A Brief History of Mathematics - Coursework Example

A Brief History of Mathematics Mathematics is a diverse adventure of ideas and concepts (Stuik, 1987, p The history of mathematics can be tracedfrom ancient history of the world. The word mathematics is derived from the Greek “máthema” that refers to knowledge, science, or learning, and “mathematikós” means “fond of learning”. With the invention of printing, the problem of getting secure texts is resolved; therefore, historians of mathematics became free to focus their editorial efforts on the correspondence or the unpublished work of mathematicians (Marsigit, n.d., p.19). It has a crucial role in the development of human societies and civilizations. From early times to modern world, mathematics served as a strategic key to the progress and development of other fields of knowledge and human civilization s a whole. Right from the beginning of this world, people needed to organize and count for their basic needs, such as, counting animals and cornstalk. Natural occurrences, such as, seasons, changing moon, days and nights urged them to keep record of days and years. Geometrical shapes and algebra developed from the need of shelter and buildings. The whole process of the development of mathematics took hundreds of years with the contributions of mathematicians and civilizations around the world, most noteworthy of them were: Egyptians, Babylonians, Greeks, Hellenistic, Muslims, and Europeans. This research paper briefly explores the history of mathematics and prominent mathematicians in different civilizations. 1. Ancient Age Mathematics During ancient times, for more than thousand years (5th century BC to 5th century A.D.), Greek mathematicians established great tradition of work in exact sciences, such as, mathematics, astronomy, and other related disciplines. However, historians could record only major figures of these exponential developments in detail (Marsigit, n.d., p.19). 2. Prehistoric Mathematics Prehistoric people would have had a general sense of amounts as they instinctively knew the differences, such as, difference between one and two antelopes. However, it took ages to takes that intellectual leap from the tangible idea of two things to the invention of symbol or word for the abstract idea of “two”. Even in today’s modern age, there are isolated hunter gatherer tribal people in Amazonia who use words for “one”, “two” and “many”, moreover, others have words for only up to five. Without proper agriculture and trade, these people don’t need formal system of numbers (Mastin, 2010). Ancient people kept track of regular occurrences and events, such as, the phases of moon, and seasons. Some of the most ancient evidence that reveal human thinking about numbers is from notched bones in Africa that dates back to 35,000 to 20,000 years ago. However, it was very basic of the math or similar activity, such as, the count or tallying (Mastin, 2010). Pre-dynastic Egyptians and Sumerians depicted their knowledge of geometric designs on their artifacts as early as the 5th millennium BC, similarly, some megalithic societies did the same in northern Europe in the 3rd millennium BC or even before that. However, their mathematical practices were more of art and decorative stuff rather than systematic treatment of figures, forms, quantities, patterns. Proper mathematical practice started largely as a reaction to bureaucratic needs when civilizations flourish and developed agriculture. They started practicing it in Sumerian and Babylonian civilizations of Mesopotamia and in ancient Egypt. The major objective was to measure plots of land and to impose taxes on people. There is some evidence of basic arithmetic and geometric notations on the petroplyphs at Knowth and Newgrange burial hills in Ireland that is estimated to be from 3500 BC and 3200 BC respectively. These samples used a recurring zigzag glyph like thing for the purpose of counting. This system was so effective and successful that it remained in use during 1st millennium Britain .Moreover, there are some evidence from 2300 BC is Stonehenge in England that represents the use of 60 and 360 in the measurement of a circle, however, there are some opposition to this conception. This practice was developed as an independent one from sexagesimal counting system of the ancient Sumerian and Babylonians (Mastin, 2010). 3. Sumerian or Babylonian Mathematics Modern day Iraq and a part of ancient Mesopotamia, called Sumer was the origin of writing system, the wheel, the arch, the plow, agriculture, irrigation and several other innovations. Therefore, this land is often referred to as Cradle of Civilization. The Sumerians invented the first ever writing system; it was a pictographic writing system referred as cuneiform script. It used wedge shaped characters engraved on baked clay tablets. Due to this writing system, we have more knowledge of ancient Sumerians and Babylonian mathematics as compared to early Egyptian one. We even have things that appear to be school exercises in anathematic and geometric problems (Mastin, 2010). Just like Egypt, the credit for the development of Sumerian mathematics goes to bureaucratic needs. As early as in 6th millennium BC, with the development of civilization and agriculture, they needed to measure land and to gather tax money from people. Moreover, they needed to narrate large numbers because they tried to record the course of night sky and invent their sophisticated lunar calendar. Sumerians were the first in the world to give symbols to the group of objects in order to describe large number of things in an easier manner. From using different tokens or symbols to signify sheaves of wheat and oil jars, they developed more abstract symbols in order to count definite number of anything. Before that in 4th millennium BC, they started using a small clay cone to represent one, a clay ball for ten, and a large cone for sixty. During 3rd millennium, they swapped these objects with their cuneiform equivalent in order to write numbers with the same stylus and used them as words in text. Sumerians were already using an elementary model of abacus in 2700 and 2300 BC (Mastin, 2010). Sumerian mathematics worked on the foundation of sexegesimal, or base sixty, numeric system that was countable physically by using the twelve knuckles on one hand, and the five fingers on the other hand. Contrary to Egyptians, Romans, and Greeks, Babylonian numbers utilized a true place-value system. In this system, digits written in the left column hold larger values. It was quite similar to modern day counting system but used a base 60 instead of base 10. It is assumed that the Babylonian developments in mathematics were influenced by the fact that 60 is a number that has several divisors, moreover, it is the smallest integer that is dividable by all other digit from 1 to 6. Similarly, our modern day use of 60 that includes, 60 seconds in a minute, 60 minutes in an hour, 360 degrees (60×6) in a circle are all evidence of the ancient Babylonian system. For same reasons, 12 that has factors of 1,2,3,4 and 6, is all time favorite multiple throughout history, for instance, 12months, 12 inches, 12 pence, and 2×12 hours(Mastin, 2010). Another breakthrough concept that neither Egyptians nor Greeks or Romans, but Babylonians came up with was the circle character of zero. However, this symbol was a placeholder rather than a number in itself. Research evidence shows the presence and development of a complex system of meteorology in Sumer dating back to 3000 BC, and multiplication and reciprocal tables, square roots, cube roots, table of squares, geometrical exercises, and division problems evidences belong to as early as 2600 BC onward. Babylonian tablets from 1800 to 1600 BC included variety of complex topics, such as, fraction, algebra, methods of solving linear, some cubic equations, even the calculation of regular reciprocal pairs. Furthermore, they introduced geometric shapes in the building structures and design, and in the dice of leisure games that became popular, such as, the ancient game of backgammon. They developed geometry to the calculation of the areas of rectangles and trapezoids and volumes of simple shapes, such as, cylinders and bricks (Mastin, 2010). 4. Egyptian Mathematics By 6000 BC Egyptians started documenting lunar phases and seasons for religious and agricultural concerns (Mastin, 2010). The oldest mathematical document was composed by an Egyptian Ahmes during 1700 B.C; it is referred as the Ahmes or Rhind Papyrus. This work was based on the works of people from about five thousand years back as Ahmes regarded it as a crux of all existing and all obscure secrets (cited in Nirenberg, 1997, p.25) Some believe that Egyptians devised the earliest form of completely developed base 10 numeration systems during 2700 BC. Rhind Papyrus is similar to an instruction manual that guides the reader for division and multiplication through clear demonstrations. It includes other mathematical information, such as, prime numbers, composite, unit fractions, arithmetic, harmonic, and geometric means. Moreover, it reveals the way to solve first order linear equations, and arithmetic and geometric series. The Berlin Papyrus that was written as early as 1300 BC reveals that ancient Egyptians knew how to solve second-order equations. They identified fractions and approximated the area of a circle by using the existing knowledge of shapes (Mastin, 2010). Egyptian mathematics concentrated on practical problems more than abstract ideas. Research evidence shows that they must have practiced algebra five thousand years ago for solving problems that involved measuring, weighing, land measurements and parceling (Nirenberg, 1997, p.25).The pyramids are a great example of Egyptian sophistication in the field of mathematics. First of all, they had discovered the golden ratio of 1:1.618. Some argue that it could have happened due to purely aesthetic reasons rather than mathematical calculations. But it is certainly evident that they were well aware of the formula for the volume of a pyramid (1/3×length×width×height) in addition to truncated or clipped pyramid. They knew the secret of right angle triangle with sides 3, 4 and 5 units before Pythagoras, in fact, the 3-4-5 right angle triangle is often referred as “Egyptian”(Mastin,2010). 5. Greek contributions in Mathematics With their invasions and resulting influence into Asia Minor, Mesopotamia and beyond, Greeks adopted all beneficial stuff they had to offer. It was particularly true for mathematical practices as they adopted these influences from Egyptians and Babylonians as well. With the advent of Hellenistic period, they started making contributions on their own which proved to be quite dramatic in the history of mathematics. Ancient system of numerals referred to as Attic or Herodianic numerals was invented during 450 BC and became part of daily lives by 7th Century BC. It was essentially a base 1 system and influenced by Egyptians and even the somewhat close of later Roman system. However, most part of Greek mathematics consisted of their contributions to geometry (Mastin, 2010). Greeks started to develop mathematical concepts during 450 BC. The atomic theory of Democritus has its roots in Zeno of Elea’s paradoxes. A more defined handling of these concepts made them realize that the rational numbers are not sufficient enough to measure all lengths. Therefore, a geometric representation of irrational numbers came into being. Different areas of study thus led to the formation of integration. Apollonius concepts and theory of conic sections represent a peak point in pure mathematical studies. In addition, the mathematical inventions and discoveries stemmed from astronomy, for instance, trigonometry. The major phase of success in mathematical studies for Greeks came from 30 BC to 200 AD as mathematical developments continued and flourished in Islamic countries, such as, Syria, India, and Iran. Though the work has no match with the Greeks, but they helped in preserving the Greek mathematics. In the later period during 11th Century, Adelard of Bath and Fibonacci brought Greek knowledge of mathematics with Muslim’s addition back into the Europe (O’Connor and Robertson, 1997). Most legendary Greek mathematician was Pythagoras of Samos who came up with Pythagoras’ Theorem (Mastin, 210). Pythagoras started studying number problems during 1700 BC. Linear equations systems were analyzed for solving number problems. Study of quadratic equations resulted in early forms of numerical algebra (O’Connor and Robertson, 1997). He proposed that all of the mathematical system can be constructed. With all the fame and accomplishment, he remained a controversial mathematician. However, his works influenced mathematics for future mathematicians to a great extent, such as, Plato and Aristotle. Plato spread his work by contributing to mathematics and opening up an Academy in Athens, moreover, he invented the five Platonic solids. Aristotle’s contributions to logic remained an authority for more than two thousand years. Plato’s student Eudoxus provided the foundation of the “method of exhaustion” application for the first time, invented a general theory of proportion, thus invigorated the Plato’s incomplete works. There is no doubt that Greek’s most important contribution was the idea of proof, and deductive method of proving or rejecting theorems. Earlier civilizations used inductive reasoning in order to establish rules. Greeks established the power of mathematics through theories and provided the foundation of systematic mathematics of Euclid and rest of the world (Mastin, 2010). By 300 BC, Euclid had developed geometry into 10 axioms from where he derived all the geometrical facts. Greece might have passed on, but her mathematical achievements contributed to the empire of Alexander the Great, sustain Byzantium, and eventually fell into the hands of Muslims when all of the Western Europe went through Dark Ages(Nirenberg, 1997,pp.36-37). 6. Hellenistic Mathematics After the arrival of Alexander the Great, mathematical progress started in the outskirts of the Greek Hellenistic empire. Alexandria became the centre of learning and Ptolemies’s library reached to the level of Athenian Academy. For the first time in history, the library patrons were professional scientists who were paid for their passion for research and science. Some of the most prominent mathematicians include Euclid, Archimedes, Diophantus, Hipparchus, Apollonius, Menelaus, and Heron (Mastin, 2010). 6.1. Roman Mathematics When Romans got hold of Greek and Hellenistic empires during 1st Century BC, mathematical progress stopped. There were no prominent mathematicians or innovations occurred in Roman Empire and Republic. They sued math for only practical reasons and not purely for the development of it (Mastin, 2010). By 1 A.D. Greece is absorbed by Rome, thus her economic ideas and merchant class as well. Most of the Romans were innumerate, however, some living in room were efficient in arithmetic of money (cited in Nirenberg, 1997, p. 37). Innovation of Roman numeral is the most prominent area of Roman mathematics as it became most influential system of numbers for trade and administration throughout Europe. Though it was decimal base 10 system, but it wasn’t directly positional and didn’t include zero. Therefore, it was an inefficient system for mathematical uses. Considering the problem in writing arithmetic by using Roman numeral notation, mathematical calculations were carried out by using an abacus. Abacus was a counting frame that was based on Babylonian and Greek abaci (Mastin, 2010). 6.2. Mayan Mathematics The Mayan civilization’s golden age is estimated to be from around 250 B.C. to A.D. 800. Even today, approximately seven million Mayan Indians are settled in some areas of Honduras, El Salvador, Guatemala, Belize, Southern Mexico, and Nicaragua (Arellano, 2003, p.2). Despite being considered Barbarian in their roots, Mayans were one of most intelligent civilizations in 1st or 2nd century of Christian era (Morley, 1915, p.2). They were expert and made influential discoveries in the field of mathematics (Ifrah, 1998, p.297). They had developed a numeral system which is considered to be more sophisticated than that of Aztec and Roman. This numeral system also referred as vigesimal system used a base 20 instead of 10.They also came up with dot and bar for representing number rather than symbols. They were expert priests, mathematicians, astronomers and scientists. However, most wise men died as a result of epidemics, colonization, and wars that hindered the further development of mathematics and other areas of knowledge (Arellano, 2003, p.2). According to research evidence, pre-classic Maya and their neighboring societies had discovered zero as early as 36 BC. Without even discovering the concept of fraction, Mayans developed tremendously accurate astronomical observations without any specific instrument but sticks. Surprisingly, they calculated solar year with more accuracy than that of Europeans. According to their results, a solar year is composed of 365.242 days which is closest to modern day calculation of 365.242198.Moreover, they accurately calculated lunar month to be of 29.5308 days comparable to modern value of 29.53059.Unfortunately, geographical isolation kept their wisdom from influencing Asians or European mathematics (Mastin, 2010). 6.3. Chinese Mathematics For historical and philosophical reasons, Chinese mathematics developed in different direction as compared to Western mathematics. It focused on algebra and practical mathematical application rather than geometry and theoretical reasoning. Most prominent Chinese mathematical work is the Nine Chapters on the Mathematical Art from 1st century AD. During Chinese Renaissance and following Mongol invasion led mathematics to its peak during 12th and 14th century (Brandenburg and Keimpe, 2007, p.1). Ancient Chinese had pervasive fascination with numbers and mathematical patterns, therefore, different numbers were believed to hold some cosmic important, for instance, magic square. One of the greatest Chinese mathematicians was Liu Hui who developed an extensive commentary on the “Nine Chapters” during 263 AD. He was the first one who left root unevaluated and driven more accurate results rather than merely approximate results. The golden age of Chinese mathematics started in 13th century when more than thirty mathematic schools started operating across China. One of the most revolutionary Chinese mathematicians was Qin Jiushao who invented solution for quadratic and cubic equations by using a method close to Isaac Newton (Mastin, 2010). 7. Indian Mathematics Without much influences from Chinese or Babylonian mathematics, Indian mathematics has also made advancement in early times of India. Mantras during early Vedic period presented the use of powers of ten from and hundred up to trillion. There are some evidences of arithmetic operations, such as, addition, subtraction, division, multiplication, square, cubes, roots, and fractions. Before Pythagoras, a text from 8th Century BC referred to as “Sulba Sutras” documented Pythagorean triples, simple version of Pythagorean theorem, geometric solutions of linear and quadratic equations. During 2nd or 3rd Century BC, Jain mathematicians identified five kinds of infinities. Indians are also given credit for the invention of zero that is found in a temple in Gwalior during 9th century; however, the credit for the use of zero as a number itself goes to 7th Century Indian mathematician Brahmagupta or another Indian Bhaskara I. Bhaskara II also contributed to different areas of mathematics, such as, cubic and quadratic equations, Diophantine equations of second order, infinitesimal calculus, and mathematical analysis of spherical trigonometry and other areas of trigonometry(Mastin,2010). 8. Islamic Mathematics Islamic contribution to the preservation of ancient mathematics and its revolutionary development for over five centuries is remarkable. Muslim scholars and mathematicians translated most of the Greek works, assimilated their knowledge, and provided critique, innovated and developed mathematical ideas and theories (Allen, 2000, p.1). Islamic mathematicians across Middle East, Central Asia, North Africa, Persia, Iberia, and India contributed significantly after 8th Century. Their influences came from Indian and Greek mathematics. By using complex geometrical patterns in architecture, they contributed to evolve mathematics to an art form. They revealed that all forms of symmetry can be represented on a two dimensional surface. It is interesting to note that their Holy book, Qur’an taught them to gather knowledge and moved them into Golden Age of Islamic science and mathematics. Muhammad Al-Khwarizmi was one of the revolutionary and greatest mathematicians of Islamic history. Perhaps his most prominent work is the advocacy of Hindu numerical system which not only revolutionized Islamic but also Western and European mathematics in later times. His other crucial accomplishments are to algebra as he invented algebraic methods of “reduction” and “balancing”, moreover, he presented a detailed account for the solution of polynomial equations solution. Overall he gave us a strong abstract mathematical language and a way of dealing with mathematical problems in more general rather than specific (Indian and Chinese) way that is still practiced throughout the world. Another prominent Muslim mathematician was Muhammad Al-Karaji who devised the theory of algebraic calculus. With this theory he broadened the body work in algebra and freed it from its geometrical roots. Al-Karaji was the one who used method of proof by using mathematical induction that he further extend to prove binomial theorem (Mastin, 2010). Omar Khayyam was another prominent Muslim mathematician and astronomer, perhaps more famous for his poetry and writings. He generalized Indian methods of taking square and cube roots and extended it to fourth, fifth, and higher roots. He conducted systematic analysis of cubic problems and revealed the existence of variety of cubic equations. The 13th Century Persian mathematician, astronomer and scientist dealt with trigonometry as an isolated discipline. He invented first extensive exposition of spherical trigonometry and classified six cases of a right triangle in spherical geometry. One of his most famous works was the invention of the law of sines for planes triangles (Mastin, 2010). Thabit ibn Qurra was a Muslim mathematician from 9th Century Arab who discovered the general formula through which amicable numbers are derived. Abul Hasan al-Uqlidisi was an Arab Muslim mathematician revealed the potential use of Arabic numeral and came up with the concept of decimal instead of fraction. Ibrahin ibn Sinan contributed Archimedes’ measurements of areas and volumes, and tangents of a circle. Persian Muslim mathematician Ibn al-Haytham or Alhazen revolutionized optics and physics; moreover, he marked the beginning of association between algebra and geometry and devised “Alhazen’s problem”. Other noteworthy Muslims mathematicians include Kamal al-Din al-Farisi, and Ibn al-Banna al-Marrakushi. During 14th to 15th Century, oppressive influence of Turkish Ottoman Empire led to the stagnation of Islamic mathematics. Later, mathematical knowledge and future developments moved to Europe (Mastin, 2010). 9. European Progress in Mathematics Major advancements in European mathematics started at the dawn of 16th Century when mathematicians like Pacioli, Cardan, Tartaglia, and Ferrari came with the concepts and algebraic solutions of cubic and quadratic equations. In that period, mathematics paved the ways of the study of universe with their revolutionary applications of mathematical ideas. The advancement in algebra proved very nourishing when it comes to psychological acceptance and related enthusiasm for mathematics as it led to the research studies in Italy, Belgium, and France. In 17th Century, John Napier and Briggs invigorated the soul of mathematics with their invention of logarithms. Cavalieri contributed to the progress of calculus with his infinitesimal methods. Descartes empowered geometry by incorporating algebraic methods in it (O’Connor and Robertson, 1997). Advancements in calculus continued with the mathematical study of probability by Fermat and his collaboration with Pascal. However, calculus remained to be the most important topic of interests and evolution during the 17th Century. Barrows works provided the foundation for Newton since he transformed calculus into a tool that leads the study of nature forward. His work included rich discoveries that represented the relation between mathematics, physics and astronomy. On the basis of his works, Newton’s theory of gravity and light brought the world into 18th Century. Furthermore, the progress of calculus cannot be completed without mentioning Leibniz. His approach towards calculus was much more meticulous than that of Newton; moreover, he provided the basis of the mathematical works of 18th Century. Leibniz’s connections and influence on Bernoulli family proved to be critical for the growth of calculus and its application in variety of areas. Another prominent mathematician of 18th century was Euler who not only contributed to the existing body of mathematics through his wide range of works, but also invented two new branches of mathematics called differential geometry and calculus of variations. Euler also contributed effectively and enhanced the works of Fermat. By the end of 18th Century, Lagrange stepped into introducing an exhaustive theory of functions and mechanics. In the beginning of the 19th Century, Laplace’s work on celestial mechanics and Monge and Carnot’s works in synthetic geometry marked the bright future of mathematics. This period witnessed the rapid progress in the field of mathematics as Fourier’s work on heat, Plucker primary work on analytic geometry, and Steiner’s synthetic geometry hold great importance in the history of mathematics (O’Connor and Robertson, 1997). Lobachevsky and Bolyai came up with non-euclidean geometry that provided the foundations for Riemann who developed the characterization of geometry. Some scholars consider Gauss to be the greatest mathematician this world has ever seen. He worked on quadratic reciprocity and integer congruences, moreover, his contributions to differential geometry proved to be revolutionary to the subject. More of his significant works contributed to magnetism and astronomy. Galois worked on equations and his approach toward the use of mathematics for studying fundamental operations opened a new avenue. Moreover, his contribution of the group concept directed mathematical research into a new direction that transmitted into the research during 20th century. Working on the foundations of Lagrange about functions, Cauchy analyzed and introduced the theory of functions of a complex variable. His contributions led to the works of Weiestrass and Riemann. Cayley made the progress to Algebraic geometry, his contributions in matrices and linear algebra was further enhanced by Hamilton and Grassmann. By the end of 19th Century, Cantor formulated the set theory. Moreover, his analysis of the concept of number influenced the works on irrational numbers by Dedekind and Weierstrass. Mathematical analysis was determined by the inquiries of mathematical physics and astronomy. Lie’s contribution in differential equations made the study of topological groups and differential topology possible. A revolution in the study of application of mathematical analysis to mathematical physics was brought by Maxwell. In addition, Maxwell, Boltzmann and Gibbs came up with the invention of statistical mechanics from which stemmed Ergodic theory. The areas of electrostatics and potential theory gave birth to the study of integral equations. The works of Hilbert and evolution of functional analysis came from the works of Fredholm(O’Connor and Robertson, 1997). 10. Nature and role of mathematics in society Ideas and perceptions about the role of mathematics have huge impact on the development of school mathematics curriculum, instruction, and research. Understanding different mathematical concepts is as important as it is to draw results and interpretations from research work (Dossey, 1992, p.39). The literature of the reform movement (National Council of Teachers of Mathematics, 1989; Mathematical Science Education Board, 1989, 1990; American Association for the Advancement of Science, 1989) in mathematics and science education represents mathematics as ever-changing and ever-growing field of study (cited in Dossey, 1992, p.39). On the other hand, some define mathematics as a static discipline, with reestablished set of concepts, skills, and principles (Fisher cited in Dossey, 1992, p.39). Over the past fifty years, rapid growth in the field of mathematics and its applications led to several scholarly research essays that investigated its nature and importance. These studies have built a rich base of conception of the nature of mathematics that includes complex ideas like axiomatic structures and general rules for solving problem. These wide ranging views have greatly influenced our ideas and perceptions about mathematics and reactions to its every increasing impact on our daily lives (Dossey, 1992, p.39). According to Steen, mathematics is just like a rich fruit bearing tree for scientists and engineers in general since they take benefits from what it has to offer and develop their theories on the basis of it. On the other hand, mathematicians consider the subject as a rainforest that is nurtured by external forces and as a result enrich human civilization. The reason behind these different perceptions is the abstract language of mathematics that isolates it from normal human activities (1988 cited in Dossey, 1992, p.39). According to research evidence (Brown, 1985; Bush 1982; Cooney, 1985…cited in Dossey, 1992, p.39) these wide ranging perceptions have influenced the way that both teachers and mathematicians adopt for teaching and development of mathematics. Some of them consider mathematics as a stationary discipline that developed in an abstract manner, while others believe it to be a dynamic discipline that continues to change with new findings through application and experimentation (Crosswhite et al., 1986). These differing conceptions about the nature and source of mathematical knowledge have presented a continuum for mathematical ideas since the time of Greeks. However, the lack of a common viewpoint of mathematics has serious implications for both teaching and practice of mathematics. Some argue that this lack of agreement is the main reason that contrasting philosophies are not even discussed. On the other hand, some assume that these differing viewpoints are conveyed to the students and help them forming their perception about the nature of mathematics (Brown, Cooney & Jones, 1990; Cooney, 1987 cited in Dossey, 1992, p.39). When it comes to relationship with history of science, history of mathematics is fundamentally different from the history of other sciences since it was not the integral part of the science in Whewellian sense. It is different due to the esoteric nature of mathematics; therefore, its history was told to only a selected group of beginners (Sarton, n.d., and p.1).In early centuries, mathematicians focused on the external use of mathematics and to explain external phenomenon, such as, universe. In today’s world, mathematics is being used for gaining knowledge about human being himself, such as, gathering data, behavioral issues, and the issues of the way human function (Nirenberg, 1997, p.23). Studying the history of mathematics is a very complex area of research. There is diverse range of research material from different aspect and different civilization. An attempt to describe it in a brief manner seems impossible. The mathematical history is perhaps as old as human himself. The mathematical journey started from simple count and number and passed from civilizations, such as, Babylonians, Sumerians, Egyptians, Indian, Greek, Islamic, and European who assimilated each other’s knowledge, innovated, and developed mathematics for the bright future that humankind has now. Works Cited Struik,J.D., 1987.A Concise History of Mathematics. 4th ed.USA: Dover Publication. Nirenberg, I.1997.Living with Math. New Orleans,LA:Pi Press. Dossey,A.J., The Nature of Mathematics: Its role and its influence.[Online] Illinois: Illinois State University. Available at: < http://storage.cet.ac.il/sharvitnew/storage/939935/424648.pdf> Marsigit, M.A.n.d.The Nature and History of Mathematics.[Online].Yogyakarta State University. Available at: ‎>[Accessed 2 May 2014]. Brandenburg,R. and Keimpe Nevenzeel.2007.The Nine Chapters on the History of Chinese Mathematics. [Online]. Groningen:University of Groningen.Available at: [Accessed 4 May 2014]. Arellano,A.2003.Maya Mathematics and Science.[Online].Available at: < https://math.ucsd.edu/programs/...of_math.../math_history_07.pdf>[Accessed 4 May 2014]. Sarton,G.,n.d. The Study of the History of Mathematics.[online].Available at: [Accessed 4 May 2014]. Morley, S.G. 1915.An Introduction to the Study of the Maya Hieroglyphs. Washington, D.C: Government printing Office. Ifrah, G. 1998.The Universal History of Numbers: From Prehistory to the Invention of the Computer. Great Britain: The Harvill Press Ltd. Allen, G.D.2000.Islamic Mathematics and Mathematicians.[online].Availabel at: [Accessed 4 May 2014]. Mastin,L.2010. The Story of Mathematics.[Online]Available at: [Accessed 4 May 2014]. Crosswhite, F J., Dossey, ]. A., Cooney, T J., Downs, F L., Grouws, D. A., McKnight, C. c., Swafford,]. ., and Weinzweig, A. 1. 1986. Second international mathematics study detailed report for the United States. Champaign, IL: Stipes. Steen, L.A .1988. The science of patterns. [Online]Northfield: Sciencemag .Availabel at:< http://www.stolaf.edu/people/steen/Papers/sci_patterns.pdf>[Accessed 3 May 2014]. O’Connor,J.J.,and Robertson,E.F. An Overview of the History of Mathematics.[online]Available at: < http://www-history.mcs.st and.ac.uk/HistTopics/History_overview.html>[Accessed 3 May 2014]. Read More

Ancient people kept track of regular occurrences and events, such as, the phases of moon, and seasons. Some of the most ancient evidence that reveal human thinking about numbers is from notched bones in Africa that dates back to 35,000 to 20,000 years ago. However, it was very basic of the math or similar activity, such as, the count or tallying (Mastin, 2010). Pre-dynastic Egyptians and Sumerians depicted their knowledge of geometric designs on their artifacts as early as the 5th millennium BC, similarly, some megalithic societies did the same in northern Europe in the 3rd millennium BC or even before that.

However, their mathematical practices were more of art and decorative stuff rather than systematic treatment of figures, forms, quantities, patterns. Proper mathematical practice started largely as a reaction to bureaucratic needs when civilizations flourish and developed agriculture. They started practicing it in Sumerian and Babylonian civilizations of Mesopotamia and in ancient Egypt. The major objective was to measure plots of land and to impose taxes on people. There is some evidence of basic arithmetic and geometric notations on the petroplyphs at Knowth and Newgrange burial hills in Ireland that is estimated to be from 3500 BC and 3200 BC respectively.

These samples used a recurring zigzag glyph like thing for the purpose of counting. This system was so effective and successful that it remained in use during 1st millennium Britain .Moreover, there are some evidence from 2300 BC is Stonehenge in England that represents the use of 60 and 360 in the measurement of a circle, however, there are some opposition to this conception. This practice was developed as an independent one from sexagesimal counting system of the ancient Sumerian and Babylonians (Mastin, 2010). 3. Sumerian or Babylonian Mathematics Modern day Iraq and a part of ancient Mesopotamia, called Sumer was the origin of writing system, the wheel, the arch, the plow, agriculture, irrigation and several other innovations.

Therefore, this land is often referred to as Cradle of Civilization. The Sumerians invented the first ever writing system; it was a pictographic writing system referred as cuneiform script. It used wedge shaped characters engraved on baked clay tablets. Due to this writing system, we have more knowledge of ancient Sumerians and Babylonian mathematics as compared to early Egyptian one. We even have things that appear to be school exercises in anathematic and geometric problems (Mastin, 2010). Just like Egypt, the credit for the development of Sumerian mathematics goes to bureaucratic needs.

As early as in 6th millennium BC, with the development of civilization and agriculture, they needed to measure land and to gather tax money from people. Moreover, they needed to narrate large numbers because they tried to record the course of night sky and invent their sophisticated lunar calendar. Sumerians were the first in the world to give symbols to the group of objects in order to describe large number of things in an easier manner. From using different tokens or symbols to signify sheaves of wheat and oil jars, they developed more abstract symbols in order to count definite number of anything.

Before that in 4th millennium BC, they started using a small clay cone to represent one, a clay ball for ten, and a large cone for sixty. During 3rd millennium, they swapped these objects with their cuneiform equivalent in order to write numbers with the same stylus and used them as words in text. Sumerians were already using an elementary model of abacus in 2700 and 2300 BC (Mastin, 2010). Sumerian mathematics worked on the foundation of sexegesimal, or base sixty, numeric system that was countable physically by using the twelve knuckles on one hand, and the five fingers on the other hand.

Contrary to Egyptians, Romans, and Greeks, Babylonian numbers utilized a true place-value system. In this system, digits written in the left column hold larger values. It was quite similar to modern day counting system but used a base 60 instead of base 10.

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