Short History Of Mathematics Pdf

History of mathematics

A proof from Euclid 's

Elements

, widely considered the most in- ﬂuential textbook of all time.

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The area of study known as the

history of mathematics

is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past . Before the modern age and the worldwide spread of knowledge, written examples of new mathematical de- velopments have come to light only in a few locales. The most ancient mathematical texts available are

Plimpton 322

( Babylonian c. 1900 BC),

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the

Rhind Mathematical Papyrus

( Egyptian c. 2000–1800 BC)

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and the

Moscow Mathematical Papyrus

(Egyptian c. 1890 BC). All of these texts concern the so-called Pythagorean theorem , whichseemstobethemostancientandwidespreadmath- ematical development after basic arithmetic and geome- try. The study of mathematics as a demonstrative discipline begins in the 6th century BC with the Pythagoreans , who coined the term "mathematics" from the ancient Greek

μάθημα

(

mathema

), meaning "subject of instruction".

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Greek mathematics greatly reﬁned the methods (espe- ciallythroughtheintroductionofdeductivereasoningand mathematical rigor in proofs ) and expanded the subject matter of mathematics.

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Chinese mathematics made early contributions, including a place value system .

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The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world to- day, likely evolved over the course of the ﬁrst millen- nium AD in India and were transmitted to the west via Islamicmathematicsthroughtheworkof Muḥammadibn Mūsā al-Khwārizmī .

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Islamic mathematics , in turn, developed and expanded the mathematics known to these civilizations.

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Many Greek and Arabic texts on mathe- matics were then translated into Latin , which led to fur- ther development of mathematics in medieval Europe . From ancient times through the Middle Ages , periods of mathematical discovery were often followed by cen- turies of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, inter- acting with new scientiﬁc discoveries, were made at an increasing pace that continues through the present day.

1 Prehistoric mathematics

The origins of mathematical thought lie in the concepts of number , magnitude , and form .

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Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. The idea of the"number"conceptevolvinggraduallyovertimeissup- ported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.

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Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time.

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The Ishango bone , found near the headwaters of the Nile river (northeastern Congo ), may be more than 20,000 years old and consists of a series of tally marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers

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or a six-month lunar calendar.

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Pe- ter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."

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The Ishango bone, according to scholar Alexander Marshack , may have inﬂuenced the later development of mathemat- ics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this, however, is disputed.

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2 BABYLONIAN MATHEMATICS

Predynastic Egyptians of the 5th millennium BC pictori- ally represented geometric designs. It has been claimed that megalithic monuments in England and Scotland , dat- ing from the 3rd millennium BC, incorporate geomet- ric ideas such as circles , ellipses , and Pythagorean triples in their design.

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All of the above are disputed how- ever, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.

2 Babylonian mathematics

Main article: Babylonian mathematics See also: Plimpton 322 Babylonian mathematicsreferstoanymathematicsofthe

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.

peoples of Mesopotamia (modern Iraq ) from the days of theearly Sumerians throughthe Hellenisticperiod almost to the dawn of Christianity .

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The majority of Baby- lonian mathematical work comes from two widely sep- arated periods: The ﬁrst few hundred years of the second millennium BC (Old Babylonian period), and the last few centuriesoftheﬁrstmillenniumBC( Seleucid period).

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It is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire , Mesopotamia, especially Baghdad , once again became an important center of study for Islamic mathe- matics . In contrast to the sparsity of sources in Egyptian mathe- matics , our knowledge of Babylonian mathematics is de- rived from more than 400 clay tablets unearthed since the 1850s.

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Written in Cuneiform script , tablets were in- scribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.

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The earliest evidence of written mathematics dates back to the ancient Sumerians , who built the earliest civiliza- tion in Mesopotamia. They developed a complex system of metrology from 3000 BC. From around 2500 BC on- wards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.

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Geometry problem on a clay tablet belonging to a school for scribes; Susa , ﬁrst half of the 2nd millennium BCE

Babylonian mathematics were written using a sexagesimal (base-60) numeral system .

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From this derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is likely the sexagesimal system was chosen because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.

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Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the decimal system.

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The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no diﬀerent than multiplying integers, similar to our modern notation.

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The notational system of the Babylonians was the best of any civilization until the Renaissance ,

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and its power allowed it to achieve remarkable computation accuracy and power; for example, the Babylonian tablet YBC 7289 gives an approximation of √2 accurate to ﬁve decimal places.

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The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.

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By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions.

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This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.

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Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and

3 the calculation of regular reciprocal pairs .

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The tablets also include multiplication tables and methods for solving linear , quadraticequations and cubicequations ,aremark- able achievement for the time.

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Tablets from the Old Babylonian period also contain the earliest known state- ment of the Pythagorean theorem .

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However, as with Egyptian mathematics, Babylonian mathematics shows noawarenessofthediﬀerencebetweenexactandapprox- imate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for proofs or logical principles.

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3 Egyptian mathematics

Main article: Egyptian mathematics Egyptian mathematics refers to mathematics written

Image of Problem 14 from the Moscow Mathematical Papyrus . The problem includes a diagram indicating the dimensions of the truncated pyramid.

in the Egyptian language . From the Hellenistic pe- riod , Greek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continuedunderthe ArabEmpire aspartof Islamicmath- ematics , when Arabic became the written language of Egyptian scholars. The most extensive Egyptian mathematical text is the Rhind papyrus (sometimes also called the Ahmes Pa- pyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000–1800 BC.

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It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, di- vision and working with unit fractions, it also contains evidence of other mathematical knowledge,

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includ- ing composite and prime numbers ; arithmetic , geometric and harmonic means ; and simplistic understandings of boththe SieveofEratosthenes and perfectnumbertheory (namely, that of the number 6).

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It also shows how to solve ﬁrst order linear equations

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as well as arithmetic and geometric series .

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Another signiﬁcant Egyptian mathematical text is the Moscow papyrus , also from the Middle Kingdom period, dated to c. 1890 BC.

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It consists of what are today called

word problems

story problems

, which were ap- parently intended as entertainment. One problem is con- sidered to be of particular importance because it gives a method for ﬁnding the volume of a frustum (truncated pyramid). Finally, the Berlin Papyrus 6619 (c. 1800 BC) shows that ancient Egyptians could solve a second-order algebraic equation .

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4 Greek mathematics

Main article: Greek mathematics Greek mathematics refers to the mathematics written in

The Pythagoreantheorem . The Pythagoreans aregenerallycred- ited with the ﬁrst proof of the theorem.

the Greek language from the time of Thales of Miletus (~600 BC) to the closure of the Academy of Athens in 529 AD.

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Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greekmathematicsoftheperiodfollowing Alexanderthe Great is sometimes called Hellenistic mathematics.

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Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cul- tures. All surviving records of pre-Greek mathematics show the 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 derive conclusions from deﬁni- tions and axioms, and used mathematical rigor to prove them.

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Greek mathematics is thought to have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although the extent of the inﬂuence is disputed, they were probably inspired by Egyptian and Babylonian mathematics . According to legend, Pythago- rastraveledtoEgypttolearnmathematics,geometry, and astronomy from Egyptian priests. Thales used geometry to solve problems such as calcu- lating the height of pyramids and the distance of ships