Matrix Theory and Linear Algebra (562420), страница 2

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By the time of Gauss, algebra had entered its modern phase. Attention shifted from solving polynomial equations to studying the structure of abstract mathematical systems whose axioms were based on the behavior of mathematical objects, such as complex numbers, that mathematicians encountered when studying polynomial equations. Two examples of such systems are groups (see Group) and quaternions, which share some of the properties of number systems but also depart from them in important ways. Groups began as systems of permutations and combinations of roots of polynomials, but they became one of the chief unifying concepts of 19th-century mathematics. Important contributions to their study were made by the French mathematicians Galois and Augustin Cauchy, the British mathematician Arthur Cayley, and the Norwegian mathematicians Niels Abel and Sophus Lie. Quaternions were discovered by British mathematician and astronomer William Rowan Hamilton, who extended the arithmetic of complex numbers to quaternions while complex numbers are of the form a + bi, quaternions are of the form a + bi + cj + dk.

Immediately after Hamilton's discovery, the German mathematician Hermann Grassmann began investigating vectors. Despite its abstract character, American physicist J. W. Gibbs recognized in vector algebra a system of great utility for physicists, just as Hamilton had recognized the usefulness of quaternions. The widespread influence of this abstract approach led George Boole to write The Laws of Thought (1854), an algebraic treatment of basic logic. Since that time, modern algebra—also called abstract algebra—has continued to develop. Important new results have been discovered, and the subject has found applications in all branches of mathematics and in many of the sciences as well.

In operating with polynomials, the assumption is that the usual laws of the arithmetic of numbers hold. In arithmetic, the numbers used are the set of rational numbers (see Number; Number Theory). Arithmetic alone cannot go beyond this, but algebra and geometry can include both irrational numbers, such as the square root of 2, and complex numbers. The set of all rational and irrational numbers taken together constitutes the set of what are called real numbers.

The sum of any two real numbers A1. a and b is again a real number, denoted a + b. The real numbers are closed under the operations of addition, subtraction, multiplication, division, and the extraction of roots; this means that applying any of these operations to real numbers yields a quantity that also is a real number.

No matter how terms are grouped in carrying out additions, the sum A2. will always be the same: (a + b) + c = a + (b + c). This is called the associative law of addition.

Given any real number A3. a, there is a real number zero (0) called the additive identity, such that a + 0 = 0 + a = a.

Given any real number A4. a, there is a number (-a), called the additive inverse of a, such that (a) + (-a) = 0.

No matter in what order addition is carried out, the sum will A5. always be the same: a + b = b + a. This is called the commutative law of addition.

Any set of numbers obeying laws A1 through A4 is said to form a group. If the set also obeys A5, it is said to be an Abelian, or commutative, group.

Laws similar to those for addition also apply to multiplication. Special attention should be given to the multiplicative identity and inverse, M3 and M4.

The product of any two real numbers M1. a and b is again a real number, denoted a·b or ab.

No matter how terms are grouped in carrying out multiplications, M2. the product will always be the same: (ab)c = a(bc). This is called the associative law of multiplication.

Given any real number M3. a, there is a number one (1) called the multiplicative identity, such that a(1) = 1(a) = a.

Given any nonzero real number M4. a, there is a number (a^-1), or (1/a), called the multiplicative inverse, such that a(a^-1) = (a^-1)a = 1.

No matter in what order multiplication is carried out, the product M5. will always be the same: ab = ba. This is called the commutative law of multiplication.

Any set of elements obeying these five laws is said to be an Abelian, or commutative, group under multiplication. The set of all real numbers, excluding zero (because division by zero is inadmissible), forms such a commutative group under multiplication.

Another important property of the set of real numbers links addition and multiplication in two distributive laws as follows:

 D1.a(b + c) = ab + ac

( D2.b + c)a = ba + ca

Any set of elements with an equality relation and for which two operations (such as addition and multiplication) are defined, and which obeys all the laws for addition A1 through A5, the laws for multiplication M1 through M5, and the distributive laws D1 and D2, constitutes a field.

Number Systems

Number Systems, in mathematics, various notational systems that have been or are being used to represent the abstract quantities called numbers. A number system is defined by the base it uses, the base being the number of different symbols required by the system to represent any of the infinite series of numbers. Thus, the decimal system in universal use today (except for computer application) requires ten different symbols, or digits, to represent numbers and is therefore a base-10 system.

Throughout history, many different number systems have been used; in fact, any whole number greater than 1 can be used as a base. Some cultures have used systems based on the numbers 3, 4, or 5. The Babylonians used the sexagesimal system, based on the number 60, and the Romans used (for some purposes) the duodecimal system, based on the number 12. The Mayas used the vigesimal system, based on the number 20. The binary system, based on the number 2, was used by some tribes and, together with the system based on 8, is used today in computer systems. For historical background, see Numerals.

Except for computer work, the universally adopted system of mathematical notation today is the decimal system, which, as stated, is a base-10 system. As in other number systems, the position of a symbol in a base-10 number denotes the value of that symbol in terms of exponential values of the base. That is, in the decimal system, the quantity represented by any of the ten symbols used—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9—depends on its position in the number. Thus, the number 3,098,323 is an abbreviation for (3 × 10⁶) + (0 × 10⁵) + (9 × 10⁴) + (8 × 10³) + (3 × 10²) + (2 × 10¹) + (3 × 10⁰, or 3 × 1). The first “3” (reading from right to left) represents 3 units; the second “3,” 300 units; and the third “3,” 3 million units. In this system the zero plays a double role; it represents naught, and it also serves to indicate the multiples of the base 10: 100, 1000, 10,000, and so on. It is also used to indicate fractions of integers: 1/10 is written as 0.1, 1/100 as 0.01, 1/1000 as 0.001, and so on.

Two digits—0, 1—suffice to represent a number in the binary system; 6 digits—0, 1, 2, 3, 4, 5—are needed to represent a number in the sexagesimal system; and 12 digits—0, 1, 2, 3, 4, 5, 6, 7, 8, 9, t (ten), e (eleven)—are needed to represent a number in the duodecimal system. The number 30155 in the sexagesimal system is the number (3 × 6⁴) + (0 × 6³) + (1 × 6²) + (5 × 6¹) + (5 × 6⁰) = 3959 in the decimal system; the number 2et in the duodecimal system is the number (2 × 12²) + (11 × 12¹) + (10 × 12⁰) = 430 in the decimal system

To write a given base-10 number n as a base-b number, divide (in the decimal system) n by b, divide the quotient by b, the new quotient by b, and so on until the quotient 0 is obtained. The successive remainders are the digits in the base-b expression for n. For example, to express 3959 (base 10) in the base 6, one writes

from which, as above, 3959₁₀ = 30155₆. (The base is frequently written in this way as a subscript of the number.) The larger the base, the more symbols are required, but fewer digits are needed to express a given number. The number 12 is convenient as a base because it is exactly divisible by 2, 3, 4, and 6; for this reason, some mathematicians have advocated adoption of base 12 in place of the base 10.

The binary system plays an important role in computer technology. The first 20 numbers in the binary notation are 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100. The zero here also has the role of place marker, as in the decimal system. Any decimal number can be expressed in the binary system by the sum of different powers of two. For example, starting from the right, 10101101 represents (1 × 2⁰) + (0 × 2¹) + (1 × 2²) + (1 × 2³) + (0 × 2⁴) + (1 × 2⁵) + (0 × 2⁶) + (1 × 2⁷) = 173. This example can be used for the conversion of binary numbers into decimal numbers. For the conversion of decimal numbers to binary numbers, the same principle can be used, but the other way around. Thus, to convert, the highest power of two that does not exceed the given number is sought first, and a 1 is placed in the corresponding position in the binary number. For example, the highest power of two in the decimal number 519 is 2⁹ = 512. Thus, a 1 can be inserted as the 10th digit, counted from the right: 1000000000. In the remainder, 519 - 512 = 7, the highest power of 2 is 2² = 4, so the third zero from the right can be replaced by a 1: 1000000100. The next remainder, 3, consists of the sum of two powers of 2: 2¹ + 2⁰, so the first and second zeros from the right are replaced by 1: 519₁₀ = 1000000111₂.

Arithmetic operations in the binary system are extremely simple. The basic rules are: 1 + 1 = 10, and 1 × 1 = 1. Zero plays its usual role: 1 × 0 = 0, and 1 + 0 = 1. Addition, subtraction, and multiplication are done in a fashion similar to that of the decimal system:

Because only two digits (or bits) are involved, the binary system is used in computers, since any binary number can be represented by, for example, the positions of a series of on-off switches. The “on” position corresponds to a 1, and the “off” position to a 0. Instead of switches, magnetized dots on a magnetic tape or disk also can be used to represent binary numbers: a magnetized dot stands for the digit 1, and the absence of a magnetized dot is the digit 0. Flip-flops—electronic devices that can only carry two distinct voltages at their outputs and that can be switched from one state to the other state by an impulse—can also be used to represent binary numbers; the two voltages correspond to the two digits. Logic circuits in computers (see Computer; Electronics) carry out the different arithmetic operations of binary numbers; the conversion of decimal numbers to binary numbers for processing, and of binary numbers to decimal numbers for the readout, is done electronically.

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.

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