The Development of Mathematics
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After the fall of Rome, the development of mathematics was taken on by the Arabs, then the Europeans. Fibonacci was one of the first European mathematicians, and was famous for his theories on arithmetic, algebra, and geometry. The Renaissance led to advances that included decimal fractions, logarithms, and projective geometry. Number theory was greatly expanded upon, and theories like probability and analytic geometry ushered in a new age of mathematics, with calculus at the forefront.
In the 17th century, Isaac Newton and Gottfried Leibniz independently developed the foundations for calculus. Calculus development went through three periods: anticipation, development and rigorization. In the anticipation stage, mathematicians were attempting to use techniques that involved infinite processes to find areas under curves or maximize certain qualities. In the development stage, Newton and Leibniz brought these techniques together through the derivative and integral. Though their methods were not always logically sound, mathematicians in the 18th century took on the rigorization stage, and were able to justify them and create the final stage of calculus.
Today, we define the derivative and integral in terms of limits. In contrast to calculus, which is a type of continuous mathematics, other mathematicians have taken a more theoretical approach.
Discrete mathematics is the branch of math that deals with objects that can assume only distinct, separated value. Discrete objects can be characterized by integers, whereas continuous objects require real numbers. Discrete mathematics is the mathematical language of computer science, as it includes the study of algorithms. Fields of discrete mathematics include combinatorics, graph theory, and the theory of computation.
People often wonder what relevance mathematicians serve today. In a modern world, math such as applied mathematics is not only relevant, it's crucial.
Applied mathematics is the branches of mathematics that are involved in the study of the physical, biological, or sociological world. The idea of applied math is to create a group of methods that solve problems in science. Modern areas of applied math include mathematical physics, mathematical biology, control theory, aerospace engineering, and math finance. Not only does applied math solve problems, but it also discovers new problems or develops new engineering disciplines.
Applied mathematicians require expertise in many areas of math and science, physical intuition, common sense, and collaboration. The common approach in applied math is to build a mathematical model of a phenomenon, solve the model, and develop recommendations for performance improvement. While not necessarily an opposite to applied mathematics, pure mathematics is driven by abstract problems, rather than real world problems.
Much of what's pursued by pure mathematicians can have their roots in concrete physical problems, but a deeper understanding of these phenomena brings about problems and technicalities. These abstract problems and technicalities are what pure mathematics attempts to solve, and these attempts have led to major discoveries for mankind, including the Universal Turing Machine, theorized by Alan Turing in The Universal Turing Machine, which began as an abstract idea, later laid the groundwork for the development of the modern computer.
Pure mathematics is abstract and based in theory, and is thus not constrained by the limitations of the physical world. According to one pure mathematician, pure mathematicians prove theorems, and applied mathematicians construct theories. Pure and applied are not mutually exclusive, but they are rooted in different areas of math and problem solving.
Though the complex math involved in pure and applied mathematics is beyond the understanding of most average Americans, the solutions developed from the processes have affected and improved the lives of all. Live Science. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date.
After the book burning of BC, the Han dynasty BC— AD produced works of mathematics which presumably expanded on works that are now lost. The most important of these is The Nine Chapters on the Mathematical Art , the full title of which appeared by AD , but existed in part under other titles beforehand.
It consists of word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying , and includes material on right triangles. The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the Song dynasty — , with the development of Chinese algebra. The most important text from that period is the Precious Mirror of the Four Elements by Zhu Shijie — , dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method.
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Even after European mathematics began to flourish during the Renaissance , European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards. Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.
Japanese mathematics , Korean mathematics , and Vietnamese mathematics are traditionally viewed as stemming from Chinese mathematics and belonging to the Confucian -based East Asian cultural sphere. The earliest civilization on the Indian subcontinent is the Indus Valley Civilization mature phase: to BC that flourished in the Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.
The oldest extant mathematical records from India are the Sulba Sutras dated variously between the 8th century BC and the 2nd century AD ,  appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.
An A-to-Z History of Mathematics
As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity. The next significant mathematical documents from India after the Sulba Sutras are the Siddhantas , astronomical treatises from the 4th and 5th centuries AD Gupta period showing strong Hellenistic influence. Around AD, Aryabhata wrote the Aryabhatiya , a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology. Several centuries later, the Muslim mathematician Abu Rayhan Biruni described the Aryabhatiya as a "mix of common pebbles and costly crystals".
In the 7th century, Brahmagupta identified the Brahmagupta theorem , Brahmagupta's identity and Brahmagupta's formula , and for the first time, in Brahma-sphuta-siddhanta , he lucidly explained the use of zero as both a placeholder and decimal digit , and explained the Hindu—Arabic numeral system.
Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. Various symbol sets are used to represent numbers in the Hindu—Arabic numeral system, all of which evolved from the Brahmi numerals. Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century, Halayudha 's commentary on Pingala 's work contains a study of the Fibonacci sequence and Pascal's triangle , and describes the formation of a matrix. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, derivatives, the mean value theorem and the derivative of the sine function.
To what extent he anticipated the invention of calculus is a controversial subject among historians of mathematics. Madhava also found the Madhava-Gregory series to determine the arctangent, the Madhava-Newton power series to determine sine and cosine and the Taylor approximation for sine and cosine functions.
Although most Islamic texts on mathematics were written in Arabic , most of them were not written by Arabs , since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. Persians contributed to the world of Mathematics alongside Arabs.
His book On the Calculation with Hindu Numerals , written about , along with the work of Al-Kindi , were instrumental in spreading Indian mathematics and Indian numerals to the West. He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,  and he was the first to teach algebra in an elementary form and for its own sake. In Egypt, Abu Kamil extended algebra to the set of irrational numbers , accepting square roots and fourth roots as solutions and coefficients to quadratic equations. He also developed techniques used to solve three non-linear simultaneous equations with three unknown variables.
One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found solutions. Further developments in algebra were made by Al-Karaji in his treatise al-Fakhri , where he extends the methodology to incorporate integer powers and integer roots of unknown quantities.
Something close to a proof by mathematical induction appears in a book written by Al-Karaji around AD, who used it to prove the binomial theorem , Pascal's triangle , and the sum of integral cubes. Woepcke,  praised Al-Karaji for being "the first who introduced the theory of algebraic calculus.
Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a paraboloid , and was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.
In the late 11th century, Omar Khayyam wrote Discussions of the Difficulties in Euclid , a book about what he perceived as flaws in Euclid's Elements , especially the parallel postulate. He was also the first to find the general geometric solution to cubic equations.
He was also very influential in calendar reform. In the 13th century, Nasir al-Din Tusi Nasireddin made advances in spherical trigonometry. He also wrote influential work on Euclid 's parallel postulate. Kashi also had an algorithm for calculating n th roots, which was a special case of the methods given many centuries later by Ruffini and Horner. During the time of the Ottoman Empire and Safavid Empire from the 15th century, the development of Islamic mathematics became stagnant.
In the Pre-Columbian Americas , the Maya civilization that flourished in Mexico and Central America during the 1st millennium AD developed a unique tradition of mathematics that, due to its geographic isolation, was entirely independent of existing European, Egyptian, and Asian mathematics. Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians.
One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by Plato 's Timaeus and the biblical passage in the Book of Wisdom that God had ordered all things in measure, and number, and weight.
Boethius provided a place for mathematics in the curriculum in the 6th century when he coined the term quadrivium to describe the study of arithmetic, geometry, astronomy, and music. He wrote De institutione arithmetica , a free translation from the Greek of Nicomachus 's Introduction to Arithmetic ; De institutione musica , also derived from Greek sources; and a series of excerpts from Euclid 's Elements.
His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works. Europe was still using Roman numerals. There, he observed a system of arithmetic specifically algorism which due to the positional notation of Hindu—Arabic numerals was much more efficient and greatly facilitated commerce. Leonardo wrote Liber Abaci in updated in introducing the technique to Europe and beginning a long period of popularizing it.
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The book also brought to Europe what is now known as the Fibonacci sequence known to Indian mathematicians for hundreds of years before that which was used as an unremarkable example within the text. The 14th century saw the development of new mathematical concepts to investigate a wide range of problems. Thomas Bradwardine proposed that speed V increases in arithmetic proportion as the ratio of force F to resistance R increases in geometric proportion.
One of the 14th-century Oxford Calculators , William Heytesbury , lacking differential calculus and the concept of limits , proposed to measure instantaneous speed "by the path that would be described by [a body] if Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion today solved by integration , stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".
Nicole Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time. During the Renaissance , the development of mathematics and of accounting were intertwined.
There is probably no need for algebra in performing bookkeeping operations, but for complex bartering operations or the calculation of compound interest , a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.