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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 = 1Å6; 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= 2I = 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
Octal figures


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 2k-|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?



Date: 2015-12-24; view: 55

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