1. Data Representation, Number Systems and Base Conversions
Computer Orgn & Arch 2011

2. Chapter Outline Numeric data representation systems
Binary, octal, and hexadecimal
Non-numeric data representation
ASCII and EBCDIC, UNICODE
Representation of numbers
Integers, floating point and Binary coded decimal
Number base conversions
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3. 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.
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4. Binary presentation of data
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.
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5. Numeric Data representation
Deals with the representation of numbers in a computer.
In digital computers, the most commonly used number
systems are:
binary
octal
hexadecimal
Number System Radix (base) Digits used
Decimal , 1, .....,9
Binary , 1
Octal , 1, , 7
Hexadecimal , 1, ..., 9, A, B, ...,
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6. There are three Numbers formats in computer world.
Integer or fixed point format
Floating point format
Binary Coded Decimal format (BCD)
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7. Numeric Data representation
Numeric 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 example: 4 bits, is 5 & is -5
Floating-point numbers - numbers with fractional components
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8. Binary Coded Decimal
When data is entered , it must be converted into some binary form before processing can begin. This is called BCD
In BCD, each figure of the number to be coded is represented by its 4 bits binary equivalent
Example 8159=

9. There are two BCD formats; Packed(condensed) and Extended(unpacked)
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10. Formats
Packed, each figure occupies half a byte. Eg 341=
Unpacked, each decimal figure is represented by a byte. Eg 341=
Hardware designed for BCD is more complex than that for binary formats
Advantage, its closer to the alphanumeric codes.

11. Number Base Conversions
Two techniques available:
Repeated division
Bit grouping
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12. 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.
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13. Number Base Conversions (Cont)
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.
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14. Number Base Conversions (Cont)
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 to octal .
to octal
To convert an octal number to binary, for each octal digit, get the equivalent 3-bit binary representation.
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15. 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
Hexadecimal
0,1,2…9
0,1,2…..9 10 A 11 B 12 C 13 D 14 E 15 F
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16. 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.
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17. Hexadecimal - Binary conversions A B C D E F
Hex to binary:
(Bit group repeating)
1D7F16 =
5AB216 =
Binary to hex:
= E30C16
= 98DF16
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18. Number Base Conversions (Cont)
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 to Hex.
To convert a hexadecimal number to binary, for each hexadecimal digit, get the equivalent 4-bit binary representation.
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19. Number Systems - Octal
An octal number can easily be converted to binary by replacing each octal digit with the corresponding group of 3 binary digits.
Octal
Binary
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20. Data representation
Human brain can process large variety of data including:
characters
numbers
images and sounds
touch, 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.
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21. Data Representation
Computers 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.
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22. 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.
4TH.
3RD.
2ND.
1ST.
POSITION
103 = 1000
102 = 100
101 = 10
100 = 1
WEIGHT
For example, = 4* * *10 +6*1
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23. Base 10 number systems
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.
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24. Number Systems - Binary
The 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.
8TH TH TH TH TH RD ND ST POSITION
27= = = = = = = =1 WEIGHT
For example: = 1* *64 + 0*32 + 1*16 +
0*8 + 0*4 + 1*2 +0*1 = 21010
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25. Converting decimal numbers to 8-bit binary numbers
Example: Convert the binary number 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 
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26. Number Systems - Octal
Octal 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 =
To convert to decimal, you can again use a weighted system = ?
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27. 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 zero
Use the remainders in reverse order
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28. Decimal to Base-n Conversions
Decimal to Octal Decimal to Hex
8 | |735
8 |84  2 (*80) |45  F (*160)
8 |10  4 (*81) | 2  D (*161)
8 | 1  2 (*82) | 0  2 (*162)
8 | 0  1 (*83)
Answer: Answer: 2DF16
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29. Computer Orgn & Arch 2011

30. Adding numbers So to add the numbers 0610=01102 and 0710=01112 (answer=1310=11012) we can write out the calculation Decimal Unsigned Binary (carry) (carry)
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31. Multiplying number Decimal Binary
For example, × =          0  0  1  0  1  0  0  1 = 41 (base10)     × 0  0  0  0  0  1  1  0 = (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)   
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32. Rules of Binary Subtraction = = 1, and borrow 1 from the next more significant bit = = 0 For example, =                  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)    
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33. 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)
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34. Alphanumeric Codes There are three main coding methods in use:
The American Standard Code for Information Interchange (ASCII)
Extended Binary Coded Decimal Interchange Code (EBCDIC)
Unicode.
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35. 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.
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36. 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)
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37. ASCII ASCII 7-bit code (128 characters) has an extended 8-bit version
used on PC’s and non-IBM mainframes
Was widely used to transfer data from one computer to another – now being replaced by Unicode
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38. ASCII
ASCII 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.
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39. ASCII Coding Examples An ASCII subset “CAB” = 43414216F
Symbol Code
“CAB” = =
“F1” = =
“3415” = =
*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.
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40. Alphanumeric Codes EBCDIC
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
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41. Alphanumeric Codes
Unicode - recent 16 bit standard - can represent 65 thousand characters, of which 49,000 have been defined, incorporates ASCII-7 as subset.
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42. Both of those are 1 byte per character
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.
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43. Representing Images Two 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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