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The history of mathematical notation  includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation's move to popularity or inconspicuousness.
Mathematical notation  comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators. The development of mathematical notation can be divided in stages. From ancient times through the post-classical age, [note 1] bursts of mathematical creativity were often followed by centuries of stagnation. As the early modern age opened and the worldwide spread of knowledge began, written examples of mathematical developments came to light.
The " symbolic " stage is where comprehensive systems of notation supersede rhetoric. Beginning in Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day.
This symbolic system was in use by medieval Indian mathematicians and in Europe since the middle of the 17th century,  and has continued to develop in the contemporary era.
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, the focus here, the investigation into the mathematical methods and notation of the past. Although the history commences with that of the Ionian schools , there is no doubt that those Ancient Greeks who paid attention to it were largely indebted to the previous investigations of the Ancient Egyptians and Ancient Phoenicians.
Numerical notation's distinctive feature, i. Our knowledge of the mathematical attainments of these early peoples, to which this section is devoted, is imperfect and the following brief notes be regarded as a summary of the conclusions which seem most probable, and the history of mathematics begins with the symbolic sections. Many areas of mathematics began with the study of real world problems , before the underlying rules and concepts were identified and defined as abstract structures.
For example, geometry has its origins in the calculation of distances and areas in the real world; algebra started with methods of solving problems in arithmetic. There can be no doubt that most early peoples which have left records knew something of numeration and mechanics , and that a few were also acquainted with the elements of land-surveying. In particular, the Egyptians paid attention to geometry and numbers, and the Phoenicians to practical arithmetic, book-keeping , navigation , and land-surveying.
The results attained by these people seem to have been accessible , under certain conditions, to travelers. It is probable that the knowledge of the Egyptians and Phoenicians was largely the result of observation and measurement , and represented the accumulated experience of many ages.
Written mathematics began with numbers expressed as tally marks , with each tally representing a single unit. The numerical symbols consisted probably of strokes or notches cut in wood or stone, and intelligible alike to all nations. The peoples with whom the Greeks of Asia Minor amongst whom notation in western history begins were likely to have come into frequent contact were those inhabiting the eastern littoral of the Mediterranean: and Greek tradition uniformly assigned the special development of geometry to the Egyptians, and that of the science of numbers [note 3] either to the Egyptians or to the Phoenicians.
The Ancient Egyptians had a symbolic notation which was the numeration by Hieroglyphics. Smaller digits were placed on the left of the number, as they are in Hindu—Arabic numerals. Later, the Egyptians used hieratic instead of hieroglyphic script to show numbers.
Hieratic was more like cursive and replaced several groups of symbols with individual ones. For example, the four vertical lines used to represent four were replaced by a single horizontal line. This is found in the Rhind Mathematical Papyrus c. The system the Egyptians used was discovered and modified by many other civilizations in the Mediterranean.
The Egyptians also had symbols for basic operations: legs going forward represented addition, and legs walking backward to represent subtraction. The Mesopotamians had symbols for each power of ten. Instead of having symbols for each power of ten, they would just put the coefficient of that number. Each digit was separated by only a space, but by the time of Alexander the Great , they had created a symbol that represented zero and was a placeholder.
The Mesopotamians also used a sexagesimal system, that is base sixty. It is this system that is used in modern times when measuring time and angles. Babylonian mathematics is derived from more than clay tablets unearthed since the s.
Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and the system of metrology from BC. From around BC onwards, 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. The majority of Mesopotamian clay tablets date from to BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of regular , reciprocal and pairs. Babylonian mathematics were written using a sexagesimal base numeral system. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors: the reciprocal of any integer which is a multiple of divisors of 60 has a finite expansion in base In decimal arithmetic, only reciprocals of multiples of 2 and 5 have finite decimal expansions.
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. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context. The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks, but the subsequent history may be divided into periods, the distinctions between which are tolerably well marked.
Greek mathematics, which originated with the study of geometry, tended from its commencement to be deductive and scientific. Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem , a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse the side opposite the right angle is equal to the sum of the areas of the squares of the other two sides.
Plato 's influence has been especially strong in mathematics and the sciences. He helped to distinguish between pure and applied mathematics by widening the gap between "arithmetic", now called number theory and "logistic", now called arithmetic. Greek mathematics greatly refined the methods especially through the introduction of deductive reasoning and mathematical rigor in proofs and expanded the subject matter of mathematics.
Abstract Mathematics  is what treats of magnitude [note 6] or quantity , absolutely and generally conferred, without regard to any species of particular magnitude, such as arithmetic and geometry , In this sense, abstract mathematics is opposed to mixed mathematics , wherein simple and abstract properties, and the relations of quantities primitively considered in mathematics, are applied to sensible objects, and by that means become intermixed with physical considerations, such as in hydrostatics , optics , and navigation.
Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time.
In the historical development of geometry, the steps in the abstraction of geometry were made by the ancient Greeks. Euclid's Elements being the earliest extant documentation of the axioms of plane geometry— though Proclus tells of an earlier axiomatisation by Hippocrates of Chios. Euclid's first theorem is a lemma that possesses properties of prime numbers. The influential thirteen books cover Euclidean geometry, geometric algebra, and the ancient Greek version of algebraic systems and elementary number theory.
It was ubiquitous in the Quadrivium and is instrumental in the development of logic, mathematics, and science. Diophantus of Alexandria was author of a series of books called Arithmetica , many of which are now lost. These texts deal with solving algebraic equations. 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. The Greeks employed Attic numeration ,  which was based on the system of the Egyptians and was later adapted and used by the Romans.
Greek numerals one through four were vertical lines, as in the hieroglyphics. Numbers six through nine were pente with vertical lines next to it. The Ionian numeration used their entire alphabet including three archaic letters. The numeral notation of the Greeks, though far less convenient than that now in use, was formed on a perfectly regular and scientific plan,  and could be used with tolerable effect as an instrument of calculation, to which purpose the Roman system was totally inapplicable.
The Greeks divided the twenty-four letters of their alphabet into three classes, and, by adding another symbol to each class, they had characters to represent the units, tens, and hundreds. When lowercase letters became differentiated from upper case letters, the lower case letters were used as the symbols for notation. Greek mathematical reasoning was almost entirely geometric albeit often used to reason about non-geometric subjects such as number theory , and hence the Greeks had no interest in algebraic symbols.
The great exception was Diophantus of Alexandria , the great algebraist. It was not completely symbolic, but was much more so than previous books.
An unknown number was called s. The Chinese used numerals that look much like the tally system. Five was an X between two horizontal lines; it looked almost exactly the same as the Roman numeral for ten.
In the history of the Chinese, there were those who were familiar with the sciences of arithmetic, geometry, mechanics, optics, navigation, and astronomy.
Mathematics in China emerged independently by the 11th century BC. Chinese of that time had made attempts to classify or extend the rules of arithmetic or geometry which they knew, and to explain the causes of the phenomena with which they were acquainted beforehand. The Chinese independently developed very large and negative numbers , decimals , a place value decimal system, a binary system , algebra , geometry , and trigonometry.
Chinese mathematics made early contributions, including a place value system. In arithmetic their knowledge seems to have been confined to the art of calculation by means of the swan-pan , and the power of expressing the results in writing.
Our knowledge of the early attainments of the Chinese, slight though it is, is more complete than in the case of most of their contemporaries. It is thus instructive, and serves to illustrate the fact, that it can be known a nation may possess considerable skill in the applied arts with but our knowledge of the later mathematics on which those arts are founded can be scarce.
Knowledge of Chinese mathematics before BC is somewhat fragmentary, and even after this date the manuscript traditions are obscure. Dates centuries before the classical period are generally considered conjectural by Chinese scholars unless accompanied by verified archaeological evidence. As in other early societies the focus was on astronomy in order to perfect the agricultural calendar , and other practical tasks, and not on establishing formal systems.
The Chinese Board of Mathematics duties were confined to the annual preparation of an almanac, the dates and predictions in which it regulated.
Ancient Chinese mathematicians did not develop an axiomatic approach, but made advances in algorithm development and algebra. The achievement of Chinese algebra reached its zenith in the 13th century, when Zhu Shijie invented method of four unknowns. As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and that of the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when The Nine Chapters on the Mathematical Art reached its final form, while the Writings on Reckoning and Huainanzi are roughly contemporary with classical Greek mathematics.
Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory.
The Pythagorean theorem for example, has been attested to the time of the Duke of Zhou. Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal ,  such as by Shen Kuo. The state of trigonometry in China slowly began to change and advance during the Song Dynasty — , where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendarical science and astronomical calculations.
Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy. Although the origin of our present system of numerical notation is ancient, there is no doubt that it was in use among the Hindus over two thousand years ago.
Maria D. I present some few results of my study on the mathematical content of the published part f. The other part f. Hungerand K. Vogel in This 15 th century Byzantine MS includes the solution of problems of pr ac ticalarithmetic, and algebra, the roots of which can be tr ac ed b ac k to antiquity and their comparison with modernmathematical methods and terminology  reveals -apart from some differences- many identities and similaritiesshowing the unbroken continuity of mathematical tradition through the centuries.
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra in fact, every proof must use the completeness of the real numbers , which is not an algebraic property. This article describes the history of the theory of equations, called here "algebra", from the origins to the emergence of algebra as a separate area of mathematics. The treatise provided for the systematic solution of linear and quadratic equations. According to one history, "[i]t is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the previous translation. The word 'al-jabr' presumably meant something like 'restoration' or 'completion' and seems to refer to the transposition of subtracted terms to the other side of an equation; the word 'muqabalah' is said to refer to 'reduction' or 'balancing'—that is, the cancellation of like terms on opposite sides of the equation.
Enter a word or phrase in the dialogue box, e. Culturally, our discomfort with the concepts of zero and infinite is reflected in such humor as 2 plus 0 still equals 2, even for large values, and popular retorts of similar tone. A like uneasiness occurs in confronting infinity, whose proper use first rests on a careful definition of what is finite. Are we mortals hesitant to admit to our finite nature? Such lighthearted commentary reflects an underlying awkwardness in the manipulation of mathematical expressions where the notions of zero and infinity present themselves.
There are several closely related results that are variously known as the binomial theorem depending on the source. Even more confusingly a number of these and other related results are variously known as the binomial formula, binomial expansion, and binomial identity , and the identity itself is sometimes simply called the " binomial series " rather than "binomial theorem. The most general case of the binomial theorem is the binomial series identity. This series converges for an integer, or. This general form is what Graham et al.
After a brief chapter on three ancient mathematicians, the remainder of the book is devoted to the lives of about forty mathematicians who worked in the seventeenth, eighteenth and nineteenth centuries. The emphasis is on mainstream mathematics following on from the work. To keep the interest of readers, the book typically focuses on unusual or dramatic aspects of its subjects' lives.
The history of mathematical notation  includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation's move to popularity or inconspicuousness. Mathematical notation  comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators.
How many elephants are in the same room as you right now? Most people would answer zero to that question if you answered something else, we should be friends. The concept of zero is familiar to us. Earlier today, my two-year-old cousin told me that his baby sister is zero years old.
Он не мог поверить в свою необыкновенную удачу. Он снова говорил с этим американцем, и если все прошло, как было задумано, то Танкадо сейчас уже нет в живых, а ключ, который он носил с собой, изъят. В том, что он, Нуматака, в конце концов решил приобрести ключ Энсея Танкадо, крылась определенная ирония. Токуген Нуматака познакомился с Танкадо много лет. Молодой программист приходил когда-то в Нуматек, тогда он только что окончил колледж и искал работу, но Нуматака ему отказал. В том, что этот парень был блестящим программистом, сомнений не возникало, но другие обстоятельства тогда казались более важными. Хотя Япония переживала глубокие перемены, Нуматака оставался человеком старой закалки и жил в соответствии с кодексом менбоко - честь и репутация.
Затем он одним движением швырнул ее на пол возле своего терминала. Сьюзан упала на спину, юбка ее задралась. Верхняя пуговица блузки расстегнулась, и в синеватом свете экрана было видно, как тяжело вздымается ее грудь. Она в ужасе смотрела, как он придавливает ее к полу, стараясь разобрать выражение его глаз. Похоже, в них угадывался страх. Или это ненависть. Они буквально пожирали ее тело.
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A history of mathematics / Carl B. Boyer and Uta Merzbach. 3rd ed. p. cm. Stephen Power, the senior editor, was unfailingly generous and diplo- matic in his.Somer G. 18.03.2021 at 08:38
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"Boyer and Merzbach distill thousands of years of mathematics into this fascinating chronicle. From the Greeks to Gödel, the mathematics is brilliant; the cast of.