Matrix Theory and Linear Algebra (562420), страница 11

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 11)

In the French and American system of notation, each number after a million is a thousand times the preceding number; in the English and German system, each number is a million times the preceding. A vigintillion is written as a 1 followed by 63 zeros in the French and American system; by 120 zeros in England and Germany.

Decimals are written in the form 1.23 in the United States, 1·23 in the United Kingdom, and 1,23 in continental Europe. In standard scientific notation, a number such as 0.000000123 is written as 1.23x10^-7.

Fermat’s Last Theorem

Fermat’s Last Theorem, in mathematics, famous theorem which has led to important discoveries in algebra and analysis. It was proposed by the French mathematician Pierre de Fermat. While studying the work of the ancient Greek mathematician Diophantus, Fermat became interested in the chapter on Pythagorean numbers—that is, the sets of three numbers, a, b, and c, such as 3, 4, and 5, for which the equation a² + b² = c² is true. He wrote in pencil in the margin, “I have discovered a truly remarkable proof which this margin is too small to contain.” Fermat added that when the Pythagorean theorem is altered to read aⁿ + b ⁿ = cⁿ, the new equation cannot be solved in integers for any value of n greater than 2. That is, no set of positive integers a, b, and c can be found to satisfy, for example, the equation a³ + b³ = c³ or a⁴ + b⁴ = c⁴.

Fermat’s simple theorem turned out to be surprisingly difficult to prove. For more than 350 years, many mathematicians tried to prove Fermat’s statement or to disprove it by finding an exception. In June 1993, Andrew Wiles, an English mathematician at Princeton University, claimed to have proved the theorem; however, in December of that year reviewers found a gap in his proof. On October 6, 1994, Wiles sent a revised proof to three colleagues. On October 25, 1994, after his colleagues judged it complete, Wiles published his proof.

Despite the special and somewhat impractical nature of Fermat’s theorem, it was important because attempts at solving the problem led to many important discoveries in both algebra and analysis.

Pascal, Blaise

Pascal, Blaise (1623-62), French philosopher, mathematician, and physicist, considered one of the great minds in Western intellectual history.

Pascal was born in Clermont-Ferrand on June 19, 1623, and his family settled in Paris in 1629. Under the tutelage of his father, Pascal soon proved himself a mathematical prodigy, and at the age of 16 he formulated one of the basic theorems of projective geometry, known as Pascal's theorem and described in his Essai pour les coniques (Essay on Conics, 1639). In 1642 he invented the first mechanical adding machine. Pascal proved by experimentation in 1648 that the level of the mercury column in a barometer is determined by an increase or decrease in the surrounding atmospheric pressure rather than by a vacuum, as previously believed. This discovery verified the hypothesis of the Italian physicist Evangelista Torricelli concerning the effect of atmospheric pressure on the equilibrium of liquids. Six years later, in conjunction with the French mathematician Pierre de Fermat, Pascal formulated the mathematical theory of probability, which has become important in such fields as actuarial, mathematical, and social statistics and as a fundamental element in the calculations of modern theoretical physics. Pascal's other important scientific contributions include the derivation of Pascal's law or principle, which states that fluids transmit pressures equally in all directions, and his investigations in the geometry of infinitesimals. His methodology reflected his emphasis on empirical experimentation as opposed to analytical, a priori methods, and he believed that human progress is perpetuated by the accumulation of scientific discoveries resulting from such experimentation.

Pascal espoused Jansenism and in 1654 entered the Jansenist community at Port Royal, where he led a rigorously ascetic life until his death eight years later. In 1656 and 1657 he wrote the famous 18 Lettres provinciales (Provincial Letters), in which he attacked the Jesuits for their attempts to reconcile 16th-century naturalism with orthodox Roman Catholicism. His most positive religious statement appeared posthumously (he died August 19, 1662); it was published in fragmentary form in 1670 as Apologie de la religion Chrétienne (Apology of the Christian Religion). In these fragments, which later were incorporated into his major work, he posed the alternatives of potential salvation and eternal damnation, with the implication that only by conversion to Jansenism could salvation be achieved. Pascal asserted that whether or not salvation was achieved, humanity's ultimate destiny is an afterlife belonging to a supernatural realm that can only be known intuitively. Pascal's final important work was Pensées sur la religion et sur quelques autres sujets (Thoughts on Religion and on Other Subjects), also published in 1670. In the Pensées Pascal attempted to explain and justify the difficulties of human life by the doctrine of original sin, and he contended that revelation can be comprehended only by faith, which in turn is justified by revelation. Pascal's writings urging acceptance of the Christian life contain frequent applications of the calculations of probability; he reasoned that the value of eternal happiness is infinite and that although the probability of gaining such happiness by religion may be small it is infinitely greater than by any other course of human conduct or belief. A reclassification of the Pensées, a careful work begun in 1935 and continued by several scholars, does not reconstruct the Apologie, but allows the reader to follow the plan that Pascal himself would have followed.

Pascal was one of the most eminent mathematicians and physicists of his day and one of the greatest mystical writers in Christian literature. His religious works are personal in their speculation on matters beyond human understanding. He is generally ranked among the finest French polemicists, especially in the Lettres provinciales, a classic in the literature of irony. Pascal's prose style is noted for its originality and, in particular, for its total lack of artifice. He affects his readers by his use of logic and the passionate force of his dialectic.

Probability

Probability, also theory of probability, branch of mathematics that deals with measuring or determining quantitatively the likelihood that an event or experiment will have a particular outcome. Probability is based on the study of permutations and combinations and is the necessary foundation for statistics.

The foundation of probability is usually ascribed to the 17th-century French mathematicians Blaise Pascal and Pierre de Fermat, but mathematicians as early as Gerolamo Cardano had made important contributions to its development. Mathematical probability began in an attempt to answer certain questions arising in games of chance, such as how many times a pair of dice must be thrown before the chance that a six will appear is 50-50. Or, in another example, if two players of equal ability, in a match to be won by the first to win ten games, are obliged to suspend play when one player has won five games, and the other seven, how should the stakes be divided?

The probability of an outcome is represented by a number between 0 and 1, inclusive, with “probability 0” indicating certainty that an event will not occur and “probability 1” indicating certainty that it will occur. The simplest problems are concerned with the probability of a specified “favorable” result of an event that has a finite number of equally likely outcomes. If an event has n equally likely outcomes and f of them are termed favorable, the probability, p, of a favorable outcome is f/n. For example, a fair die can be cast in six equally likely ways; therefore, the probability of throwing a 5 or a 6 is 2/6. More involved problems are concerned with events in which the various possible outcomes are not equally likely. For example, in finding the probability of throwing a 5 or 6 with a pair of dice, the various outcomes (2, 3, ... 12) are not all equally likely. Some events may have infinitely many outcomes, such as the probability that a chord drawn at random in a circle will be longer than the radius.

Problems involving repeated trials form one of the connections between probability and statistics. To illustrate, what is the probability that exactly five 3s and at least four 6s will occur in 50 tosses of a fair die? Or, a person, tossing a fair coin twice, takes a step to the north, east, south, or west, according to whether the coin falls head, head; head, tail; tail, head; or tail, tail. What is the probability that at the end of 50 steps the person will be within 10 steps of the starting point?

In probability problems, two outcomes of an event are mutually exclusive if the probability of their joint occurrence is zero; two outcomes are independent if the probability of their joint occurrence is given as the product of the probability of their separate occurrences. Two outcomes are mutually exclusive if the occurrence of one precludes the occurrence of the other; two outcomes are independent if the occurrence or nonoccurrence of one does not alter the probability that the other will or will not occur. Compound probability is the probability of all outcomes of a certain set occurring jointly; total probability is the probability that at least one of a certain set of outcomes will occur. Conditional probability is the probability of an outcome when it is known that some other outcome has occurred or will occur.

If the probability that an outcome will occur is p, the probability that it will not occur is q = 1 - p. The odds in favor of the occurrence are given by the ratio p:q, and the odds against the occurrence are given by the ratio q:p. If the probabilities of two mutually exclusive outcomes X and Y are p and P, respectively, the odds in favor of X and against Y are p to P. If an event must result in one of the mutually exclusive outcomes O₁,O₂,..., O_n, with probabilities p₁,p₂,..., p_n, respectively, and if v₁,v₂,...v_n are numerical values attached to the respective outcomes, the expectation of the event is E = p₁v₁ + p₂v₂ + ...p_nv_n. For example, a person throws a die and wins 40 cents if it falls 1, 2, or 3; 30 cents for 4 or 5; but loses $1.20 if it falls 6. The expectation on a single throw is 3/6 × .40 + 2/6 × .30 - 1/6 × 1.20 = .10.

The most common interpretation of probability is used in statistical analysis. For example, the probability of throwing a 7 in one throw of two dice is 1/6, and this answer is interpreted to mean that if two fair dice are randomly thrown a very large number of times, about one-sixth of the throws will be 7s. This concept is frequently used to statistically determine the probability of an outcome that cannot readily be tested or is impossible to obtain. Thus, if long-range statistics show that out of every 100 people between 20 and 30 years of age, 42 will be alive at age 70, the assumption is that a person between those ages has a 42 percent probability of surviving to the age of 70.

Mathematical probability is widely used in the physical, biological, and social sciences and in industry and commerce. It is applied in such diverse areas as genetics, quantum mechanics, and insurance. It also involves deep and important theoretical problems in pure mathematics and has strong connections with the theory, known as mathematical analysis, that developed out of calculus.

Calculus (mathematics)

Calculus (mathematics), branch of mathematics concerned with the study of such concepts as the rate of change of one variable quantity with respect to another, the slope of a curve at a prescribed point, the computation of the maximum and minimum values of functions, and the calculation of the area bounded by curves. Evolved from algebra, arithmetic, and geometry, it is the basis of that part of mathematics called analysis.

Calculus is widely employed in the physical, biological, and social sciences. It is used, for example, in the physical sciences to study the speed of a falling body, the rates of change in a chemical reaction, or the rate of decay of a radioactive material. In the biological sciences a problem such as the rate of growth of a colony of bacteria as a function of time is easily solved using calculus. In the social sciences calculus is widely used in the study of statistics and probability.

Calculus can be applied to many problems involving the notion of extreme amounts, such as the fastest, the most, the slowest, or the least. These maximum or minimum amounts may be described as values for which a certain rate of change (increase or decrease) is zero. By using calculus it is possible to determine how high a projectile will go by finding the point at which its change of altitude with respect to time, that is, its velocity, is equal to zero. Many general principles governing the behavior of physical processes are formulated almost invariably in terms of rates of change. It is also possible, through the insights provided by the methods of calculus, to resolve such problems in logic as the famous paradoxes posed by the Greek philosopher Zeno.

Характеристики

Список файлов учебной работы