computer orgn & arch 20111 data representation, number systems and base conversions

Computer Orgn & Arch 2011 1

Data Representation, Number Systems and Base

Conversions


Chapter Outline

Numeric data representation systemsBinary, octal, and hexadecimal

Non-numeric data representation ASCII and EBCDIC, UNICODE

Representation of numbersIntegers, floating point and Binary coded decimal

Number base conversions


Number systems and computers

The BINARY number system is a positional number system with similar properties to the decimal system.

However, we often need to convert binary to a number system we are familiar with (decimal) or a system that is easier for humans to use.

Computer programs and data are often represented using octal and hexadecimal number systems because they are a short hand way of representing binary numbers.


Sometimes binary numbers need to be converted to Hexadecimal (hex) numbers which reduces a long string of binary digits to a few hexadecimal characters. This makes it easier to remember and to work with the numbers.

Binary presentation of data


Numeric Data representation Deals with the representation of numbers in a

computer.In digital computers, the most commonly used

numbersystems are:

• binary• octal• hexadecimal

Number System Radix (base) Digits used

Decimal 10 0, 1, .....,9

Binary 2 0, 1

Octal 8 0, 1, ......, 7

Hexadecimal 16 0, 1, ..., 9, A, B, ...,

There are three Numbers formats in computer world.

Integer or fixed point formatFloating point formatBinary Coded Decimal format (BCD)



Numeric Data representationNumeric data can be• Integers - fixed-point numbers (no fractional

part) Unsigned integers. the standard binary encoding already given. Supports only positive values

Signed Magnitude integers. This is another way or representing negative integers. Involves treating the most significant (left most) bit in the word as a sign bit .Positive-0, negative -1. example: 4 bits, 0101 is 5 &1101 is -5

• Floating-point numbers - numbers with fractional components

Binary Coded DecimalWhen data is entered , it must be converted into some binary form before processing can begin. This is called BCDIn BCD, each figure of the number to be coded is represented by its 4 bits binary equivalentExample 8159=1000 0001 0101 1001

There are two BCD formats; Packed(condensed) and Extended(unpacked)


FormatsPacked, each figure occupies half a byte. Eg 341= 0011 0100 0001Unpacked, each decimal figure is represented by a byte. Eg 341=00000011 00000100 00000001Hardware designed for BCD is more complex than that for binary formatsAdvantage, its closer to the alphanumeric codes.


Number Base Conversions

Two techniques available:

Repeated division Bit grouping


Number Base Conversions (Cont)

Repeated Division Technique

To convert a number in base to the equivalent number in base , divide the given number by base using base arithmetic.



Bit Grouping Technique

Employed for conversion between octal and binary and for the conversion between hexadecimal and binary.Each hexadecimal digit corresponds to a group of four binary bits.

Each octal digit corresponds to a group of three binary bits.



Binary to Octal and Octal to Binary To convert a binary number to octal, simply split up the binary number into 3-bit groups starting from the least significant bit and get the corresponding octal digits.

Qn . Convert 1001101 to octal . 1001101.0111 to octal

To convert an octal number to binary, for each octal digit, get the equivalent 3-bit binary representation.


Number Systems - Hexadecimal

The hexadecimal system is a base-16 system. It contains the digits 0 to 9 and the letters A to F. The letters A to F represent the decimal numbers 10 to 15.Decimal Hexadecim

al

0,1,2…9 0,1,2…..9

10 A

11 B

12 C

13 D

14 E

15 F


Number Systems - Hexadecimal

Conversion to binary is done the same way as octal to binary, but binary digits are organized into groups of 4.

Conversion from binary to hexadecimal involves breaking the bits into groups of 4 and replacing them with the hexadecimal equivalent.


HEX BINARY 0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001 A 1010 B 1011 C 1100 D 1101 E 1110 F 1111

Hex to binary:(Bit group repeating)

1D7F16 = 0001 1101 0111 11112

5AB216 = 0101 1010 1011 00102

Binary to hex:

1110 0011 0000 11002 = E30C16

1001 1000 1101 11112= 98DF16

Hexadecimal - Binary conversions



Binary to hex and hex to binary To convert a binary number to hexadecimal, simply split up the binary number into 4-bit groups starting from the least significant bit and get the corresponding hexadecimal digits.

Qn. Convert 1001101 to Hex.

To convert a hexadecimal number to binary, for each hexadecimal digit, get the equivalent 4-bit binary representation.


Octal 0 1 2 3 4 5 6 7

Binary 000 001 010 011 100 101 110 111

Number Systems - OctalAn octal number can easily be converted to binary by replacing each octal digit with the corresponding group of 3 binary digits.


Data representationHuman brain can process large variety of data including:

charactersnumbersimages and soundstouch, smell and taste

Current technology limits data that computer can efficiently manipulate to numeric data. Therefore, ALL data - no matter how complex - must be represented (encoded) in the computers memory in numeric form.


Data RepresentationComputers represent data using binary numbers because:

It is easy to represent binary numbers as electrical states or signals which can be processed by two-state (on-off) electrical devices (eg transistors)

. Standardization; organizations have created standard data encoding methods for communication among computer systems and their components.


POSITION

WEIGHT

4TH. 3RD. 2ND. 1ST.

103 = 1000 102 = 100 101 = 10 100 = 1

For example, 491610 = 4*1000 + 9*100 + 1*10 +6*1

Number Systems - Decimal

The decimal system is a base-10 system. There are 10 distinct digits (0 to 9) used to represent any quantity. For an n-digit number, the value that each digit represents depends on its weight or position. The weights are based on powers of 10.


Base 10 uses the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 which are combined to represent all possible numeric values. The decimal number system is based on powers of 10. Each column position of a value, from right to left, is multiplied by the number 10, which is the base number, raised to a power, which is the exponent. The power that 10 is raised to depends on its position to the left of the decimal point.

Base 10 number systems


8TH 7TH 6TH 5TH 4TH 3RD 2ND 1ST POSITION

27=128 26=64 25=32 24=16 23=8 22=4 21=2 20=1 WEIGHT

For example: 110100102 = 1*128 + 1*64 + 0*32 + 1*16 + 0*8 + 0*4 + 1*2 +0*1 = 21010

Number Systems - BinaryThe binary system is a base-2 system. There are 2 distinct digits (0 and 1) to represent any quantity.

For an n-digit number, the value that each digit represents depends on its weight or position. The weights are based on powers of

2.


Converting decimal numbers to 8-bit binary numbers

Example: Convert the binary number 01110000 to a decimal number.

Note: Work from right to left. Remember that anything raised to the 0 power is 1. Therefore 20 = 1

0 x 20 = 0 0 x 21 = 0 0 x 22 = 0 0 x 23 = 0 1 x 24 = 16 1 x 25 = 32 1 x 26 = 64 + 0 x 27= 0

––––––––––– 112


Number Systems - OctalOctal and hexadecimal systems provide a shorthand way to deal with the long strings of 1’s and 0’s in binary.Octal is a base-8 system using the digits 0 to 7. To convert to decimal, you can again use a weighted system eg. 75128 = 7*83 + 5*82 + 1*81 + 2*80 = 391410

To convert to decimal, you can again use a weighted system.

3216 = ?


Decimal to Base-n Conversions

To convert from decimal to a different number base such as octal, binary or hexadecimal involves repeated division by that number base.

Keep dividing until the quotient is zeroUse the remainders in reverse order


Decimal to Base-n Conversions

Decimal to Octal Decimal to Hex

8 |674 16 |735 8 |84 2 (*80) 16 |45 F (*160)

8 |10 4 (*81) 16 | 2 D (*161) 8 | 1 2 (*82) 16 | 0 2 (*162) 8 | 0 1 (*83) 0 0Answer: 12428 Answer: 2DF16

Adding numbers

So to add the numbers 0610=01102 and 0710=01112 (answer=1310=11012) we can write

out the calculation Decimal

Unsigned Binary 1 (carry)

06 +07 13

110 (carry) 0110 +0111 1101


For example,00101001 × 00000110 = 11110110

0 0 1 0 1 0 0 1 = 41 (base10)

× 0 0 0 0 0 1 1 0 = 6 (base 10)

0 0 0 0 0 0 0 0

0 0 1 0 1 0 0 1

0 0 1 0 1 0 0 1

0 0 1 1 1 1 0 1 1 0 = 246 (base 10)


Multiplying number Decimal Binary

Rules of Binary Subtraction0 - 0 = 0

0 - 1 = 1, and borrow 1 from the next more significant bit 1 - 0 = 1 1 - 1 = 0

For example,00100101 - 00010001 = 00010100

0

borrows

0 0 1 10 0 1 0 1 =37 (base10)

- 0 0 0 1 0 0 0 1 =17 (base 10)

0 0 0 1 0 1 0 0 =20 (base10)



Non-numeric Data

Representation

Alphanumeric codes are used when the computing device has to handle letters and special symbols as well as decimal digits.

Two dominant coding schemes are:

• ASCII (American Standard Code for Information Interchange)

• EBCDIC (Extended Binary Coded decimal Interchange Code)


Alphanumeric Codes

There are three main coding methods in use:

1. The American Standard Code for Information Interchange (ASCII)

2. Extended Binary Coded Decimal Interchange Code (EBCDIC)

3. Unicode.


Alphanumeric Data

Alphanumeric (character) data such as names and addresses are represented by assigning a unique binary code or sequence of bits to represent each character. As each character is entered from a keyboard (or other input device) it is converted into its binary code.


Alphanumeric Data

Character code sets contain two classes of codes:Printable (normal characters)

Non-printable ie. characters used as control codes. For example:

CTRL G CTRL Z .Most computers use ASCII codes to represent text, which makes it possible to transfer data from one computer to another. ASCII (128 characters)

http://www.webopedia.com/TERM/C/computer.html

http://www.webopedia.com/TERM/T/text.html

http://www.webopedia.com/TERM/D/data.html


ASCII

ASCII 7-bit code (128 characters)has an extended 8-bit version used on PC’s and non-IBM mainframesWas widely used to transfer data from one computer to another – now being replaced by Unicode


ASCIIASCII is a code for representing English characters as numbers, with each letter assigned a number from 0 to 127.

Each character has a unique pattern of eight binary digits assigned to represent it.

http://www.webopedia.com/TERM/C/character.html


ASCII Coding ExamplesAn ASCII subset

A 41 B 42 C 43 D 44 E 45 F 46 0 30 1 31 2 32 3 33 4 34 5 35 6 36 7 37

Symbol Code“CAB” = 43414216

= 0100 0011 0100 0001 0100 00102

“F1” = 463116

= 0100 0110 0011 00012

“3415” = 3334313516

= 0011 0011 0011 0100 0011 0001 0011 01012

*Note that this is a text string and no arithmetic may be done on it. A postcode is a good example of storing numbers as text.


Alphanumeric CodesEBCDIC

an 8-bit code (256 characters)used on mainframe IBM machines.The binary code for text as well as communications and printer control from IBM

Both ASCII and EBCDIC are inadequate for representing all international characters. eg Chinese characters


Alphanumeric Codes

Unicode - recent 16 bit standard - can represent 65 thousand characters, of which 49,000 have been defined, incorporates ASCII-7 as subset.

Both of those are 1 byte per character. ( ASCII & EBCDIC)

Unicode is a coding where characters require 2 bytes per character.

Windows can use unicode, but it is not required.It is useful if extra characters not supported in the 1 byte character sets are needed in the same document.



Representing ImagesTwo popular techniques used

Bitmap techniques . image composed of pixels is known as a bitmapped image Vector techniques. Vector graphics is the use of geometrical primitives such as points, lines, curves, and shapes or polygon(s), which are all based on mathematical equations, to represent images in computer graphics.

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