. The History of Mathematics As of today, there are plenty of new developments in the area of Mathematics. In at least some Mathematics courses, it doesn’t cross the mind of a student about the history that Mathematics has. There is much more to Mathematics than theorems and rules. The history of Mathematics has progressed through the years even into the present day. It was before the earliest records, drawings were used to indicate the knowledge of Mathematics and measurement of time based on the stars. There were paleontologists that had discovered ochre rocks in a South African cave that was adorned with scratched geometric patterns that had been dated back to c. 70,000 BC. South Africa wasn’t the only place to have some sort of Mathematical history. Prehistoric artifacts were even discovered in Africa and France. These prehistoric artifacts were dated back between 35,000 BC and 20, 000 BC, which indicated the early attempts to quantify time. Now, the history of Mathematics was now progressing even more in the Ancient Near East (c.1800-500 BC), such as Mesopotamia and Egypt. In Mesopotamia (which today is Iraq), there was evidence of written Mathematics. The evidence dated back to the ancient Sumerians whom built the earliest civilization in Mesopotamia. It was the Sumerians who developed the complex system of metrology from 3000 BC. Then from 2500 BC and on, the Sumerians wrote the multiplication tables on tablets that were made out of clay. Not only did the Sumerians write their multiplication tables on the clay tablets, but they also wrote geometrical exercises and division problems. Also, in c.1800-500 BC the Babylonians had their Mathematics written in sexagesimal (which is a base 60) numeral system. This is where the present day use of 60 seconds in a minute, 60 minutes in a hour, and 360 degrees in a circle comes from. In c. 900 BC – AD 200, the ancient Indian Mathematics was also developing. It was the early Iron when Verdic Mathematics began with the Shatapatha Brahmana (c. 800-500 BC), which was the approximation of the value of pi to two decimal places. There was also the Sulba Sutras (c. 800-500 BC) which were the geometry texts in which it used irrational numbers, prime, the rule of three, and cube roots. Not only did the Sulba Sutras provide geometry text, but it also calculated the square root of two to five decimal places, gave a method for squaring the circle, solved linear equations and quadratic equations, developed the Pythagorean triples algebraically and gave a statement and numerical proof of the Pythagorean theorem. Then the there was the Greek and Hellenistic Mathematics in c. 550 BC- AD 300. The Greek Mathematics supposedly started with Thales (c. 624- c. 546 BC) and Pythagoras (c.582- c. 507 BC). Thales and Pythagoras influences were often disputed, but there is a possibility that both of them were influenced by the ideas of Egypt, Mesopotamia, and even possibly India. Legend has it, that Pythagoras traveled to Egypt in order to learn Mathematics, Geometry, and Astronomy from Egyptian priests. Geometry was used by Thales to solve problems like computing the height of the pyramids and the distance of the ships from the shore. Pythagoras is the one who is credited as to being the first person with verification of the Pythagorean Theorem. Also, in the Greek and Hellenistic Mathematics, Aristotle was involved. He (384- c. 322) was the first to write down the laws of logic. Euclid (c. 300 BC) was the initial example of the format in which is still used in Mathematics to this day, definition, axiom, theorem, proof. Euclid also studied conics. In Euclid’s book Elements , it became a well known book among the educated people in the West until the middle twentieth century. Along with recognizable theorems of geometry, like the Pythagorean Theorem, Elements incorporated a proof that the square root of two is irrational. Not only did Elements include that proof, but also that there were infinitely many prime numbers. Then in ca. 230 BC, Sieve of Eratosthenes was used to determine prime numbers. It wasn’t until c. 500- AD 1300, which was the Classical Chinese Mathematics started. In 212 BC China, the Emperor Qin Shi Huang (Shi Huang-ti) demanded that books outside of Qin state be burned. This demand wasn’t obeyed. The book that survived was the oldest Mathematical work I Ching. I Ching was from the Western Zhou Dynasty (from 1046 BC). In this book, eight binary three-tuples (trigrams) and sixty-four binary six tuples (hexagrams) were used for philosophical, Mathematical, and/or mystical purposes. However, there was also the oldest surviving work on Geometry in China. The book Mo Jing, came from the philosophical Mohist canon of c. 330 BC and complied by the followers of Mozi (470 BC-390 BC). In Mo Jing, it illustrated the various outlooks of many fields that were related with physical science, and offered a little bit of information on Mathematics too. Aside from books that involved Mathematics in China, the Chinese also made use of the complex combinatorial diagram recognized as the magic square. The magic square was expressed in ancient times and perfected by Yang Hui (1238-1398 AD). Also, Zu Chongzhi, of the Southern and Northern Dynasties calculated the value of pi to seven decimal places, which stayed the most precise value of pi for approximately one thousand years. Diverging from the Classical Chinese Mathematics, there is the Classical Indian Mathematics in c. 400-1600. In c. 400, the Surya Siddhanta introduced trigonometric functions of sine, cosine, and inverse sing, and put down the rules to determine the true motions of the luminaries, which corresponded to their definite arrangement in the sky. Then in 499, Aryabhata introduced the versine function , produced the first trigonometric tables of sine, developed techniques and algorithms of algebra, infinitesimals, differential equations, and attained whole number solutions to linear equations by a method that is alike to the modern method. Aryabhata also introduced a precise astronomical calculation that was based on the heliocentric system of gravitation. He also figure the value of pi to the fourth decimal place. In the fourteenth century, Madhava figured the value of pi to the eleventh decimal place. The history of Mathematics didn’t stop there. There were Arabic and Islamic Mathematics in c. 800-1500. It was the Islamic Arab Empire who were established across the Middle East, Central Asia, North Africa, Iberia, and parts of India in the eighth century all made noteworthy contributions to Mathematics. In fact, some of the most significant Islamic mathematicians were Persian. Around 1000 AD, in a book written by Al-Karaji appeared the first known proof by mathematical induction. The book was used to prove the binomial theorem, Pascal’s triangle, and the sum of integral cubes. Aside from Al-Karaji, there was Omar Khayyam, a 12th century poet who was a mathematician. He wrote a book titled Discussions of the Difficulties in Euclid. Khayyam’s book discussed the errors inn Euclids book in the particular area such as the parallel postulate. Therefore, he laid the foundations for analytic geometry and non Euclidean geometry. Khayyam was the first to find the general geometric solution to cubic equations. In the 13th century, it was Nasir al-Din Tusi a Persian mathematician who made advances in spherical trigonometry. Then in the 15th century, Ghiyahth al-Kashi figured the value of pi to the sixteenth decimal place. He also had an algorithm for computing nth root. During this period, there were other achievements of Muslim mathematicians. Some achievements were the development of algebra and algorithms, the invention of spherical trigonometry, adding the decimal point notation to the Arabic numerals, and the discovery of all modern trigonometric functions besides sine just to name a few. In c. 500- 1400, the Medieval European mathematics progressed. The Medieval European’s interest in Mathematics was driven by concerns that were quite different from those of modern Mathematics. It was the belief that Mathematics provided the main insight of the created order of nature, that was often justified by Plato’s Timaeus and the passage from the bible that God had “ordered all thins in measure, and number, and weight” (Wisdom 11:21). Also, in Medieval European in the early middle ages (c. 500-1100), Boethius gave a place for Mathematics in the curriculum when he thought of the word “quadrivium.” He used this word to express the study of arithmetic, geometry, astronomy, and music. Mathematics was reborn in Europe in 1100-1400. During the 12th century , there were European scholars who went to Spain and Sicily in search of scientific Arabic texts. The texts that these European scholars were looking for were al-Khwarizmi’s al-Jabr wa-al Muqabilah. al-Jabr wa-al Muqabilah translated in Latin by Robert of Chester, and the finished text of Elements which was also translated into an assortment of versions by Adelard of Bath, Herman of Carinthia, and Gerard of Creomona. With these new sources, it sparked a renewal of Mathematics. Thomas Bradwardine proposed that speed (which is represented as V) gets bigger in arithmetic proportion as the ratio of force (which is represented as F) to the resistance (which is represented as R) increases in geometric proportion. He expressed this by a series of certain instances, even though the logarithm wasn’t yet invented, his end result had to be expressed by V= log(F/R). This analysis is an example of transferring a particular Mathematical method that was used by al-Kindi and Arnald of Villanova to measure the nature of compound medicines to a different physical problem. Europe’s Mathematics would now shift into early modern Mathematics in c. 1400-1600. While Europe was in its dawn of the Renaissance, still Mathematics was limited by awkward notation in which Roman numerals were used and relationships were expressed by using words instead of symbols. Symbols weren’t used because there was no plus or equal sign and no use of x as an unknown. The 16th century European Mathematics started to make advancements without precedent anywhere in the world, so far as of today. One of the first advancements were the general solution of cubic equations. Towards the end of the century, Regionmontanus (1436-1476) and Francois Vieta (1540-1603) along with along with others Mathematics would now be written using Hindu-Arabic numbers and in some what of a form that is not a lot different from the notation that is used today. Europe in the 17th century would now be seeing an exceptional explosion of mathematical and scientific ideas. Galileo observed Jupiter’s moons in orbit using a telescope that was based on an imported toy from Holland. Then a Dane by the name of Tycho Brahe, collected a vast amount of data that illustrated the positions of planets in the sky. One of Galileo’s students Johannes Kelper (German) started to work with this data. John Napier, wanted to help Kelper’s calculations, so in part Napier was the first to investigate natural logarithms while in Scotland. Now Kelper had succeed in formulating mathematical laws of planetary motion. René Descartes a French mathematician and philosopher (1596-1650) developed the analytic geometry who allowed those orbits to be plotted on a graph in Cartesian coordinates. Adding to earlier works by other mathematicians, there was Isaac Newton, an Englishman who discovered the laws of physics which in turn explained Kelper’s Law’s and brought all the concepts that are now known as calculus. The 18th century brought even more historical events in Mathematics. The knowledge of the natural numbers are preserved in monolithic structures, it is the older than any surviving written text. One way to see the development of the assorted number systems of modern Mathematics is to see the new numbers studied and investigated to answer the questions about arithmetic performed on older numbers. During the prehistoric times, fractions answered the question: what number, when multiplied by 3, gives the answer 1? In the countries of India and China, and then later on in Germany, negative numbers were developed to answer the question: what is the result when a larger number is subtracted from a smaller number? In addition, the discovery of zero may have some sort of similar question: what is the result when a number is subtracted from itself? Still there was one more question which was: what kind of number is the square root of two? Greeks had already known that it wasn’t a fraction and this question could have possibly played a part in the on going development of fractions. John Napier (1550-1617) came up with a better answer when he invented decimals. Napier’s invention was later perfected by Simon Stevin. With the use of decimals and an idea that anticipated the idea of the limit, Napier had also studied a new constant in which Leonhard Euler (1707-1783) was named e. Euler came with other mathematical terms and notations, such as the square root of minus one with symbol i. He even popularized the use of the Greek letter π to represent the ratio of a circle’s circumference to its diameter. The 19th century lead to Mathematics becoming abstract. One of the greatest mathematicians of all time lived during the 19th century. He was Carl Friedrich Gauss (1777-1855) and he gave the first acceptable proofs of the fundamental theorem of algebra and of the quadratic reciprocity law. Also, in this century there was the development of the two forms of non- Euclidean geometry in which the parallel postulate of Euclidean geometry no longer holds. There two mathematicians who separately discovered the hyperbolic geometry. Those mathematicians were Russian Nikolai Ivanovich Lobachevsky and his rival, Hungarian Janos Bolyai. The hyperbolic geometry is where the uniqueness of parallels no longer holds. In this geometry the sum of the angles in a triangle add up to less than one hundred and eighty degrees. Later on, elliptic geometry was developed by a German mathematician Bernhard Riemann. In elliptic geometry, no parallel could be found and the angles in the triangle would add up to more than one hundred and eighty degrees. There was also a new form that was developed in the 19th century. This new form of algebra was called Boolean algebra, which was invented by a British mathematician George Boole. Boolean algebra was a system in which only consisted of the numbers zero and one. Today, this system has important applications in computer science. The Boolean algebra wasn’t the only new development in this century, for the first time, the limits of Mathematics were explored. A Norwegian by the name of Niels Henrik Abel and Frenchman Évariste Galois both proved that there is no general algebraic method for solving polynomial equations of a degree that is more than four. Afterward, in the 20th century the profession of Mathematics became even more significant. There were hundreds of new Ph.D’s in Mathematics that were honored every year. Also, jobs were accessible both in teaching and industry. David Hilbert presented twenty three unsolved problems in Mathematics at the International Congress of Mathematics in the 1900. The unanswered problems have extended over the many areas of Mathematics and have developed a vital focus for a great deal of Mathematic in the 20th century. As of today, ten of those unsolved problems have been resolved, there are seven that are partly resolved, and two problems are still open. Still, there are four remaining unsolved problems that are too loose to be stated as whether resolved or not. Not only did the 20th century have account for hundreds of new Ph.D’s and twenty unsolved problems, but over three thousand theorems were developed in the 1910’s by Srinivasa Aiyangar Ramanujan (1887-1920). Ramanujan theorems consisted of propties of highly composite numbers, the partition function and its asymptotics, and mock theta functions. Then famous assumptions from the past generated a new and more powerful techniques. Both Wolfgand Haken and Kenneth Appel used a computer to provide evidence of the four color theorem in 1976. While working alone in his office, Andrew Wiles verified Fermant’s last theorem in 1995. It was the 20th century where entirely new areas of Mathematics like mathematical logic, topology, complexity theory, and game theory modified the types of questions that could be resolve by the mathematical methods. Still today, Mathematics history is still in the making. In about mid-March of 2007, there was a team of researchers throughout North American and Europe who used network of computers to map E8. Even though its not know exactly how the understanding of E8 can be useful, the discovery has become a great achievement for teamwork and computation technology in contemporary Mathematics. Even though, Mathematics may not seem like it has a lot of history, it really does. The history of Mathematics has progressed so much, that even today there are more developments and discoveries still going on. Without the many things that have been found throughout history, Mathematics wouldn’t have its rules, theorems, and pi just to name a few. If it wasn’t for Mathematics history, there would certainly be no Mathematics courses, people’s jobs wouldn’t be the same, technology wouldn’t function correctly, and daily routines would be inadequate. Where would the world be without Mathematics?