Автор: Пользователь скрыл имя, 21 Ноября 2012 в 11:32, реферат
Turing Model
The idea of a universal computational device was first described by Alan Turing in 1937. He proposed that all computation could be performed by a special kind of a machine, now called a Turing machine. Although Turing presented a mathematical description of such a machine, he was more interested in the philosophical defini¬tion of computation than in building the actual machine. He based the model on the actions that people perform when involved in computation. He abstracted these actions into a model for a computational machine that has really changed the world.
Several number systems have been used in the past and can be categorized into two groups: positional and non-positional systems. Our main goal is to discuss the positional number systems, but we also give examples of non-positional systems.
Positional number systems
In a positional number system, the position a symbol occupies in the number determines the value it represents.
In which S is the set of symbols, b is the base (or radix), which is equal to the total number of the symbols in the set S, and Si, is the symbol in position i. Note that we| have used an expression that can be extended from the right or from the left. In other words, the power of b can be 0 to k - 1 in one direction and -1 to -l in the other direction. The terms with non-negative powers of b are related to the inte-gral part of the number, while the terms with negative power of b are related to the fractional part of the number. The ± sign shows that the number can be either positive or negative.
The decimal system (base 10)
The first positional number system we discuss in this chapter is the decimal systern. The word decimal is derived from the Latin root decern (ten). In this system the base b = 10 and we use ten symbols to represent a number. The set of symbols is S = {0,1,2,3,4,5,6,7,8,9}. As we know, the symbols in this system are often referred to as decanal digits or just digits. In this chapter, we use ± to show that a number can be positive or negative, but remember that these signs are not stored in computers—computers handle the sign differently.
Computers store positive and negative numbers differently.
An integer (an integral number with no fractional part) in the decimal system is familiar to all of us—we use integers in our daily life. In fact, we have used them so much that they are intuitive.
The binary system (base 2)
The second positional number system we discuss in this chapter is the binary system. The word binary is derived from the Latin root bini (or two by two). In this system the base b = 2 and we use only two symbols, S = {0, 1}. The symbols this system are often referred to as binary digits or bits (binary digit). Data and programs are stored in the computer using binary pa terns, a string of bits. This is because the computer is made of electronic switche that can have only two states, on and off. The bit 1 represents one of these states and the bit 0 the other.
The hexadecimal system (base 16)
Although the binary system is used to store data in computers, it is not convenient for representation of numbers outside the computer, as a number in binary no" tion is much longer than the corresponding number in decimal notation. Howev the decimal system does not show what is stored in computer as binary directly: there is no obvious relationship between the number of bits in binary and the num ber of decimal digits. Conversion from one to the other is not fast, as we will see shortly.
To overcome this problem, two positional systems were devised: hexadecimal and octal. We first discuss the hexadecimal system, which is more common, word hexadecimal is derived from the Greek root hex (six) and the Latin decern (ten). To be consistent with decimal and binary, it should really have called sexadecimal, from the Latin roots sex and decern. In this system the base b=16 and we use sixteen symbols to represent a number. The set of symbols is S = {0,1,2, 3, 4, 5, 6, 7, 8. 9, А, В, C, D, E, F}. Note that the symbols А, В, C, D, E, F (uppercase or lowercase) arc equivalent to 10, 11, 12, 13, 14, and 15 respectively. The symbols in this system are often referred to as hexadecimal digits.
The octal system (base 8)
The second system that was devised to show the equivalent of the binary system outside the computer is the octal system. The word octal is derived from the Latin root octo (eight). In this system the base b = 8 and we use eight symbols to represent a number. The set of symbols is S = {0,1,2,3,4,5,6,7}. The symbols in this system are often referred to as octal digits.