Information representation options in the Computer

All information (data) is provided in the form of binary codes. For convenience of operation the following terms designating sets of binary places (tab. 1.1) are entered. These terms are normally used as units of measure of information volumes, storable or processed in a computer.

Table 1.1 – Binary sets

Quantity of binary places in group	1	8	16	8*1024	8*10242	8*10243	8*10244
Unit of measurement name	bit	byte	paragraph	Kilobyte (Kbyte)	Megabyte (Mb)	Gigabyte (Gbyte)	Terabyte (Tbyte)

The sequence of several bits or bytes is often called the Bit data field in number (in a word, in the field, etc.) They are numbered from right to left, since 0th discharge.

In the PC fields of constant and variable length can be processed. Numbers from the fixed comma have a format of a word and a half-word, number from a floating comma – a format of a double and expanded word more often. Fields of variable length can have any size from 0 to 256 bytes, but surely equal to an integral number of bytes.

Binary-coded decimal numbers can be provided to the PC by fields of variable length: in the so-called packed and unpacked formats.

The continuous and discrete information

To transfer the message from the source to the receiver the material substance – information medium is necessary. The message transferred by means of the carrier is called a signal. Generally the signal is a physical process changing in time. Such process can contain different characteristics (for example, by transmission of electrical signals tension and current intensity can change). The characteristic used for submission of messages is called the signal parameter.

When the parameter of a signal accepts serial in time a finite number of values (thus all of them can be enumerated), the signal is called the discrete, and the message transferred by means of such signals – the discrete message. Information transferred by a source, in this case also is called the discrete. If the source works out the continuous message (respectively signal parameter – the continuous function from time), the relevant information is called the continuous.

Data media.

The units of measure of information serve for information volume measurement – the value calculated logarithmic. It means that when some objects are considered as one, the quantity of possible statuses is multiplied, and the number of information develops. It is not important if there is a speech about random variables in mathematician, registers of digital memory in technique or quantum systems in physics.

More often measurement of information concerns the volume of computer memory and the data volume transferred on inter-chip digital links. The whole number of bits responds the quantity of statuses equal to powers of the number «two».

The special name has 4 bits – nibble (half-byte, a tetrad, four binary places) which contain in itself number of information containing in one hexadecimal digit.

In the order of popular numbers in information is 8 bits or byte. The increasing information volumes calculated in computer technologies are directly given to byte (instead of bit). Such values as a machine word, etc., bytes making as units of measure are never used.

Operations over data

Any information can be measured in bits and consequently irrespective of on what physical principles and in what numeration system the digital computer (binary, ternary, decimal, etc.), numbers, text information, images, a sound, video functions and it is possible to provide other types of data sequences of bit lines or binary numbers. It allows the computer to manipulate data under condition of sufficient capacity of storage system (for example, for storage of the text of the novel of the average size it is necessary about one megabyte).

It is necessary to distinguish classification of memory and classification of storage devices (memories). The first classifies memory on the functionality, the second – on the technical implementation. Here the first memory is considered – the hardware types of memory (implemented on the memory) and the data structures implemented in most cases program are involved in it.

Available dataful operations: memory only for reading (read-only memory, ROM), memory for reading/record

Memory on programmable and reprogrammable read-only memories has no standard place in this classification. It belongs to the memory subspecies «only for reading», or select in a separate form. Also it is offered to refer memory to this or that form on characteristic frequency of its copying in practice: the types in which information often changes in the course of operation belong to RAM, the types intended for storage of rather invariable data belong to ROM.

Control questions:

1. What is the informatics?

2. What is the main objective of informatics?

3. What does the theoretical informatics study?

4. What does programming study?

5. What does an information system study?

6. What does machine intelligence study?

7. What does the ADP equipment study?

8. What is the economic information?

9. What is data?

10. What is the information?

11. How is the information classified?

12. What are the list properties of information?

13. What is the numeration system?

14. What types of numeration systems do you know?

15. Why is the binary numeral system used in a computer?

16. What is the data field?

17. What is the bit?

18. What is the byte?

19. What is the sampling?

20. What is the signal?

Topic 2. Bases of discrete mathematics.

Relation and set functions. Bases of logic, logician of statements, logic sheaves, validity tables.

In the course of processing of binary information the processor executes arithmetical and logical actions. Therefore to receive the ideas of device of the computer it is necessary to get acquainted with the main logic gates of the computer. To understand the principle of operation of these elements it is necessary to know the main initial concepts of the formal logic. The term «logic» occurs from the Ancient Greek «logus» with the meaning of «word, thought, concept, reasoning, law»

Relation and set functions.

The relation from the set A in the set B    is called the function from A into B, if

.

The fact that f – function from the set A into the set B is registered as f:   or .

Instead of the record  we use . Such property of the relation is called as uniqueness or functionality.

If b=f(a) then a  is called the argument, and b – the function value.

Let

, then is called the function of the definition range.

Let

, then  is called the area of function values.

If , that function is called the total, and if  – the partial.

The narrowing of the set function  into the set  is called the function narrowing, ,it is defined as follows: 

.

 is called the function of n  arguments or n – local function.

Properties of function: Let . Then function  is called the injective, if

, the subjective, if 

, objective, if it injective and subjective

Properties of the relations

Let , i.e.  – the binary relation on the set A

1)  – reflexive relation, if   

It is read as: for any element from the set А the steam (а, а) belongs to the relation that means that any element from the set А is in the relation with itself.

For example, we will consider the relation  ={(a,b)}a is a group mate of b,

 , A – great number of students of technical school, then condition     means that any student of technical school is a groupmate of himself that obviously isn't correct, it means that the relation  isn't reflexive.

Let's consider the relation    , , A – set of all real numbers, then condition        means that any real number is more or equal to itself , it is obviously correct, it means that the relation   is reflexive.

2)  – the symmetric relation, if   .

It is read as follows: for any elements from the set А, if the pair  belongs to the relation and the pair  belongs to the relation , that means that if an element a is in relation with b, then element b is in relation with a.

Let's consider the relation ={(a,b)}a is a groupmate of b,  , A – great number of students of technical school, then condition     means that if the student of technical school  a is the groupmate of the student b, the student b studies in one group with a, it is obviously correct, the relation   is symmetric.

Let's consider the relation

,  , A–set of all real numbers, then condition       means that if the condition is implemented, the condition is implemented too, that isn't correct, the relation   isn't the symmetric.

3)  – antisymmetric relation, if   .

It is read as follows: for any elements from the set A, if the pair  belongs to the relation  and the pair  belongs to the relation , then ,  that means that the relation   can't contain the pair  at the same time with the pair , if the element a is distinct from an element b.

Let's consider the relation ,  , A – set of all real numbers, then condition     means that if the conditions  and are implemented, the condition is implemented too, it means that the relation is antisymmetric.

Let's consider the relation ,  , A – set of all real numbers, then the condition     means that if conditions  and are implemented, the condition is implemented too, that isn't correct, it means that the relation isn't antisymmetric.

4)  – transitive relation, if

. It is read as follows: for any elements from the set A, if the pair  belongs to the relation , the pair  belongs to the relation and the pair  belongs to the relation , it means that if the element a is in the relation with b and the element b is in relation with c, then the element a is in relation with c.

Let's consider the relation ={(a,b)}a is a groupmate of b,  , A – great number of students of technical school, then condition      means that if the student of technical school  a studies in one group with b and student  b studies in one group with с, then student  а studies in one group with с, that is obviously correct, means, the relation   is transitive.

5)  – complete or linear relation, if

    .

It is read as follows: for any element from the set А, if , then the pair  belongs to the relation  ,  that means that for any two different elements a is in the relation with b or the element b is in relation with a .

Let's consider the relation

={(a,b)}a is a groupmate of b,  ,

A – great number of students of technical school, then condition

      means that for any two students of technical school a and b, or student a is the groupmate of the student b or student b is the groupmate of the student a,  that obviously it is not true, it means that the relation  isn't full.

Let's consider the relation ,  ,

 A – set of all real numbers, then condition

      means that for any two various numbers  a and b or the condition is implemented or the condition is implemented, that, truly, means, the relation    is full.

Functions

The relation from the set A into the set B   is called the function from А into В, if

.

The fact that f – function from the set A into the set B is registered as f:   or .

Instead of record  let's use . Such property of the relation is called unambiguity or functionality.

If  then  is an argument and  is a value of function.

Let , then  is called the function range of definition.

Let , then  is called the area of values of function.

If , then the function is called as total and if  – the function is partial.

Function narrowing  on the set  is called ,it is defined as follows: .  is called the function of n  arguments or n – local function.

Properties of the function

Let . Then function f is called injective, if , then the function F is subjective, if

, if it is injective and subjective.

Concept of the set

The concept of the set is one of the main concepts of mathematics. It has no exact definition and has, as a rule, it is explained with the help of examples.

Let's make the following intuitive definition of concept of the set:

Set – a certain set of objects. Objects of the set are called the set elements.

For example. The set of houses in one street, a set of natural numbers, a great number of students of group etc.

Sets are usually designated by capital Latin letters А, В, С, D, X, Y…, set elements are determined by small Latin letters – a, b, c, d, x, y…

To designate the fact that the object x is a set element of A, we use the following symbols:  x А (it is read as: x belongs to А), x А designates that the object x isn't a set element of A (it is read as: x doesn't belong to А).

The set not containing any element is called an empty set (it is designated as Ø).

The sets with elements of which we make a concrete set is called the universal set (it is designated as U).

For example. U – great number of people on the earth, А – students of group E-102.

Sets can be represented by means of circles which are called Euler's circles, the universal set can be designated a rectangle.

Ways of presentation of sets

To establish the set, it is necessary to specify, what elements belong to it. It can be done in the various ways: 1) Enumeration of all elements of the set in braces.

Example: A= {Astana, Atbasar, Karaganda}

2) A characteristic predicate which describes the property of all elements entering into the set. The characteristic predicate is registered after the colon or the symbol « |».

For example. Р(x) = x N x < 8 – characteristic predicate.

M = {x: Р(x)} or M = {x: x N x < 8}. The set M can be represented the enumeration of its elements: M = {1, 2, 3, 4, 5, 6, 7}

For example. В = {x | x – natural number} = {2, 4, 6, 8, …}

If the set consists of a small amount of elements, it is convenient to represent it by enumeration of all elements, if there are a lot of elements or the set has infinite number of elements, it is established by means of the characteristic predicate.

The following numerical sets are known from the school course: N – set of natural numbers, N = {1, 2, 3, 4,…}; Z – set of integers, Z = {…, – 3,  – 2, –1, 0, 1, 2, 3, 4,…}; Q – set of rational numbers,

Bases of logic, logician of statements, logic sheaves, validity tables.

The algebra of logic (Boolean algebra) is the section of the mathematics which appeared in the XIX century thanks to the efforts of English mathematician D. Boulle. Logic, (from Greek’s logos – a word, concept, a reasoning, reason), or Formal logic – a science about laws and operations of the correct thinking.

First the Boolean algebra had no practical value. However in the XX century it found an application in the description of functioning and development of various electronic schemes. Laws and the device of algebra of logic began to be used at the design of various parts of computers (memory, the processor). Though it not unique scope of this science.

The algebra of logic, first, studies methods of establishment of the validity or falsity of difficult logic statements by means of algebraic methods. Secondly, the Boolean algebra does it in such a manner that the difficult logic statement is described by the function which result of calculation can be either truth or lie (1 or 0). Thus arguments of function (simple statements) also can have only two values: 0 or 1.

The simple logic statement is phrases of type «two more than one», «5.8 is an integer». In the first case we have truth and in the second it is lie. The algebra of logic doesn't concern an essence of these statements. If someone decides that the statement «Earth square» is true, the algebra of logic will accept it as the fact. The matter is that the Boolean algebra is engaged in calculations of result of difficult logic statements on the basis of in advance known values of simple statements.

Logic operations. Disjunction, conjunction and denial

In a natural language we use the various unions and other parts of speech. For example: «and», «or», «or», «not», «if», «then». Example of difficult statements: «it has knowledge and skills», «it will arrive to Tuesday, or on Wednesday», «I will play when I will make lessons», «5 isn't equal to 6». How do we solve it?, What were we told: the truth or not? Somehow logically, even somewhere unconsciously, proceeding from the previous life experience, we understand that the truth at the union «i» comes in case of truthfulness of both simple statements. It is necessary to one to become lie and all difficult statement will be false. And here, at a sheaf «or» there should be the truth only one simple statement, and then all expression becomes true.

The Boolean algebra shifted this life experience on the mathematics device, formalized it, and entered rigid rules of receiving unequivocal result. The unions began to be called here as logic operators.

The algebra of logic provides the set of logic operations. However three of them deserve special attention since with their help it is possible to describe all the others, and, therefore, to use less various devices when designing schemes. Such operations are conjunction (And – logic multiplication), disjunction (OR – logic addition) and denial (NOT). Often conjunction designate &, a disjunction – ||, and denial – line over a variable designating the statement. At conjunction the truth of difficult expression arises only in case of the validity of all simple expressions of which the difficult consists. In all other cases difficult expression will be false.

At a disjunction the truth of difficult expression comes at the validity at least one simple expression entering into it or two at once. It happens that difficult expression consists of more than two simple. In this case it is enough, that one idle time was true and then all statement will be true.

Denial is a monadic operation, because it is carried out in the relation to one simple expression or in the relation to the result of difficult expression. As a result of denial the new statement opposite initial turns out. The rules of seniority of logic operations:

1. Denial (inversion) – logic action of the first step.

2. Conjunction – logic action of the second step.

3. A disjunction – logic action of the third step.

4. Implication – logic sequence.

5. Equivalence – equivalence.

If in the logic expression the actions of various steps are used, the first steps are carried at the beginning, then the second and only after that the third step are held. Any deviation from this order should be designated by brackets.

The validity table for function implication (logic following):

А	В	А=>В
0	0	1
0	1	1
1	0	0
1	1	1

The validity table for function equivalence (equivalence):

А	В	А<=>В
0	0	1
0	1	0
1	0	1
1	1	1

Control questions:

1. What is the uniqueness?

2. What is the relation?

3. What is the set?

4. What is the discrete mathematics?

5. Why does the discrete mathematics make the basis of ADP equipment?

6. What is the graph?

7. What does the oriented graph represent?

8. What is the mixed graph?

9. What are the additional characteristics of graphs?

10. What are the methods of submission of graphs?

Topic 3. Main concepts of architecture of the computer.

Review and history of architecture of computers. Logic elements of the computer: logic gates, triggers, counters, registers. Submission of numerical data and notation. Sign representations and representations in an additional code.

Review and history of architecture of computers.

The computer facilities are the most important component of process of calculations and data processing. The first adaptations for calculations were the calculating sticks which are used today in initial classes of many schools for training the account. Such adaptations were used by the dealers and accountants of that time.

Gradually from the elementary adaptations for the account difficult devices were used more and more: abacuses (scores), slide rule, mechanical arithmometer, and electronic computer. Naturally, the productivity and speed of the account of modern computers already surpass the possibilities of the most outstanding counter.

Early adaptations and devices for the account

When the people bothered to keep count by means of bending of fingers, they invented abacuses. The mankind learned to use the elementary calculating adaptations one thousand years ago. The necessity to define the quantity of subjects used in barter was the most demanded. The use of a weight equivalent of a changed subject that didn't demand exact recalculation of quantity of its components was one of the simplest decisions. For these purposes the elementary balance scales, one of the first devices for quantitative determination of weight were used.

Yupana (abacuses of monks) allegedly used numbers of Fibonacci. The principle of equivalence was widely used and in another, familiar for many, the elementary calculating devices of Abacuses or Scores. The quantity of counted-up subjects corresponded to number of the moved bones of this tool.

Beads were rather difficult adaptation for the account; they were applied in the practice of many religions. The believer as counted the number of the said prayers on the grains of beads, and after the pass of a cycle of beads he moved on a separate tail the special grains counters meaning number of counted circles.

Asterisks and gears were heart of mechanical devices for the account. With the invention of cogwheels there were also much more difficult devices of performance of calculations. The Anti Kiter mechanism found at the beginning of the XX century on the place of crash of the antique vessel approximately in the 65th year BC (on other sources in 80 or even to the 87th year BC) even was able to model the movement of planets. However, with antiquity leaving the skills of creation of such devices were forgotten; about one and a half thousand years ago people learned to create again similar mechanisms.

In 1623 Wilhelm Shikard created «Counting hours» — the first mechanical calculator which was able to carry out four arithmetic actions. The device was called «Counting hours» because the operation of the mechanism was based on use of asterisks and gears in real clocks. Practical use is the invention found in hands of friend Shikard, the philosopher and astronomer Johann Kepler.

It was followed by Blaise Pascal's machine («Pascal», 1642) and Gottfried Wilhelm Leibniz.

Approximately in 1820 Charles Xavier Thomas created the first successful, serially let out mechanical calculator – Thomas's Arithmometer which could put, read, multiply and divide. Generally it was based on Leibniz's work. The mechanical calculators considering decimal numbers were used to the 1970th.

Leibniz also described binary notation, the central component of all modern computers. However up to the 1940th, many subsequent developments (including Charles Babbage's machine and even ENIAC, 1945) were based on the decimal system which was more difficult in realization.

John Nipper noticed that multiplication and division of numbers can be executed by addition and subtraction, respectively, logarithms of these numbers. Real numbers can be presented by intervals of length on a ruler, and it laid down in the basis of calculations by means of a slide rule that allowed carrying out the multiplication and division much quicker. Slide rules were used by several generations of engineers and other professionals till to emergence of pocket calculators. Engineers of Apollo program sent the person on the Moon after the execution of all calculations on slide rules, many of which demanded accuracy in 3-4 signs.

For drawing up of the first logarithmic tables of Nipper needed the implementation of all sets of operations of multiplication, and at the same time he developed Nipper’s sticks.

Punched card system

In 1801 Joseph Mary Jakkar developed the weaving loom in which the embroidered pattern was defined by punched cards. A series of cards could be replaced, and change of a pattern didn't demand changes in mechanics of the machine. It was an important milestone in the history of programming.

In 1838 Charles Babbage passed over from the development of the differential machine to design of more difficult analytical machine which principles of programming directly went back to Jakkar's punched cards.

In 1890 the Bureau of Census of the USA used punched cards and mechanisms the sorting (tabulators) developed by Herman Hollerith to process the data flow of ten years' census transferred under the mandate according to the Constitution. The company Hollerith finally became IBM kernel. This corporation developed technology of punched cards in the powerful tool for business data processing and let out the extensive line of the specialized equipment for their record. By 1950 the IBM technology became ubiquitous in the industry and the government. The prevention printed on the majority of cards, «not to turn off, not to braid and not to tear» became the motto of a post-war era.

In many computer solutions the punched card were used to (and after) the end of the 1970th. For example, students of engineering and scientific specialties at many universities around the world could send their program teams in the local computer center in the form of a set of cards, one card for a program line, and then the turn for processing should wait, for compilation and implementation of the program. Subsequently after listing of any results noted by the identifier of the applicant, they were located in a final tray out of the computer center. In many cases these results included only error message listing in syntax of the program, demanding other cycle editing – compilation – execution.

The 1835-1900th: the first programmed machines

Defining feature of «the universal computer» is a programmability that allows the computer to emulate any other calculating system only replacement of the kept sequence of instructions.

In 1835 Charles Babbage described the analytical machine. In «analytical» the principles which have become fundamental to computer facilities are put:

1. automatic performance of operations

2. work on the program entered «under way»

3. need of the special device of memory for data storage (Babbage called it «warehouse»)

It was the project of the computer of general purpose with the application of punched cards as the carrier of entrance data and the program, and also the steam engine as a power source. Use of the gear wheel for performance of mathematical functions was one of key ideas.

The part differential of Babbage machine collected after his death by his son from parts found in laboratory.

The use of punched cards for the machine calculating and printing logarithmic tables with big accuracy (that is for the specialized machine) was his initial idea. Further these ideas were developed for the machine of general purpose – for his «analytical machine».

Before the Second World War mechanical and electric analog computers were considered as the most modern machines, and many considered that as future of computer facilities. Analog computers used advantages of that mathematical properties of the phenomena of small scale – the provision of wheels or electric tension and a current – are similar to mathematics of other physical phenomena, for example such as ballistic trajectories, inertia, a resonance, energy transfer, the inertia moment, etc. They modeled these and other physical phenomena values of electric tension and a current.

British «Colossus»

The British Colossus was used for breaking the German codes during the Second World War. «Colossus» became the first completely electronic computer. A large number of electro vacuum lamps were used, input of information was carried out from a punched tape. «Colossus» could be adjusted on performance of various operations of Boolean logic, but it wasn't full-turing machine. Besides Colossus Mk I, nine more Mk II models were collected. Information on existence of this machine was kept in a secret till 1970th Winston Churchill personally signed the order on car destruction on a part, not exceeding in the size of a human hand. Because of the privacy, «Colossus» isn't mentioned in many works on stories of computers.

American developments

In 1937 Claude Shannon showed that there is a compliance one - to - one between concepts of Boolean logic and some electronic schemes which received the name «logic gates» which are everywhere used now in digital computers. Working in MTI in the main work, it showed that electronic communications and switches could represent expression of Boolean algebra. So the work of A Symbolic Analysis of Relay and Switching Circuits was created on the basis for practical design of digital schemes.

ENIAC carried out ballistic calculations and consumed capacity in 160 kW. ENIAC (Electronic Numerical Integrator and Computer – the Electronic numerical integrator and the calculator) – the first large-scale electronic digital computer which could be reprogrammed for the solution of a full range of tasks, often known as the first electronic computer of general purpose, publicly proved applicability of electronics for large-scale calculations. It became the key moment in development of computers, first of all because of a huge gain in speed of calculations, but also and because of the appeared possibilities for miniaturization. Created under the direction of John Mochli and J. Presper Eckert, this machine was in 1000 of times quicker, than all other machines of that time. Development «ENIAC» lasted from 1943 to 1945. When this project was offered, many researchers were convinced that among thousand fragile electro vacuum lamps many would burn down so often that «ENIAC» too much time would stand idle under repair and by that, it would be almost useless. Nevertheless, by the real machine it was possible to carry out some thousand operations a second within several hours, before the next failure because of the burned-down lamp.

Having processed Eckert and Mochli's ideas, and also, having estimated the restrictions «ENIAC», John von Neumann wrote widely quoted report describing the project of the computer (EDVAC) in which both the program, and data were stored in uniform universal memory. Principles of creation of this machine became known under the name «von Neumann’s architecture» and formed a basis for development of the first rather flexible, universal digital computers.

The first generation of computers with von Neumann's architecture

In 1956 IBM sold for the first time the device for information storage on magnetic disks – RAMAC (Random Access Method of Accounting and Control). It used 50 metal disks in diameter 24 inches, on 100 paths from each party. The device stored to 5 MB of data and coasted for $10 000 for MB. (In 2006, similar devices of data storage – hard disks – cost about $0,001 for Mb.)

The 1950th – the beginning of the 1960th: second generation

The first Soviet serial semi-conductor Computer of the spring and Snow steel were produced from 1964 to 1972. Peak productivity of the Snow Computer made 300 000 operations in a second. Machines were made on the basis of transistors with clock frequency of 5 MHz. In total 39 Computers were produced.

It is considered that the best domestic COMPUTER of the 2nd generation BESM-6 was created in 1966. In architecture BESM-6 the principle of combination of performance of teams (to 14 single-address machine teams could be at different stages of performance) for the first time was widely used. Mechanisms of interruption, protection of memory and other innovative decisions allowed using BESM-6 in a multiprogramming mode and a mode of division of time. The COMPUTER had 128 KB of random access memory on ferrite cores and external memories on magnetic drums and a tape. BESM-6 worked with clock frequency of 10 MHz and records the productivity – about 1 million operations a second. In total 355 COMPUTERS were produced.

The 1960th and further: the third and subsequent generations

Integrated chips contain many hundreds millions transistors. The integrated scheme is a micro miniature chain of a certain functional purpose which by means of special technology takes place on very small silicon (or any another suitable on properties) a plate – a basis. The area of such scheme is 1-3 cm2, but on the functionality the integrated scheme is equivalent to hundreds and thousands transistor elements.

Rapid growth of use of computers began with so-called «the 3rd generation» computers. The beginning of the invention of integrated schemes independent from each other were invented by Nobel Prize winner Jack Kilby and Robert Noyce. Later it led to the microprocessor invention of Tad Hoff (Intel Company).

During the 1960th a certain overlapping of technologies of the 2nd and 3rd generations was observed. At the end of 1975, in Sperry Univac production of cars of the 2nd generation, such as UNIVAC 494 was preceded.

Emergence of microprocessors led to development of microcomputers – small inexpensive computers which the small companies or certain people could own. Microcomputers, representatives of the fourth generation, first of which appeared in the 1970th, became the universal phenomenon in the 1980th and later. Steve Wozniak, one of founders of Apple Computer, became known as the developer of the first mass home computer, and later – the first personal computer. Computers on the basis of microcomputer architecture with the possibilities added from their big colleagues, now dominate in the majority of segments of the market.

Logic elements of the computer: logic gates, triggers, counters, register.

Logic bases of the computer

The mathematical apparatus of algebra of logic as the main notation in the computer is binary in which figures 1 and 0, and values of logic variables too two are used, it is very convenient for the description of how hardware of the computer functions: «1» and «0».

There are two conclusions of this fact:

• The same devices of the computer can be applied to processing and storage as the numerical information presented in binary notation, and logic variables

• At a stage of designing of hardware the algebra of logic allows to simplify considerably the logic functions describing functioning of schemes of the computer, and, therefore, to reduce number of elementary logic elements from which the main knots of the computer consist.

Data and teams in memory of the computer and in registers of the processor are represented in the form of binary sequences of various structure and length.

The logic element of the computer is a part electronic logic schemes which realizes elementary logic function.

Logic elements of computers are electronic schemes And, OR, NOT, And-NOT, OR-NOT and others (called also gates), and also the trigger.

By means of these schemes it is possible to realize any logic function describing operation of devices of the computer. Usually a gate has from two to eight entrances and one or two exits, the unit are coded by higher level of tension, than a zero (or on the contrary).

Each logic element has the symbol which expresses its logic function, but doesn't point what electronic scheme is realized in it. It simplifies the record and understanding of difficult logic schemes.

The work of logic elements is described by means of validity tables. The validity table is a tabular submission of the logic scheme (operation) in which all possible combinations of values of the validity of entrance signals (operands) together with value of the validity of a target signal (result of operation) for each of these combinations are listed.

Scheme AND. The scheme Also realizes the conjunction of two or more logic values. A symbol on block diagrams of the scheme And with two entrances and its table of the validity are presented below.

Unit on the scheme exit AND will be in only case when on all entrances there will be units. When at least on one entrance there is a zero, on an exit also there will be a zero. Operation of conjunction on function charts is designated by a sign «&», being an abbreviated notation of the English word And.

Scheme OR

The scheme OR realizes a disjunction of two or more logic values. A symbol on block diagrams of the scheme OR with two entrances and its table of the validity are presented below.

The value of a disjunction is equal to the unit if the sum of values of operands is more or equal to 1. When at least on one entrance of the scheme OR there is a unit, on its exit also there will be a unit. Communication between an exit of z of this scheme and entrances x and y is described by a ratio: z= x v y (it is read as «x or y»).

Scheme NOT. The scheme NOT (inverter) realizes denial operation. A scheme symbol NOT and its table of the validity are presented below. If there is O on an entrance of the scheme, there is 1 on an exit. When there is 1 on an entrance, there is O on an exit.

Scheme And-NOT. The scheme And – doesn't consist of an element AND and the inverter and carries out denial of result of the scheme of AND.

Conditional meaning of the scheme And-NOT and its table of the validity are presented below.

The scheme OR-NOT. The scheme OR - doesn't consist of an element OR and the inverter and carries out denial of result of the scheme OR. A symbol of the scheme OR-NOT and its table of the validity are presented below.

The trigger is the electronic scheme which is widely applied in registers of the computer to reliable storing of one category of a binary code. The trigger has two steady conditions one of which corresponds to binary unit, and another – to binary zero.

The term the trigger occurs from the English word trigger – a latch, a trigger. To designate this scheme in English the term flip-flop is more often used that in transfer means «clap». This onomatopoeic name of the electronic scheme indicates its ability almost instantly to pass («be thrown») from one electric condition into another and on the contrary.

Submission of numerical data and notation.

Arithmetic bases of the computer

The computer can process numerical, text, graphic, sound and video information.

All these types of information are coded in sequence of electric impulses: there is an impulse (1), there is no impulse (0), i.e. in sequence of zero and units. Such coding of information in the computer is called the binary coding and logic sequences of zero and units – computer language. These figures can be considered as two equiprobable conditions (events). At record of binary figure the choice of one of two possible conditions (by one of two figures) is realized and, therefore, it bears the number of information equal to 1 bit.

It is important that each figure of a machine binary code bears information in 1 bit. Thus, two figures bear information of 2 bits, three categories – 3 bits etc. The number of information is equal in bits to number of figures of a binary machine code. Numbers can be written down in a natural or exponential form. Usual record of numbers, for example, 3, 14, 2001 etc. is called the natural form habitual to us.

The exponential form of numbers is usually used for record or very big, or very small numbers which contain a large number of non-significant zero in a usual natural form (for example, 1000000 = 1E6; 0,000001 = 0,1E-5).

In programming languages and in computer appendices at record of numbers in an exponential form instead of the basis of notation 10 a letter E is written, instead of a comma there is a point, the sign of multiplication isn't put (for example, 1000000 = 1Е6; 0,000001 = 0.1E-5). For example, coding of one symbol needs 1 byte of information. If symbols are considered as possible events, it is possible to calculate, what quantity of various symbols can be coded: N= 2^I = 2⁸ = 256.

Such quantity of symbols is quite enough for submission of text information, including capital and capital letters of the Russian and Latin alphabet, figures, signs, graphic symbols etc. Coding is that to each symbol the unique decimal code from 0 to 255 or a binary code corresponding to it from 00000000 to 11111111 is put in compliance. Thus, the person distinguishes symbols on their tracing, and the computer – on their code.

Notation is a sign system in which numbers are registered by certain rules by means of the symbols of some alphabet called figures.

Transfer of numbers to decimal notation

Transformation of the numbers presented in binary, octal and hexadecimal notations, in decimal to execute quite easily. For this purpose it is necessary to write down number in a full form and to calculate its value.

Transfer of numbers from decimal system in binary, octal and hexadecimal

Transfer of numbers from decimal system into binary, octal and hexadecimal can be carried out in the various ways. Let's consider one of algorithms of transfer of numbers from decimal system into binary, thus it is necessary to consider that algorithms of transfer of integers and proper fractions will differ.

The algorithm of transfer of the whole decimal number in binary will be the following:

1. Consistently to carry out division of initial whole decimal number and received whole private on the system basis (on 2) until we will receive private less than a divider, i.e. less than 2.

2. To receive required binary number for what to write down the received remains in return sequence.

Transfer of numbers from binary notation in octal and hexadecimal and back

To transfer the whole binary number to octal it is necessary to break into groups in three figures, from right to left, and then to transform each group to octal figure. If in the last left group it appear less, than three categories, it is necessary to add it at the left with zero.

Binary triads	000	001	010	011	100	101	110	111
Octal figures	0	1	2	3	4	5	6	8

Transfer procedure is the same, as well as for the octal system, only each hexadecimal figure is represented 4 binary figures, for example: 1–0001; 3–0011; 9–1001; A–1010; D–1101; F–1111

To transfer the numbers from octal and hexadecimal notations in binary it is necessary to transform figures of number to groups of binary numbers. To transfer octal system into binary it is necessary to transform it to group of three bits (triad), and at transformation of hexadecimal number – in group of four categories (tetrode).

Arithmetic operations in item notations

Arithmetic operations in all item notations are carried out on same well-known rules: addition, subtraction, multiplication, division

Sign representations and representations in an additional code.

Representation of integers in the computer.

Integers are the elementary numerical data with which the COMPUTER operates. There are two representations: without sign (only for non-negative integers) and with a sign. It is obvious that it is possible to represent negative numbers only in a sign. Integers in the computer are stored in a format with the fixed comma.

Representation of integers in whole types without sign.

Without signs representation all categories of a cell are taken away under representation of the number. For example, in a byte (8 bits) it is possible to present without signs numbers from 0 to 255. Therefore, if it is known that the numerical size is non-negative, it is more favorable to consider it as the size without signs.

Representation of integers in sign whole types.

For representation with a sign the most senior (left) bit is taken away under a number sign, other categories – under number. If the number is positive, in the sign category 0 is located, if negative, the number 1 is located. For example, in a byte it is possible to present sign numbers from -128 to 127.

True form of number.

Number representation in a habitual form «sign» «size» at which the senior category of a cell is taken away under a sign, and the others – under record of number in binary system is called the true form of binary number. For example, the true form of binary numbers 1001 and-1001 for an 8-digit cell is equal to 00001001 and 10001001 respectively. Positive numbers in the COMPUTER are always represented by means of a true form. The true form of number completely coincides with record of the number in a car cell. The true form of a negative number differs from a true form of the corresponding positive number only to contents of the sign category. But negative integers aren't represented in the COMPUTER by means of a true form, for their representation the so-called additional code is used.

Additional code of number.

The additional code of positive number is equal to a true form of this number. The additional code of a negative number of m is equal to 2^k-|m|, where k - number of categories in a cell. As it was already told, at representation of non-negative numbers in a without signs format all categories of a cell are taken away under number.

Representation of material numbers in the computer.

For representation of material numbers in modern computers the way of representation from a floating comma is accepted. This way of representation relies on the normalized (exponential) record of real numbers. As well as for integers, at representation of real numbers in the computer the binary system is used more often, therefore, previously the decimal number should be translated into binary system.

Control questions:

1.What feature does the machine of Bebbidzha have?

2.When twas the electronic digital computing machine created?

3.When was the transistor invented?

4.What was the element basis of the computer processor of first generation?

5.What was the element basis of the computer processor of second generation?

6.What was the element basis of the computer processor of third generation?

7.What was the element basis of the computer processor of fourth generation?

8.What was the element basis of the computer processor of fifth generation?

9.What are the logical elements of the computer?

10.How many stable conditions does the trigger have?

11.How are data in memory of the computer stored?

12.What are the numeration systems?

13.What can you say about the arithmetical basis of the computer?

14.What is the radix notation?

15.How do the numbers from one numeration system to another transfer?

16.How are arithmetical operations in radix notations executed?

17.Why do bit operations make a basis of digital technique?

18.How is a submission of text and symbolic data in the binary code held?

19.What is the numeration system?

20.What is the difference between nonpositional and positional numeration systems?

21.How does the torn record of multidigit number differ from the contracted?

22.What are the advantages and binary numeral system shortcomings?

23.How is the integral number written in one numeration system transformed into the number in other numeration system?

24.How is the fractional part of number transformed upon the transition from one numeration system to another?

25.What forms of representation of numbers in the computer exist?

26.What is the representation of binary numbers in direct, reverse and additional codes?

27.What can you say about addition, subtraction, multiplication and division of binary numbers?

28.What format are the real numbers represented in computer memories?

Topic 4. The organization of machine

Von Neumann's principles, the control device, instruction sets and types of commands. Input-output and interruptions. Devices of storage of the computer. Memory hierarchy. The base memory and operation organization. Virtual storage. Input-output devices. The review of the modern hardware support.