MA 634 - Finanl
Newton's method is in essence the same as the Chinese-Arabic method named for Horner; but the great advantage of the Newtonian method is that it applies equally to equations involving transcendental functions.
True
Riemann found a function f(x) discontinuous at infinitely many points in an interval and yet the integral of f(x) exists and defines a continuous function that for the infinity of points in question fails to have a derivative.
True
The Dirichlet function does not have a Riemann integral for any interval
True
The thirteenth century presents such a striking advance over the earlier Middle Ages that it has occasionally been viewed as "the greatest of centuries." With the work of Leonardo of Pisa, Western Europe had come to rival other civilizations in the level of its mathematical achievement.
True
Infinite series played a large role in the early work of Leibniz.
True
Isadore of Miletus was one of the last directors of the Platonic Academy at Athens. The school had lasted 900 years. When Justinian became emperor in the East in 527, he was threatened by the learning at the academy and closed the schools and dispersed the scholars.
True
It is now fairly clear that Newton's discovery antedated that of Leibniz by about ten years, but that the discovery by Leibniz was independent of that of Newton.
True
With the introduction in Hindu notation, of the tenth numeral,a round goose egg for zero, the modern system of numeration for integers was completed. Although the Medieval Hindu forms of the ten numerals differ considerably from those in use today, the principles of the system were established. The new numeration, which we generally call the Hindu system, is merely a new combination of three basic principles, all of ancient origin: _____. Not one of these three was due originally to the Hindus, but it presumably is due to them that the three were first linked to form the modern system of numeration.
(1) a decimal base; (2) a positional notation; and (3) a ciphered form for each of the ten numerals
There was a 50 year gap between the work of Bolzano and Wiertrass, but there is a celebrated theorem with both names. The Bolzano-Weierstrass theorem states that: a bounded set S containing infinitely many elements (such as points or numbers)contains at least one limit point. This theorem is illustrated by the sequence of rational numbers xn = (-1)n·n/(n+1) that has no limit but has limit point(s ) ______ , these points are limit points of the set {xn}.
1 and -1
By combining the two series x /1 − x = x + x 2 + . . . and x /x − 1 = 1 + 1 /x + 1/ x 2 + . . . Euler concluded that . . . + 1/ x 2 + 1/ x + 1 + x + x 2 + . . . __________.
=0
According to Boyer,the death of Hypatia marked the close of __ as a mathematical center and the death of Boethius marked the end of ancient mathematics in the Western Roman Empire. Even though work continued for a little longer in ______due to the commentator Simplicius, there were no great original mathematicians left.
Alexandria, Athens
The theory of functions of a complex variables was developed by the leading French mathematician, ____ (1789-1857).
Augustin-Louis Cauchy
La Geometrie introduces curves plotted directly from its equation, and the author fully understood the meaning of negative coordinates.
False
The death of ___ may be taken to mark the end of ancient mathematics in the Western Roman Empire, as the death of ____ had marked the close of Alexandria as a mathematical center. Work continued for a little longer in Athens due to commentator Simplicius, but there was not a great original mathematician left.
Boethius, Hypatia
The Precious Mirror (ancient Chinese origin) opens with a diagram of the arithmetic triangle (see below),first discovered in the West by Fibonacci.
False
The Precious Mirror opens with a diagram of the arithmetic triangle, previously discoveredin the West by Pascal. In Chu's arrangement we have the coefficients of binomial expansions through the eighth powergiven in rod numerals and a round zero symbol.
False
The purpose of Riemann's quest for badly behaving functions was to demonstrate that the integral required a definition better than Cauchy's rigorous, geometrical definition for area under the curve.
False
______was an Itallian Renaissance man, a mathematician, physician, inventor, astrologer, astronomer, biologist, physicist, philosoper, writer, chemist, and a gambler. He was arguably the most influential mathematician of his time.
Cardono
Modern texts use _____ definition of infinitely small quantities in terms of limits.
Cauchy's
_____ concept of the integral as a limit of a sum, rather than from the antiderivative, is the origin of one of the intuitively appealing modern generalizations of the integral.
Cauchy's
_____ was quick to publish his achievements, and this is one reason that the chief characteristic of nineteenth-century mathematics — the introduction of rigor — is attributed to him, rather than to ______, despite the high standard of logical precision that he latter set for himself.
Cauchy; Gauss
With the introduction, in the Hindu notation, of the tenth numeral, a round goose egg for zero, the modern system of numeration for integers was completed. Although the Medieval Hindu forms of the ten numerals differ considerably from those in use today, the principles of the system were established. The new numeration, which we generally call the Hindu system, is merely a new combination of three basic principles, all of ancient origin: (1) a decimal base; (2) a positional notation; and (3) a ciphered form for each of the ten numerals. All three were due originally to the Hindus.
False
______________La geometrie was tucked into one of the appendages of his book Discours de la methode. The opening statement "Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain lines is sufficient for its construction."is noteworthy since it shows how far Cartesian geometry (now called analytic geometry) still had to go.
Descarte's
___ was willing to write with questionable logic on almost any aspect of mathematics, pure or applied, but ______favored pure mathematics in elegant form with due attention to rigorous proofs.
Euler; Cauchy
From Pythagoras to Boethius, pure mathematics consisted of algebraic proofs while applied mathematics consisted of geometric arguments. Mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260).
False
In the first edition of Principia Newton admitted that Leibniz was in possession of a method similar to his, but in the third edition of 1726following the bitter quarrel between adherents of the two men concerning the independence and priority of the discovery of the calculus, Newton deleted the reference to the calculus of Leibniz. It is now fairly clear that Newton's discovery came after that of Leibniz by about ten years and that the discovery by Leibniz was independent of that of Newton.
False
It is one of the ironies of history that the chief advantage of positional notation —its applicability to fractions — almost entirely escaped the users of the Hindu-Arabic numerals for the first thousand years of their existence. In this respect ______ was as much to blame as anyone, for he used three types of fractions — common, sexagesimal, and unit — but not ___ fractions. In the Liber abaci, in fact, the two worst of these systems — unit fractions and common fractions — are extensively used.
Fibonacci, decimal
______ was the undisputed mathematical center during the second third of the seventeenth century. The leading figures were ________ (1596- 1650) and _______ (1601-1665),
France; Rene Descartes; Pierre de Fermat
Gauss and Bolyai tried in vain to prove the parallel postulate.____came to believe that not only was no proof possible, but that a geometry quite different from that of Euclid might be developed, had he developed and published his thoughts on the parallel postulate, he would have been hailed as the inventor of non-Euclidean geometry, but his silence on the subject resulted in credit going to others.
Gauss
Adam Riese (1492-1559) was the most influential German writer in the move to replace computation by counters and Roman numerals with computation by pen using _____. Riese's writing cites the Algebra of ______ .
Hindu-Arabic numerals; al-Khowarizmi
Even though he was a Christian scientist, Philoponus argued against many of the widely held Aristotelian theories. For example, like Galileo later, he denied that the speed acquired by a freely falling body is proportional to its weightas evidenced by his statement:
If you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a very small one
Byzantine scientists preserved and continued the legacy of the great Ancient Greek mathematicians and put mathematics in practice. In early Byzantine (5th to 7th century) the architects and mathematician ____ and Anthemius of Tralles used complex mathematical formulas to construct the great Hagia Sophia church, a technological breakthrough for its time and for centuries afterwards due to its striking geometry, bold design and height.
Isidore of Miletus
Byzantine scientists preserved and continued the legacy of the Greek mathematicians, putting mathematics into practical use. From the 5th to the 7th century, early Byzantium mathematician(s) and architect(s) ________ used complex mathematical formulas to construct the Hagia Sophia. It is known for its striking geometry, bold design and height.
Isidore of Miletus and Anthemius of Tralles
The discoveries of a great mathematician, such as Newton, do not auto- matically become part of the mathematical tradition. They may be lost to the world unless other scholars understand them and take enough interest to look at them from various points of view, clarify and generalize them, and point out their implications. Newton, unfortunately, was hypersensitive and did not communicate freely, arid consequently the method of fluxions was not well known outside of England. Leibniz, on the other hand, found devoted disciples who were eager to learn about the differential and integral calculus and to transmit the knowledge to others. Foremost among the enthusiasts were --- (1654-1705) and ____(1667-1748), each as quick to offend as to feel offended.
Jacques Bernoulli; Jean Bernoulli
Simplicius and others looked to Athens for haven and established the "Athenian Academy in Exile". The date 529 may be taken as the close of European mathematical development in antiquity. After this Greek science developed in ___ for the next 600 years. According to Boyer, the spirit of mathematics languished during this time.
Near and Far Eastern countries
It may fairly be said that Euler did for the infinite analysis of ____ what ____ had done for the geometry of Eudoxus and Theaetetus, or what Vie`te had done for the algebra of ________.
Newton and Leibniz; Euclid; al-Khwarizmi and Cardan
The impossibility proof on the quintic, one of the most celebrated theorems in mathematics, was produced by ___ when he was only nineteen; yet in 1826, when he visited Legendre, Cauchy, and other mathematicians in Paris, he still had not been offered an appropriate academic post. On this visit to Paris, he wrote to a friend: "Every beginner has a great deal of difficulty in getting noticed here."
Niels Henrik Abel
___ discovered the polyhedral formula __________, where v, f, and e are the number of vertices, faces, and edges, respectively, of a simple polyhedron.
Rene Descartes, v + f = e + 2
The theory of numbers deals primarily with integers or ratios of integers, called rational numbers. ____ Real analysis deals with either rational or irrational nos.
Such numbers always are roots of ax + b = 0 with integral coefficients. Such number can always be expressed in the form m/n, where m and n are integers.
Neither Bernoulli nor Taylor was aware that both had been anticipated by Gregory in the discovery of "___."
Taylor's series
In 1837 Lejeune Dirichlet suggested a very broad definition of function: If a variable y is so related to a variable x that whenever a numerical value is assigned to x there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.
This is close to the modern view of correspondence between 2 sets.
Liber abaci (or book of the abacus)has a misleading title. It is not on the abacus; it is a very thorough treatise on algebraic methods and problems in which the use of the Hindu-Arabic numerals is strongly advocated.
True
Gottfried Wilhelm Leibniz (1646-1716) was born at Leipzig, where at fifteen he entered the university and at seventeen earned his bachelor's degree. He studied theology, law, philosophy, and mathematics at the university, and he sometimes is regarded as the last scholar to achieve universal knowledge.
True
Mathematical activity was on the rise during the middle of the fifteenth century. Europe was recovering from the Black Death, and the invention of printing with movable type made academic writing more available.
True
Heron of Alexandria is best known in the history of mathematics for the formula, bearing his name. The formula follows the geometric convention of his day, rather than the trigonometric one of today, to find the area of any triangle, with only the side lengths. To calculate the area, first you calculate the semiperimeter, that is half the sum of the sides. Then subtract each side from the semiperimeter. Find the product of the resulting differences and the semiperimeter and take the square root.
True
In the De analyst, Newton showed for the first time in the history of mathematics that an area was found through the inverse of what we call differentiation.,
True
To indicate the completely arbitrary nature of the rule of correspondence, Dirichlet proposed a very badly behaved function: when x is rational , let y = c and when x is irrational let y = d ≠ c.
There is no value of x for which this function is continuous. This came to be known as the Dirichlet function. This function is continuous nowhere.
In the influential Chinese mathematical book, "Nine chapters on the Mathematical Art", the area of a circle is found by three fourths times the diameter squared which can be simplified to 3r2. Comparing this to the modern formula for the area of a circle, we get their working value for pi which was 3.
True
The three Newtonian books that are best known today are the Principia, the Method of Fluxions, and the Opticks; there is also a fourth work which in the eighteenth century appeared in a greater number of editions than did the other three, and it, too, contained valuable contributions. This was the Arithmetica universalis, a work composed between 1673 and 1683, perhaps for Newton's lectures at Cambridge, and first published in 1707. This influential treatise contains the formulas, usually known as "Newton's identities," for the sums of the powers of the roots of a polynomial equation.
True
When Latin translations of Al-Khowarizmi's work appeared in Europe, careless readers began to attribute not only the book, but also the numeration, to the author. A loose pronunciation of his name came to be known as ___. The title of his most important book, Al-jabr, led to the word ___.
algorithm, algebra
Euler took the differential calculus and the method of fluxions and made them part of a more general branch of mathematics that ever since has been known as ___—the study of infinite processes.
analysis
The Fibonacci sequence has been found to have many beautiful and significant properties. For instance, it can be proved that ____.
any two successive terms are relatively prime
The Fibonacci sequence has been found to have many beautiful and significant properties. For instance, it can be proved that ____.
any two successive terms are relatively prime; As we go further out in the sequence, the ratios of adjacent terms begins to approach a fixed limiting value of 1.618034 . . .
The Summa of Pacioli, completed in 1487, was material in four fields : ____________ and double-entry bookkeeping.
arithmetic, algebra, Euclidean geometry
From Pythagoras to Boethius, pure mathematics consisted of ______ while applied mathematics consisted of _____. Mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260).
artithmetic and geometry; music and astronomy
Fibonacci's treatment of the cubic equation x3 + 2x2 + lOx = 20 was similar to a modern approach in that he showed the impossibility of a root ______. This meant that he could not solvethe equation exactly by algebraic means. Although we do not know how he did it, he was able to find an approximation for the positive root as a _______.This was the most accurate European approximation of an irrational number as the root of an algebraic equation up to that time.
as a ratio of integers; sexagesimal fraction
On July 10, 1796, Gauss confided to his diary the discovery that every integer is the sum of ______.
at most three triangular numbers
Newton stated that he had discovered ___________. It is difficult to see why its discovery took so long.Cardan and Pascal were aware of it partially but since they did not make use of the exponential notation of Descartes, they did not make the relatively simple transition from an integral power to a fractional one. Stevin and Girard had suggested fractional powers, but did not really use them. Hence it was only with Wallis that fractional exponents came into common use. It remained for Newton to supply the needed mathematical details as part of his method of infinite series.
binomial theorem
In 1676 Leibniz again visited London, bringing with him his ___.
calculating machine
Fibonacci's book, Practica geometriae,
contained a proof that the medians of a triangle divide each other in the ratio 2 to 1; continued a Babylonian and Arabic tendency to use algebra to solve geometrical problems; contained a 3D analogue of the Pythagorean theorem
For purposes of the history of mathematics, our textbook author considers the beginning of the Middle ages as year 529,roughly the ______, and the end at the year 1436, roughlythe ______.
death of Boethius, death of Joan of Arc (Hyaptia)
For purposes of the history of mathematics, our textbook author considers the beginning of the Middle ages as year 529,roughly the ______, and the end at the year 1436, roughlythe ______.
death of Boethius, death of al-Kashi
Euler was, without any doubt, the individual most responsible for methods used today in introductory college courses in the solution of ______, and even many of the specific problems appearing in current textbooks can be traced back to the great treatises.
differential equations
Leibniz always had a keen appreciation of the importance of good notations as a help to thought, and his choice in the case of the calculus was especially happy. After some trial and error he fixed on ___ for the smallest possible differences (differentials) in x and y,
dx and dy
The three symbols e, π, , and i, for which Euler was in large measure responsible, can be combined with the two most important integers, 0 and 1, in the celebrated equality _________which contains the five most significant numbers (as well as the most important relation and the most important operation) in all of mathematics. The equivalent of this equality, in generalized form, had been included by Euler in 1748 in his best-known textbook, Introductio inAnalysin Infinitorum.
e i x + 1 = 0
Johann Widman (born ca. 1460), published a commercial arithmetic, Rechenung auff alien Kauffmanschqffi, the oldest book in which our familiar + and — signs appear in print. They were used to indicate ______, they later became symbols of the familiar arithmetic operations.
excess and deficiency in warehouse measures
Around A.D. 265, Chinese mathematician Liu Huicreated a simple polygon-based iterative algorithm forfinding the value of pi, which gave four accurate digits.Later, around A.D. 480, Zu Chongzhi adopted Liu Hui'smethod and achieved seven digits of accuracy.This record may have held for 800 years. The method resembled the method of _____ made rigorous by ___ circa 350 BC
exhaustion ,Eudoxus
From Euler's time onward, the idea of ______became fundamental in analysis.
function
The Chou Pei, a Chinese astronomy text, indicates that in China, as in Egypt, _____arose from measuring. Although the Chou Pei did feature an algebraic treatment of the ___.
geometry, "Pythagorean" Theorem
Riemann was a many-sided mathematician with a fertile mind, contributing not only to geometry and the theory of numbers, but also to analysis. In analysis, _____ .
he created the first rigorous definition of an integral on an interval
Today, Diophantine analysis is the area of study where _____ solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable.
integer (whole number)
Diophantus is sometimes is called "the father of algebra," but this title more appropriately belongs to al-Khowarizmi. The algebra of al-Khowarizmi is thoroughly rhetorical, this means ___
it was written out in words rather than symbols.
Simplicius and others looked to Persia for haven and established "Athenian Academy in Exile" The date 529 may be taken as the close of European mathematical development in antiquity. After this Greek science developed in Near and Far Eastern countries for the next 600 years. The spirit of mathematics ____.
languished
Oresme had argued that everything measurable can be represented by a __________; and a mathematics of _________, both from a theoretical and a practical standpoint, flourished during the early Renaissance period.
line, mensuration
Robert Bacon is cited in our text for the statement,"Neglect of ____ works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of this world."
mathematics
The Chou Pei, a Chinese astronomy text, indicates thatin China, as in Egypt, geometry arose from _______; and, as in Babylonia, Chinese geometry was essentially only an exercise in ______. There seem to be some indications in the Chou Pei of the Pythagorean theorem, a theorem treated _______ by the Chinese.
measuring magnitudes, lengths, areas, and volumes; arithmetic or algebra; algebraically
The nineteenth century is known as the Golden Age in mathematics, it saw the addition of ______ to the subject during these one hundred years.
n-dimensional spaces, non-Euclidean geometries, noncommutative algebras, infinite processes
Diophantus considered ____ or _____ solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation 4 = 4x + 20 'absurd' because it would lead to a negative value for x. One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. Diophantus would consider which of the following equations absurd?
negative or irrational roots
The idea of ___ numbers seems not to have causedmuch difficulty for the Chinese since they were accustomed to calculating with two sets of rods — a red set for ____ and a black set for _____. Nevertheless, they did not accept the notion that a(n) ____ number might be a solution of an equation.
negative, positive coefficients, negative coeficients, negative
Lambert in 1770 and Legendre in 1794 had shown pi and pi squared are irrational , but this proof had not put an end to the age old question of the squaring of the circle. The matter was finally put to rest in 1882 in a paper where Lindemann showed that pi is also a transcendental number. Lindemann in his proof first demonstrated that the equation eix+1=0 cannot be satisfied if x is algebraic. Inasmuch as Euler had shown that the value x = pi does satisfy the equation, it must follow that pi is ___. Here, finally , was the answer to the classical problem of the quadrature of the circle. In order for the quadrature of the circle to be possible with Euclidean tools, the number pi would have to be the root of an algebraic equation with a root expressible in square roots. Since pi is not algebraic, the circle cannot be squared according to the classical rules.
not-algebraic
Gauss described mathematics as the queen of the sciences and _____________as the queen of mathematics ; but Gauss in the early nineteenth century virtually abandoned his Queen of Mathematics, his attention distracted by too many other subjects. Later in life he expressed regret that he had abandoned his first love.
number theory
The civilizations of China and India are _____ than those of Greece and Rome, ____ than those in the Nile and Mesopotamian valleys.
older, although not older
An interest in patterns led the author of the Nine Chapters to solve the system of simultaneous linear equations 3x + 2y + z = 39 2x + 3y + z = 34 x + 2y + 3z = 26 by ___.
performing column operations on the matrix
Both Euler and d'Alembert wrote problems of life expectancy, the value of an annuity, lotteries, and other aspects of social science. ____, after all, had been among the chief interests of Euler's friends Daniel and Nicolaus Bernoulli.
probability
Cardano's Book on Games of Chance was the first systematic treatment of _____.
probability
Dedekind saw that the domain of rational numbers can be extended to form a continuum of real numbers if one assumes what now is known as the Cantor-Dedekind axiom-that the points on a line can be put into one to one correspondence with the real numbers. Arithmetically expressed this means that for every division of the rational numbers into two classes A and B such that every number of the first class, A, is less than every number of the second class, B, there is one and only one real number producing this Schnitt, or Dedekind cut. If A has a largest number, or if B contains a smallest number, the cut defines a(n) ___ ; but if A has no largest number and B no smallest, then the cut defines a(n) ___. If, for example, we put in A all negative rational numbers and also all positive rational numbers whose squared are less than two, and in B all positive rational numbers whose squares are more than two, we have subdivided the entire field of rational numbers - defining an irrational number - in this case square root of 2. Now, Dedekind pointed out, the fundamental theorems on limits can be proved rigorously without geometry.
rational number, irrational number
Without doubt the problem in the Liber abaci that has most inspired future mathematicians was the following : How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on? This celebrated problem gives rise to the "Fibonacci sequence" 1, 1, 2, 3, 5, 8, 13, 21, . . . , u„, . . . which can be represented generally as, ___
s n = s n-1 + s n-2
Early in the 20th century, Bertrand Russell found that since either of Dedekind's two classes A and B is uniquely determined by the other, just one is sufficient to determine a real number. Thus ___.
square root of 2 can be defined as that segment of the rationals made up of all positive rational numbers whose squares are less than 2 and also of all negative rational numbers every real number is nothing more than a segment of the rational number system
As he studied infinitesimal analysis, he said that a light burst upon him when he realized that the determination of the __ to a curve depended on the ratio of the differences in the y coordinates and the x coordinates, as these became infinitely small, and that ___ depended on the sum of infinitely thin rectangles making up the area.
tangents, quadratures
For Leibniz, finding ____ called for the use of the calculus differentialis, and finding ____required the calculus summatorius or the calculus integralis; from these phrases arose our words "differential calculus" and "integral calculus."
tangents; quadratures
Heron of Alexandria is best known in the history of mathematics for the formula, bearing his name. The formula follows the geometric convention of his day, rather than the trigonometric one of today, to find the area of any triangle, with only the side lengths. To calculate the area, first you calculate the semiperimeter, ____. Then subtract each side length from the ____. Find the product of the resulting differences and the semiperimeter and take the square root.
that is half the sum of the sides, semiperimeter
According to Boyer, zero may have started in ____and transferred to India after the ______ had been established there.
the Greek world in Alexandria, decimal positional system
Euler's imaginative treatment of series led him to some striking relationships between analysis and the theory of numbers.
true
It is one of the ironies of history that the chief advantage of positional notation —its applicability to fractions — almost entirely escaped the users of the Hindu-Arabic numerals for the first thousand years of their existence. In this respect Fibonacci was as much to blame as anyone, for he used three types of fractions — common, sexagesimal, and unit — but not decimal fractions. In the Liber abaci, in fact, the two worst of these systems — ___ — are extensively used.
unit fractions and common fractions
Lambert, like Saccheri, tried to prove the parallel postulate but was aware of his lack of success. He wrote, Proofs of the Euclidean postulate can be developed to such an extent that apparently a mere trifle remains. But a careful analysis shows that in this seeming trifle lies the crux of the matter; ____ No one else came so close to the truth without actually discovering non-Euclidean geometry.
usually it contains either the proposition that is being proved or a postulate equivalent to it.
Isadore of Miletus was one of the last directors of the Platonic Academy at Athens. The school had lasted 900 years. When Justinian became emperor in the East in 527, he ___.
was threatened by the learning at the academy and closed the schools and dispersed the scholars
Leonardo of Pisa (1180-1250), better known as Fibonacci, wrote Liber abaci
when he was 20 years old, problems which advocated the use of Hindu-Arabic numerals, not a book about the abacus, a thorough treatis on algebraic methods
The most striking aspect of Riemann's work was a strongly intuitive and geometrical background in analysis. This contrasts sharply with the arithmetizing tendencies of the Weierstrass. His approach was a method of discovery whereas Wierstrass was a method of demonstration. It was Riemann's intuitive genius that produced concepts such as curvature of a Riemannian space or manifold, ____ .
without which the theory of general relativity could not have been formulated.
The strictly analytic treatment of the trigonometric functions was established by The Introductio.The sine, for example, was no longer a line segment; it was simply a number or a ratio—the ordinate of a point on a unit circle, or the number defined by the series _____for some value of z.
z − z 3/ 3 ! + z 5/ 5 ! − . . .
The reason why there were three cases ( ax2 + bx = c, ax2 = bx + c, and ax2 + c = bx) to Diophantus' analysis, while today we have only one case, is that he did not have any notion for ____ and he avoided negative coefficients by considering the given numbers a, b, c to all be positive in each of the three cases above.
zero