Matrix Theory and Linear Algebra (Несколько текстов для зачёта), страница 3

Numerals

Numerals, signs or symbols for graphic representation of numbers. The earliest forms of number notation were simply groups of straight lines, either vertical or horizontal, each line corresponding to the number 1. Such a system is inconvenient when dealing with large numbers, and as early as 3400 bc in Egypt and 3000 bc in Mesopotamia a special symbol was adopted for the number 10. The addition of this second number symbol made it possible to express the number 11 with 2 instead of 11 individual symbols and the number 99 with 18 instead of 99 individual symbols. Later numeral systems introduced extra symbols for a number between 1 and 10, usually either 4 or 5, and additional symbols for numbers greater than 10. In Babylonian cuneiform notation the numeral used for 1 was also used for 60 and for powers of 60; the value of the numeral was indicated by its context. This was a logical arrangement from the mathematical point of view because 60 ⁰ = 1, 60¹ = 60, and 60² = 3600. The Egyptian hieroglyphic system used special symbols for 10, 100, 1000, and 10,000.

The ancient Greeks had two parallel systems of numerals. The earlier of these was based on the initial letters of the names of numbers: The number 5 was indicated by the letter pi; 10 by the letter delta; 100 by the antique form of the letter H; 1000 by the letter chi; and 10,000 by the letter mu. The later system, which was first introduced about the 3rd century bc, employed all the letters of the Greek alphabet plus three letters borrowed from the Phoenician alphabet as number symbols. The first nine letters of the alphabet were used for the numbers 1 to 9, the second nine letters for the tens from 10 to 90, and the last nine letters for the hundreds from 100 to 900. Thousands were indicated by placing a bar to the left of the appropriate numeral, and tens of thousands by placing the appropriate letter over the letter M. The late Greek system had the advantage that large numbers could be expressed with a minimum of symbols, but it had the disadvantage of requiring the user to memorize a total of 27 symbols.

The system of number symbols created by the Romans had the merit of expressing all numbers from 1 to 1,000,000 with a total of seven symbols: I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, and M for 1000. Roman numerals are read from left to right. The symbols representing the largest quantities are placed at the left; immediately to the right of those are the symbols representing the next largest quantities, and so on. The symbols are usually added together. For example, LX = 60, and MMCIII = 2103. When a numeral is smaller than the numeral to the right, however, the numeral on the left should be subtracted from the numeral on the right. For instance, XIV = 14 and IX = 9.  represents 1,000,000—a small bar placed over the numeral multiplies the numeral by 1000. Thus, theoretically, it is possible, by using an infinite number of bars, to express the numbers from 1 to infinity. In practice, however, one bar is usually used; two are rarely used, and more than two are almost never used. Roman numerals are still used today, more than 2000 years after their introduction. The Roman system's one drawback, however, is that it is not suitable for rapid written calculations.

The common system of number notation in use in most parts of the world today is the Arabic system. This system was first developed by the Hindus and was in use in India in the 3rd century bc. At that time the numerals 1, 4, and 6 were written in substantially the same form used today. The Hindu numeral system was probably introduced into the Arab world about the 7th or 8th century ad . The first recorded use of the system in Europe was in ad976.

The important innovation in the Arabic system was the use of positional notation, in which individual number symbols assume different values according to their position in the written numeral. Positional notation is made possible by the use of a symbol for zero. The symbol 0 makes it possible to differentiate between 11, 101, and 1001 without the use of additional symbols, and all numbers can be expressed in terms of ten symbols, the numerals from 1 to 9 plus 0. Positional notation also greatly simplifies all forms of written numerical calculation.

Axiom

Axiom, in logic and mathematics, a basic principle that is assumed to be true without proof. The use of axioms in mathematics stems from the ancient Greeks, most probably during the 5th century bc, and represents the beginnings of pure mathematics as it is known today. Examples of axioms are the following: “No sentence can be true and false at the same time” (the principle of contradiction); “If equals are added to equals, the sums are equal”; “The whole is greater than any of its parts.” Logic and pure mathematics begin with such unproved assumptions from which other propositions (theorems) are derived. This procedure is necessary to avoid circularity, or an infinite regression in reasoning. The axioms of any system must be consistent with one another, that is, they should not lead to contradictions. They should be independent in the sense that they cannot be derived from one another. They should also be few in number. Axioms have sometimes been interpreted as self-evident truths. The present tendency is to avoid this claim and simply to assert that an axiom is assumed to be true without proof in the system of which it is a part.

The terms axiom and postulate are often used synonymously. Sometimes the word axiom is used to refer to basic principles that are assumed by every deductive system, and the term postulate is used to refer to first principles peculiar to a particular system, such as Euclidean geometry. Infrequently, the word axiom is used to refer to first principles in logic, and the term postulate is used to refer to first principles in mathematics.

Vector (mathematics)

Vector (mathematics), quantity having both magnitude and direction. For example, an ordinary quantity, or scalar, can be exemplified by the distance 6 km; a vector quantity can be exemplified by the term 6 km north. Vectors are usually represented by directed line segments, such as  in the diagram below; the length of the line segment is a measure of the vector quantity, and its direction is the same as that of the vector.

The simplest use of vectors and calculation by means of vectors is illustrated in the diagram, drawn to represent a boat moving across a stream. Vector a, or , indicates the motion of the boat in the course of a given interval of time if it were moving through still water; vector b, or , shows the drift or flow of the current during the same period of time. The actual path of travel of the boat under the influence of its own propulsion and of the current is represented by vector c, or . By the use of vectors any type of problem involving the motion of an object being acted on by several forces can be solved graphically.

This method of problem solution, known as vector addition, is performed as follows. A vector representing one force is drawn from the origin O in the proper direction. The length of the vector is made to agree with any convenient arbitrary scale, such as a given number of centimeters to the kilometer. In the diagram the rate of rowing was 2.2 km/h, the time rowed was 1 hr, and the scale is 1 cm to 1 km. Therefore, the line  is drawn as 2.2 cm to equal 2.2 km. The current speed of 6 km/h is then represented by a vector  that is 6 cm long, indicating a distance of 6 km that the current moved during 1 hr. This second vector is drawn with its origin at the end of vector a in a direction parallel to the flow of the current. The point B at the end of the second vector represents the actual position of the boat at the end of 1 hr of travel, and the actual distance traveled is represented by the length (in this case, about 6.4 km) of the vector c, or .

Problems in vector addition and subtraction such as the one above can be easily solved by graphic methods and can also be calculated by means of trigonometry. This type of calculation is useful in solving problems in navigation and motion as well as in mechanics and other branches of physics. In present-day advanced mathematics, a vector is considered an ordered set of quantities with appropriate rules of manipulation. Vector analysis, that is, the algebra, geometry, and calculus of vector quantities, enters into the applied mathematics of every field of science and engineering.

Sequence and Series

Sequence and Series, in mathematics, an ordered succession of numbers or other quantities, and the indicated sum of such a succession, respectively.

A sequence is represented as a₁, a₂..., a_n, .... The as are numbers or quantities, distinct or not; a₁ is the first term, a ₂ the second term, and so on. If the expression has a last term, the sequence is finite; otherwise, it is infinite. A sequence is established or defined only if a rule is given that determines the nth term for every positive integer n; this rule may be given as a formula for the nth term. For example, all the positive integers, in natural order, form an infinite sequence; this sequence is defined by the formula a_n=n. The formula a_n = n² determines the sequence 1, 4, 9, 16, .... The rule of starting with 0, 1, then letting each term be the sum of the two preceding terms determines the sequence 0, 1, 1, 2, 3, 5, 8, 13, ...; this is known as the Fibonacci sequence.

Important types of sequences include arithmetic sequences (also known as arithmetic progressions) in which the differences between successive terms are constant, and geometric sequences (also known as geometric progressions) in which the ratios of successive terms are constant. Examples arise when a sum of money is invested. If the money is invested at a simple interest of 8 percent, then after n years an initial principal of P dollars grows to a_n = P + n × (0.08)P dollars. Since (0.08) P dollars is added each year, the amounts a_n form an arithmetic progression. If the interest is instead compounded, the amounts present after a sequence of years form a geometric progression, g_n = P × (1.08)ⁿ. In both of these cases, it is clear that a_n and g_n will eventually become larger than any preassigned whole number N, however large N may be.

Terms in a sequence, however, do not always increase without limit. For example, as n increases, the sequence a_n = 1/n approaches 0 as a limiting value, and b_n = A + B/n approaches A. In any such case some finite number L exists such that whatever tolerance e is specified, the values of the sequence all eventually lie within a distance e of L. For example, in the case of the sequence 2 + (-1)ⁿ/2n,L = 2. Even if e is as small as 1/10,000, it can be seen that if n is greater than 5000, all values of n are within e of 2. The number L is called the limit of the sequence, since even though individual terms of the sequence may be bigger or smaller than L, the terms eventually cluster closer and closer to L. When the sequence has a limit L, it is said to converge to L. For the sequence a_n, for example, this is written as lim a_n = L, which is read as “the limit of a_n as n goes to infinity is L.”

The term series refers to the indicated sum, a₁ + a₂ + ... + a_n, or a₁ + a₂ + ... + a_n + ..., of the terms of a sequence. A series is either finite or infinite, depending on whether the corresponding sequence of terms is finite or infinite.

The sequence s₁ = a_1,s₂ = a₁ + a₂, s₃ = a₁ + a₂ + a₃, ..., s_n = a₁ + a₂ + ... + a_n, ..., is called the sequence of partial sums of the series a₁ + a₂ + ... + a_n + .... The series converges or diverges as the sequence of partial sums converges or diverges. A constant-term series is one in which the terms are numbers; a series of functions is one in which the terms are functions of one or more variables. In particular, a power series is the series a₀ + a₁(x - c) + a₂(x - c)² + ... + a_n(x - c)ⁿ + ..., in which c and the as are constants. In the case of power series, the problem is to describe what values of x they converge for. If a series converges for some x, then the set of all x for which it converges consists of a point or some connected interval. The basic theory of convergence was worked out by the French mathematician Augustin Louis Cauchy in the 1820s.

The theory and application of infinite series are important in virtually every branch of pure and applied mathematics.

Logarithm