Matrix Theory and Linear Algebra (Несколько текстов для зачёта), страница 14

Important philosophical works by Leibniz include Essays in Theodicy on the Goodness of God, the Liberty of Man, and the Origin of Evil (2 volumes, 1710; translated in Philosophical Works,1890), Monadology (1714; published in Latin as Principia Philosophiae,1721; translated 1890), and New Essays Concerning Human Understanding (1703; published 1765; translated 1916). The latter two greatly influenced German philosophers of the 18th century, including Christian von Wolff and Immanuel Kant.

Digital Logic

Digital Logic, also called binary logic, in computer science, a strict set of rules for representing the relationships and interactions among numbers, words, symbols, and other data stored or entered in the memory of a computer. Digital logic is the heart of the operational function for all modern digital computers. The system uses binary arithmetic, in which a sequence of 1s and 0s (called bits) are used to represent a number. These bits are combined in meaningful ways through the operation of digital logic and physically describe electrical voltage states in a computer’s circuitry. Digital logic uses the bit value 1 to represent a transistor with electric current flowing through it and the bit value 0 to represent a transistor with no current flowing through it.

The instructions that direct a computer’s operation are known as machine code, and they are written as a sequence of binary digits. These binary digits, or bits, switch specific groups of transistors, called gates, on or off (see Transistor). There are three basic logic states, or functions, for logic gates: AND, OR, and NOT. An AND gate takes the value of two input bits and tests them to see if they are both equal to 1. If they are, the output from the AND gate is a 1, or true. If they are not, the AND gate will output a 0, or false. An OR gate tests two input bits to see if either of the bits is equal to 1. If either input bit is equal to 1, the gate outputs a 1; if both input bits are 0, it outputs a 0. A NOT gate negates the input bit, so an input of 1 results in an output of 0 and vice versa.

Combinations of logic gates in open or closed positions can be used to represent and execute operations on data. A series of logic gates together form a logic circuit. The output of a logic circuit can provide input to another circuit or produce the result of an operation. Extremely complex operations can be performed using combinations of the AND, OR, and NOT functions.

Binary logic was first proposed by 19th-century British logician and mathematician George Boole, who in 1847 invented a two-valued system of algebra that represented logical relationships and operations. This system of algebra, called Boolean Algebra, was used by German engineer Konrad Zuse in the 1930s for his Z1 calculating machine. It was also used in the design of the first digital computer in the late 1930s by American physicist John Atanasoff and his graduate student Clifford Berry. During 1944 and 1945 Hungarian-born American mathematician John von Neumann suggested using the binary arithmetic system for storing programs in computers. In the 1930s and 1940s British mathematician Alan Turing and American mathematician Claude Shannon also recognized how binary logic was well suited to the development of digital computers.

Complex Number

I

INTRODUCTION

Complex Number, in mathematics, the sum of a real number and an imaginary number. An imaginary number is a multiple of i, where i is the square root of -1. Complex numbers can be expressed in the form a + bi, where a and b are real numbers. They have the algebraic structure of a field in mathematics. In engineering and physics, complex numbers are used extensively to describe electric circuits and electromagnetic waves (see Electromagnetic Radiation). The number i appears explicitly in the Schrödinger wave equation (see Schrödinger, Erwin), which is fundamental to the quantum theory of the atom. Complex analysis, which combines complex numbers with ideas from calculus, has been widely applied to subjects as different as the theory of numbers and the design of airplane wings.

II

HISTORY

Historically, complex numbers arose in the search for solutions to equations such as x²=-1. Because there is no real number x for which the square is -1, early mathematicians believed this equation had no solution. However, by the middle of the 16th century, Italian mathematician Gerolamo Cardano and his contemporaries were experimenting with solutions to equations that involved the square roots of negative numbers. Cardano suggested that the real number 40 could be expressed as

Swiss mathematician Leonhard Euler introduced the modern symbol i for

in 1777 and expressed the famous relationship e^i=-1 which connects four of the fundamental numbers of mathematics. For his doctoral dissertation in 1799, German mathematician Carl Friedrich Gauss proved the fundamental theorem of algebra, which states that every polynomial with complex coefficients has a complex root. The study of complex functions was continued by French mathematician Augustin Louis Cauchy, who in 1825 generalized the real definite integral of calculus to functions of a complex variable.

III

PROPERTIES

For a complex number a+ bi,a is called the real part and b is called the imaginary part. Thus, the complex number -2 + 3i has the real part -2 and the imaginary part 3. Addition of complex numbers is performed by adding the real and imaginary parts separately. To add 1 + 4i and 2 - 2i, for example, add the real parts 1 and 2 and then the imaginary parts 4 and -2 to obtain the complex number 3 + 2i. The general rule for addition is

(a+ bi) + (c+di) = (a+ c) + (b+d)i

Multiplication of complex numbers is based on the premise that i×i=-1 and the assumption that multiplication distributes over addition. This gives the rule

(a+bi) × (c+di) = ( ac-bd) + (ad+bc)i

For example,

(1 + 4i) × (2 - 2i) = 10 + 6i

If z=a+bi is any complex number, then, by definition, the complex conjugate of z is

and the absolute value, or modulus, of z is

For example, the complex conjugate of 1 + 4i is 1 - 4i, and the modulus of 1 + 4i is

A basic relationship connecting absolute value and complex conjugate is

IV

THE COMPLEX PLANE

In the same way that real numbers can be thought of as points on a line, complex numbers can be thought of as points in a plane. The number a+bi is identified with the point in the plane with x coordinate a and y coordinate b. The points 1 + 4i and 2 - 2i are plotted in Figure 1 and correspond to the points (1,4) and (2,-2). In 1806 Swiss bookkeeper Jean Robert Argand was one of the first people to express complex numbers geometrically as points in the plane. For this reason, Figure 1 is sometimes referred to as an Argand diagram. If a complex number in the plane is thought of as a vector joining the origin to that point, then addition of complex numbers corresponds to standard vector addition. Figure 1 shows the complex number 3 + 2i obtained by adding the vectors 1 + 4i and 2 - 2i.

Since points in the plane can be written in terms of the polar coordinates r and θ (see Coordinate System), every complex number z can be written in the form

z= r (cos θ+i sin θ)

Here, r is the modulus, or distance to the origin, and θ is the argument of z or the angle that z makes with the x axis. If z=r (cos θ+i sin θ) and w=s (cos φ+i sin φ) are two complex numbers in polar form, then their product in polar form is given by

zw=rs (cos (θ+φ) +i sin (θ+φ))

This has a simple geometric interpretation that is illustrated in Figure 2.

V

SOLUTIONS TO POLYNOMIALS

There are many polynomial equations that have no real solutions, such as

x2+ 1 = 0

However, if x is allowed to be complex, the equation has the solutions x=±i, where i and - i are roots of the polynomial x²+ 1. The equation

x2- 2x+ 2 = 0

has the solutions x= l ±i. In his fundamental theory of algebra, Gauss showed that every nontrivial (having at least one nonzero root) polynomial with complex coefficients must have at least one complex root. From this it follows that every complex polynomial of degree n must have exactly n roots, although some roots may be the same. Consequently, every complex polynomial of degree n can be written as a product of exactly n linear, or first-degree, factors.

Calculator

Calculator, handheld device that performs mathematical calculations. A calculator can also be a program on a computer that simulates a handheld calculator or offers more sophisticated calculation features.

A standard calculator is rectangular in shape and has a keypad through which numbers and operations are entered, as well as a display on which the entered numbers and the results of calculations are shown. Modern calculators can perform many types of mathematical computations, as well as permit the user to store and access data from memory. Common handheld calculators have the ability to use complicated geometric, algebraic, trigonometric, statistical, and calculus functions. Many can also be programmed for specialized tasks. Calculators operate on electrical power supplied by either batteries, solar cells, or standard electrical current. Modern calculators have digital displays, usually using some form of LCD (liquid crystal display).

Calculator programs are common accessories included with most personal computer operating systems. For example, both the Macintosh and Windows operating systems have a simple desktop calculator program.

While the calculator is a very modern invention, machines able to perform addition and subtraction have existed for centuries. The abacus, a simple instrument for carrying out arithmetic operations, has been used by many cultures and dates back to ancient times. Another calculating instrument called the slide rule was invented in the early 1600s by the English mathematician William Oughtred. The slide rule was based on logarithms and made multiplication and division much easier. Engineers and scientists used slide rules until the introduction of calculators in the early 1970s.

The invention of the calculating machine is commonly credited to the French mathematician Blaise Pascal. In 1642 Pascal created a machine to free his father, who was a tax collector, from the tedious task of adding columns of numbers. Pascal’s machine used a complicated arrangement of numbered wheels connected by gears, which could add and subtract numbers up to nine digits long.

Until the late 1960s, hundreds of different types of calculating machines were invented that utilized many different technologies. Two of the most popular were the stepped-drum and pin-wheel mechanisms. These early calculators, even when refined for the desktop, were large, heavy, and expensive, compared to today’s calculators. Earlier models required the user to press a hand lever to turn the calculating mechanism. Later versions used electrical power to turn the mechanism. Most early calculators used paper tape to print typewritten numbers, operations, and results. For the most part, these mechanical devices could only perform four basic operations: addition, subtraction, multiplication, and division.

In 1967 a team of three engineers from Texas Instruments, Inc. invented the portable, electronic, handheld calculator. Jack Kilby, widely known as the inventor of the integrated circuit (IC), or computer chip, along with Jerry Merryman and James Van Tassel, built an IC-based, battery-powered miniature calculator that could add, subtract, multiply, and divide. This basic calculator could accept 6-digit numbers and display results as large as 12 digits. The prototype of this device is now displayed in the Smithsonian Institution, based in Washington, D.C.

Proof, Mathematical

Proof, Mathematical, an argument that is used to show the truth of a mathematical assertion. In modern mathematics, a proof begins with one or more statements called premises and demonstrates, using the rules of logic, that if the premises are true then a particular conclusion must also be true.

The accepted methods and strategies used to construct a convincing mathematical argument have evolved since ancient times and continue to change. Consider the Pythagorean theorem, named after the 5th century bc Greek mathematician and philosopher Pythagoras, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Many early civilizations considered this theorem true because it agreed with their observations in practical situations. But the early Greeks, among others, realized that observation and commonly held opinion do not guarantee mathematical truth. For example, before the 5th century bc it was widely believed that all lengths could be expressed as the ratio of two whole numbers. But an unknown Greek mathematician proved that this was not true by showing that the length of the diagonal of a square with an area of 1 is the irrational number  (see Number).