Encoding Systems: Combining Bits to form Bytes


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1 Encoding Systems: Combining Bits to form Bytes Alphanumeric characters are represented in computer storage by combining strings of bits to form unique bit configuration for each character, also called byte Characters are translated into bytes according to an encoding system (EBCDIC, ASCII) Parity checking ensures that data transmission is complete and accurate Numbering System
2 Numbering Systems and Computers The two primary numbering systems used in conjunction with computer are binary and decimal Decimal is translated into binary on input and binary is translated into decimal on output The hexadecimal numbering system is used in reading and reviewing binary output in the form of a memory dump Numbering System 2
3 Numbering Systems (cont.) Name Base Digits Hexadecimal >.. 9 and A.. F Decimal.. 9 Octal Binary 2, Numbering System 3
4 Numbering Systems(cont.) Converting Decimal to Binary Number Remainder (37) =() 2 Numbering System 4
5 Numbering Systems (cont.) Converting Binary to Decimal () 2 =X2 + X2 +X2 2 + x2 3 + x2 4 +x2 5 = = ( 37) Numbering System 5
6 Numbering Systems(cont.) Converting Decimal to Octal Number Remainder (37) =(45) 8 Numbering System 6
7 Numbering Systems(cont.) Converting Binary to Octal ( 37) = (45) 8 = () 2 Notice the special relation between binary and octal, if we take every three bits we will find that: () 2 = x2 +x2 +x2 2 = (5) 8 () 2 = x2 +x2 +x2 2 = (4) 8 Numbering System 7
8 Numbering Systems (cont.) Converting Decimal to Hexadecimal Number ( 37) = (25) 6 Remainder Numbering System 8
9 6 3 Numbering Systems (cont.) Converting Decimal to Hexadecimal Number Remainder 6 5 (F) (3) = (F) 6 Numbering System 9
10 Numbering Systems(cont.) Converting Binary to Hexadecimal ( 37) = (25) 6 = () 2 Notice the special relation between binary and Hexadecimal, if we take every four bits we will find that: () 2 = x2 +x2 +x2 2 + x2 = (5) 6 () 2 = x2 +x2 +x2 2 + x2 = (2) 6 Numbering System
11 Computer Representation of Information Basic unit of information is the Bit or Binary digit. With a single bit, we can represent two distinct values and. With two bits, we can represent four distinct values:,,, and. In general, with m bits, we can represent 2m distinct values. A byte is a grouping of 8 bits. A word is a grouping of either 6, 32, or 64 bits, depending on the computer system. A word is typically 32 bits on most systems, a half word is 6 bits, and a double word is 64 bits. Numbering System
12 Representation of Characters Characters are typically represented by single byte. The ASCII standard (American Standard Code for Information Interchange) defines unique binary codes for English letters, digits, and special symbols. The ASCII standard defines only 28 characters, with decimal codes to 27. A single byte can represent up to 256 characters. The remaining 28 characters, coded 28 to 255, can be used for a second language. For example, the ASCII standard can be extended for Arabic letters. Some languages, such as Chinese, Japanese, and Korean, have more than 256 characters. Characters can be encoded using 2 bytes, rather than a single byte. Numbering System 2
13 Representation of Characters Binary Decimal Character Representation Code space 32! A 65 B 66 Z 9 [ 9 a 97 b 98 Numbering System 3
14 Representation of Integers Unsigned Integers Represented using a fixed number of bits or bytes With m bits, we can have 2 m distinct values:,, 2,, 2 m. Using byte to represent an integer, m = 8, 2 8 distinct values:,, 2,, 2 8 = 255. Using 2 bytes to represent an integer, m = 6, 2 6 distinct values:,, 2,, 2 6 = Using 4 bytes to represent an integer, m = 32, 2 32 distinct values:,, 2,, 2 32 = Signed Integers Unsigned representation of integers does not allow negative numbers. We need to divide the 2 m distinct values into positive and negative values. Two methods are used to represent signed integers: SignMagnitude Representation Two s Complement Representation Numbering System 4
15 SignMagnitude Representation Signed integers can be represented as a sign and a magnitude. Using m bits to represent a signed integer: the most significant bit is used to represent the sign. sign = is positive. sign = is negative. The least significant m bits are used to represent the magnitude. With m bits for the magnitude, the range of the magnitude is from to 2 m. For an mbit signed integer in signmagnitude representation, the range of values is from 2 m to + 2 m. When m = 8, the range of values is from 27 to +27. Examples using 8bit signmagnitude representation (m = 8) +29 = 2 29 = 2 + = 2 = 2 There are two different representations of (+ and ). Addition of a positive and a negative number must be treated as a subtraction problem. The sign and the magnitude of both operands must be checked to obtain the correct result Numbering System 5
16 Two s Complement Representation Signmagnitude representation is natural for humans, but not for computers. A better representation for computers is the 2 s complement representation. 2 s complement of an mbit number N = (Bitwise complement of N) + Bitwise complement of N is called the s complement of N. Examples of 8bit numbers and their 2 s complement representation: + = 2 = 2 s complement of 2 = 2 + = = 2 29 = 2 s complement of 2 = 2 + = = 2 s complement of 2 = 2 + = 2 For an mbit signed integer in 2 s complement notation, the range of integer values is from 2 m to 2 m. When m = 8, the range is from 28 to 27. When m = 6, the range is from to When m = 32, the range is from to Numbering System 6
17 Comparison Table for 8bit Number Representations 8bit Binary Decimal equivalent when Representation Unsigned SignMag 2 s Comp When 8bit binary representation is interpreted as unsigned: Decimal equivalent is counted from = 2 to 255 = 2 When interpreted as signmagnitude: Positive values are counted from + = 2 to +27 = 2 Negative values are counted from 27 = 2 to = 2 When interpreted as 2 s complement: Positive values are counted from = 2 = to +27 = 2 Negative values are counted from 28 = 2 to = 2 Numbering System 7
18 Binary Addition a b c carry sum Examples: carry carry sum +2 sum carry carry sum 8 sum + 27 Numbering System 8
19 Binary Addition The addition of 3 bits a + b + c produces a sum bit and a carry bit. Signed Integers are represented in 2 s complement notation. Addition of integers in 2 s complement notation results in a signed integer. Although a carry out is produced in the 2nd and 4th computations, it is ignored. Numbering System 9
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