4]), including composite andprime numbers; arithmetic, geometric and harmonic means; andsimplistic understandings of both the Sieve of Eratosthenes and perfectnumber theory (namely, that of the number 6)[5]. It also shows how tosolve first order linear equations [6] as well as arithmetic and geometricseries [7].Also, three geometric elements contained in the Rhind papyrus suggestthe simplest of underpinnings to analytical geometry: (1) first andforemost, how to obtain an approximation of ? accurate to within lessthan one percent; (2) second, an ancient attempt at squaring the circle;and (3) third, the earliest known use of a kind of cotangen
  • 16. Ancient Indian mathematics (c. 900BC—AD 200)Vedic mathematics begins in the early Iron Age, with the Shatapatha Brahmana (c. 9th centuryBC), which approximates the value of ? to 2 decimal places and the Sulba Sutras (c. 800-500 BC)were geometry texts that used irrational numbers, prime numbers, the rule of three and cuberoots; computed the square root of 2 to five decimal places; gave the method for squaring thecircle; solved linear equations and quadratic equations; developed Pythagorean triplesalgebraically and gave a statement and numerical proof of the Pythagorean theorem.Between 400 BC and AD 200, Jain mathematicians began studying mathematics for the solepurpose of mathematics. They were the first to develop transfinite numbers, set theory,logarithms, fundamental laws of indices, cubic equations, quartic equations, sequences andprogressions, permutations and combinations, squaring and extracting square roots, and finiteand infinite powers. The Bakshali Manuscript written between 200 BC and AD 200 includedsolutions of linear equations with up to five unknowns, the solution of the quadratic equation,arithmetic and geometric progressions, compound series, quadratic indeterminate equations,simultaneous equations, and the use of zero and negative numbers. Accurate computations forirrational numbers could be found, which includes computing square roots of numbers as largeas a million to at least 11 decimal places.
  • 17. Greek mathematicsPythagoras of Samos Greek mathematics refers to mathematics written in Greek between about600 BCE and 450 CE.[12] Greek mathematicians lived in cities spread over the entire EasternMediterranean, from Italy to North Africa, but were united by culture and language. Greekmathematics is sometimes called Hellenistic mathematics.Thales of MiletusGreek mathematics was much more sophisticated than the mathematics thathad been developed by earlier cultures. All surviving records of pre-Greek mathematics showthe use of inductive reasoning, that is, repeated observations used to establish rules of thumb.Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to deriveconclusions from definitions and axioms.[13]Greek mathematics is thought to have begun with Thales (c. 624—c.546 BC) and Pythagoras (c.582—c. 507 BC). Although the extent of the influence is disputed, they were probably inspiredby the ideas of Egypt, Mesopotamia and perhaps India. According to legend, Pythagorastravelled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests. Somesay the greatest of Greek mathematicians, if not of all time, was Archimedes (287—212 BC) ofSyracuse. According to Plutarch, at the age of 75, while drawing mathematical formulas in thedust, he was run through with a spear by a Roman soldier. Ancient Rome left little evidence ofany interest in pure mathematics.
  • 18. Chinese mathematics (before c. 4thcentury BC— AD 1300)From the Western Zhou Dynasty (from 1046 BC), the oldestmathematical work to survive the book burning is the I Ching,which uses the 8 binary 3-tuples (trigrams) and 64 binary 6-tuples (hexagrams) for philosophical, mathematical, and/ormystical purposes. The binary tuples are composed of brokenand solid lines, called yin female and yang male respectively(see King Wen sequence).The oldest existent work on geometry in China comes from thephilosophical Mohist canon of c. 330 BC, compiled by thefollowers of Mozi (470 BC-390 BC). The Mo Jing describedvarious aspects of many fields associated with physical science,and provided a small wealth of information on mathematics aswell.
  • 19. Classical Indian mathematics (c. 400—1600)AryabhataThe Surya Siddhanta (c. 400) introduced the trigonometric functions of sine, cosine, and inverse sine, and laiddown rules to determine the true motions of the luminaries, which conforms to their actual positions in the sky. Thecosmological time cycles explained in the text, which was copied from an earlier work, corresponds to an average siderealyear of 365.2563627 days, which is only 1.4 seconds longer than the modern value of 365.25636305 days. This work wastranslated to Arabic and Latin during the Middle Ages.Aryabhata in 499 introduced the versine function, produced the first trigonometric tables of sine, developed techniques andalgorithms of algebra, infinitesimals, differential equations, and obtained whole number solutions to linear equations by amethod equivalent to the modern method, along with accurate astronomical calculations based on a heliocentric system ofgravitation. An Arabic translation of his Aryabhatiya was available from the 8th century, followed by a Latin translation inthe 13th century. He also computed the value of ? to the fourth decimal place as 3.1416. Madhava later in the 14th centurycomputed the value of ? to the eleventh decimal place as 3.14159265359.In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmaguptas identity andBrahmaguptas formula, and for the first time, in Brahma-sphuta-siddhanta, he lucidly explained the use ofzero as both a placeholder and decimal digit and explained the Hindu-Arabic numeral system. It was from atranslation of this Indian text on mathematics (around 770) that Islamic mathematicians were introduced tothis numeral system, which they adapted as Arabic numerals. Islamic scholars carried knowledge of thisnumber system to Europe by the 12th century, and it has now displaced all older number systemsthroughout the world. In the 10th century, Halayudhas commentary on Pingalas work contains a study ofthe Fibonacci sequence and Pascals triangle, and describes the formation of a matrix.
  • 20. Islamic mathematics (c. 800—1500)The Islamic Arab Empire established across the Middle East, CentralAsia, North Africa, Iberia, and in parts of India in the 8th centurymade significant contributions towards mathematics. Although mostIslamic texts on mathematics were written in Arabic, they were not allwritten by Arabs, since much like the status of Greek in the Hellenisticworld, Arabic was used as the written language of non-Arab scholarsthroughout the Islamic world at the time. Some of the most importantIslamic mathematicians were Persian.
  • 21. Done byBattula pavan

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