34. Translate into English.
Алгебра – це точна, стисла та універсальна наука.
Самі слова використовують у їх символічному змісті.
У середньовічній Європі “мінус” та “плюс” позначалися повними словами.
Скорочення перетворилося у символ.
В своєму розвитку алгебра пройшла декілька ступенів.
Сучасна алгебра об’єднує велику кількість самостійних дисциплін.
Метод аналізу математичних моделей посідає провідне місце серед інших методів дослідження.
П'єр Ферма
(1601-1665)
П
'єр
Ферма − видатний французький
математик, один із основоположників
аналітичної геометрії і теорії чисел.
Він є автором робіт
в області теорії ймовірності,
оптики, численних
нескінченно-малих величин.
У 1637 році П'єр Ферма сформулював так звану Велику теорему Ферма, яка була доведена американським математиком Ендрю Уайлсом лише у 1995 році. Теорема стверджує, що для будь-якого натурального n>2 i xyz<>0 рівняння хn+уn=zn не можна розв’язати в цілих (і раціональних) числах.
Pierre Fermat - the famous French mathematician, one of the founders of analytical geometry and number theory. He is the author of works in the field of probability theory, optics, multiple-infinitelysmall quantities.
In 1637 Pierre Fermat formulated the so-called Great Fermat theorem which was proved by the American mathematician Andrew Wiles in 1995 only. The theorem states that for any natural number n> 2 i xyz <> 0 equation hn + un = zn is impossible to resolve in whole (and rational) numbers.
SPEAKING
35. A. Discuss the following questions in small groups.
What counting systems do you know?
Analyse the advantages of the 12 system (used in UK and US) over the decimal system and vice versa.
What problems must the Romans have had with their system?
What are the specialized uses of Roman numerals today?
Could we manage just cardinal numbers, rather than having both cardinal and ordinal?
B. Prepare a short talk for your classmates. Choose the topic from the given: "Maya numerals", "Babylonian numerals", "Quipus".
36. A. Look at the portraits of famous mathematicians, name them and tell about their contribution to science.
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a) Augusta Ada King Byron, Countess of Lovelace
Augusta Ada King, Countess of Lovelace (10 December 1815 – 27 November 1852), born Augusta Ada Byron, was an English writer chiefly known for her work on Charles Babbage's early mechanical general-purpose computer, the analytical engine. Her notes on the engine include what is recognised as the first algorithm intended to be processed by a machine; thanks to this, she is sometimes considered the "World's First Computer Programmer".
She was the only legitimate child of the poet Lord Byron (with Anne Isabella Milbanke). She had no relationship with her father, who died when she was nine. As a young adult, she took an interest in mathematics, and in particular Babbage's work on the analytical engine. Between 1842 and 1843, she translated an article by Italian mathematician Luigi Menabrea on the engine, which she supplemented with a set of notes of her own. These notes contain what is considered the first computer program—that is, an algorithm encoded for processing by a machine. Though Babbage's engine has never been built, Lovelace's notes are important in the early history of computers. She also foresaw the capability of computers to go beyond mere calculating or number-crunching while others, including Babbage himself, focused only on these capabilities.
b) Sofya Vasilyevna Kovalevskaya, (born January 15, 1850, Moscow, Russia—died February 10, 1891, Stockholm, Sweden), mathematician and writer who made a valuable contribution to the theory of partial differential equations. She was the first woman in modern Europe to gain a doctorate in mathematics, the first to join the editorial board of a scientific journal, and the first to be appointed professor of mathematics.
In 1868 Kovalevskaya entered into a marriage of convenience with a young paleontologist, Vladimir Kovalevsky, in order to leave Russia and continue her studies. The pair traveled together to Austria and then to Germany, where in 1869 she studied at the University of Heidelberg under the mathematicians Leo Königsberger and Paul du Bois-Reymond and the physicist Hermann von Helmholtz. The following year she moved to Berlin, where, having been refused admission to the university on account of her gender, she studied privately with the mathematician Karl Weierstrass. In 1874 she presented three papers—on partial differential equations, on Saturn’s rings, and on elliptic integrals—to the University of Göttingen as her doctoral dissertation and was awarded the degree, summa cum laude, in absentia. Her paper on partial differential equations, the most important of the three papers, won her valuable recognition within the European mathematical community. It contains what is now commonly known as the Cauchy-Kovalevskaya theorem, which gives conditions for the existence of solutions to a certain class of partial differential equations. Having gained her degree, she returned to Russia, where her daughter was born in 1878. She separated permanently from her husband in 1881.
In 1883 Kovalevskaya accepted Magnus Mittag-Leffler’s invitation to become a lecturer in mathematics at the University of Stockholm. She was promoted to full professor in 1889. In 1884 she joined the editorial board of the mathematical journal Acta Mathematica, and in 1888 she became the first woman to be elected a corresponding member of the Russian Academy of Sciences. In 1888 she was awarded the Prix Bordin of the French Academy of Sciencesfor a paper on the rotation of a solid body around a fixed point.
Kovalevskaya also gained a reputation as a writer, an advocate of women’s rights, and a champion of radical political causes. She composed novels, plays, and essays, including the autobiographical Memories of Childhood (1890) and The Nihilist Woman (1892), a depiction of her life in Russia.
c) Euclid of Megara & Alexandria (ca 322-275 BC) Greece/Egypt
Euclid may have been a student of Aristotle. He founded the school of mathematics at the great university of Alexandria. He was the first to prove that there are infinitely many prime numbers; he stated and proved the unique factorization theorem; and he devised Euclid's algorithm for computing gcd. He introduced the Mersenne primes and observed that (M2+M)/2 is always perfect (in the sense of Pythagoras) if M is Mersenne. (The converse, that any even perfect number has such a corresponding Mersenne prime, was tackled by Alhazen and proven by Euler.) He proved that there are only five "Platonic solids," as well as theorems of geometry far too numerous to summarize; among many with special historical interest is the proof that rigid-compass constructions can be implemented with collapsing-compass constructions. Although notions of trigonometry were not in use, Euclid's theorems include some closely related to the Laws of Sines and Cosines. Among several books attributed to Euclid are The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory of mirrors), a book on spherical geometry, a book on logic fallacies, and his comprehensive math textbook The Elements. Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. Apparently Desargues' Homology Theorem (a pair of triangles is coaxial if and only if it is copolar) was proved in one of these lost works; this is the fundamental theorem which initiated the study of projective geometry. Euclid ranks #14 on Michael Hart's famous list of the Most Influential Persons in History. The Elements introduced the notions of axiom and theorem; was used as a textbook for 2000 years; and in fact is still the basis for high school geometry, making Euclid the leading mathematics teacher of all time. Some think his best inspiration was recognizing that the Parallel Postulate must be an axiom rather than a theorem.
There are many famous quotations about Euclid and his books. Abraham Lincoln abandoned his law studies when he didn't know what "demonstrate" meant and "went home to my father's house [to read Euclid], and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies."
d) J.-L. Lagrange
Joseph-Louis Lagrange (born Giuseppe Lodovico Lagrangia) was a brilliant man who advanced to become a teen-age Professor shortly after first studying mathematics. He excelled in all fields of analysis and number theory; he made key contributions to the theories of determinants, continued fractions, and many other fields. He developed partial differential equations far beyond those of D. Bernoulli and d'Alembert, developed the calculus of variations far beyond that of the Bernoullis, and developed terminology and notation (e.g. the use of f'(x) and f''(x) for a function's 1st and 2nd derivatives). He proved a fundamental Theorem of Group Theory. He laid the foundations for the theory of polynomial equations which Cauchy, Abel, Galois and Poincaré would later complete. Number theory was almost just a diversion for Lagrange, whose focus was analysis; nevertheless he was the master of that field as well, proving difficult and historic theorems including Wilson's theorem (pdivides (p-1)! + 1 when p is prime); Lagrange's Four-Square Theorem (every positive integer is the sum of four squares); and that n·x2 + 1 = y2 has solutions for every positive non-square integer n.
Lagrange's many contributions to physics include understanding of vibrations (he found an error in Newton's work and published the definitive treatise on sound), celestial mechanics (including an explanation of why the Moon keeps the same face pointed towards the Earth), the Principle of Least Action (which Hamilton compared to poetry), and the discovery of the Lagrangian points (e.g., in Jupiter's orbit). Lagrange's textbooks were noted for clarity and inspired most of the 19th-century mathematicians on this list. Unlike Newton, who used calculus to derive his results but then worked backwards to create geometric proofs for publication, Lagrange relied only on analysis. "No diagrams will be found in this work" he wrote in the preface to his masterpiece Mécanique analytique.
Lagrange once wrote "As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection." Both W.W.R. Ball and E.T. Bell, renowned mathematical historians, bypass Euler to name Lagrange as "the Greatest Mathematician of the 18th Century." Jacobi bypassed Newton and Gauss to call Lagrange "perhaps the greatest mathematical genius since Archimedes."
e) G.W. Leibniz
Gottfried Wilhelm von Leibniz (1646-1716) Germany
Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his co-invention of the infinitesimal calculus) was immense. His childhood IQ has been estimated as second-highest in all of history, behind only Goethe. Descriptions which have been applied to Leibniz include "one of the two greatest universal geniuses" (da Vinci was the other); "the most important logician between Aristotle and Boole;" and the "Father of Applied Science." Leibniz described himself as "the most teachable of mortals."
Mathematics was just a self-taught sideline for Leibniz, who was a philosopher, lawyer, historian, diplomat and renowned inventor. Because he "wasted his youth" before learning mathematics, he probably ranked behind the Bernoullis as well as Newton in pure mathematical talent, and thus he may be the only mathematician among the Top Ten who was never the greatest living algorist or theorem prover. We won't try to summarize Leibniz' contributions to philosophy and diverse other fields including biology; as just three examples: he predicted the Earth's molten core, introduced the notion of subconscious mind, and built the first calculator that could do multiplication. (And his political influence may have been huge: he was a special consultant to both the Holy Roman and Russian Emperors, and was helped arrange for the son of his patron Sophia Wittelsbach, only distantly in line for the British throne, to be crowned King George I of England.)
Leibniz pioneered the common discourse of mathematics, including its continuous, discrete, and symbolic aspects. (His ideas on symbolic logic weren't pursued and it was left to Boole to reinvent this almost two centuries later.) Mathematical innovations attributed to Leibniz include the symbols ∫, df(x)/dx; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. He invented more mathematical terms than anyone, including "function," "analysis situ," "variable," "abscissa," "parameter," and "coordinate." His works seem to anticipate cybernetics and information theory; and Mandelbrot acknowledged Leibniz' anticipation of self-similarity. Like Newton, Leibniz discovered The Fundamental Theorem of Calculus; his contribution to calculus was much more influential than Newton's, and his superior notation is used to this day. As Leibniz himself pointed out, since the concept of mathematical analysis was already known to ancient Greeks, the revolutionary invention was notation("calculus"), because with "symbols [which] express the exact nature of a thing briefly ... the labor of thought is wonderfully diminished."
Leibniz' thoughts on mathematical physics had some influence. He developed laws of motion that gave different insights from those of Newton. His cosmology was opposed to that of Newton but, anticipating theories of Mach and Einstein, is more in accord with modern physics. Mathematical physicists influenced by Leibniz include not only Mach, but perhaps Hamilton and Poincaré themselves.
Although others found it independently (including perhaps Madhava three centuries earlier), Leibniz discovered and proved a striking identity for π: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
f) James Clerk Maxwell (1831-1879) Scotland
Maxwell published a remarkable paper on the construction of novel ovals, at the age of 14; his genius was soon renowned throughout Scotland, with the future Lord Kelvin remarking that Maxwell's "lively imagination started so many hares that before he had run one down he was off on another." He did a comprehensive analysis of Saturn's rings, developed the kinetic theory of gases, explored knot theory, and more. He advanced the theory of color, and produced the first color photograph. Maxwell was also a poet. One Professor said of him, "there is scarcely a single topic that he touched upon, which he did not change almost beyond recognition."
Maxwell did little work in pure mathematics, so his great creativity in mathematical physics might not seem enough to qualify him for this list. However, in 1864 James Clerk Maxwell stunned the world by publishing the equations of electricity and magnetism and showing that light itself is linked to the electro-magnetic force. This, along with Darwin's theory of evolution, is considered one of the greatest discoveries of the 19th century; and Maxwell himself, along with Newton and Einstein, is frequently named as one of the three greatest physicists ever. He ranks #24 on Hart's list of the Most Influential Persons in History.
B. You want to create an Internet site about the greatest mathematicians of all times. First, you need to write a list of the greatest mathematicians (not less than 15). Continue the list: Isaac Newton, Archimedes, Carl Gauss, ........, ......., .........,
C. You are Pythagoras of Samos. Prove the Pythagorean theorem.
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The Pythagorean theorem:
The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
D. In pairs ask and answer questions based on the text "Who Created the Quadratic Formula?" (Further Reading, UNIT 7).
37. You are a mathematician at Oxford University. It is your first lecture. The theme of your lecture is: "Introduction in Mathematics". Think about the points that you would like to cover at this lecture. You may also use the information from the text "Mathematics − the language of science".
LISTENING