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Data and representation of computer systems
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Data Representation Computer Organization and Assembly Language
(^) Introduction (^) Numbering Systems (^) Binary & Hexadecimal Numbers (^) Base Conversions (^) Integer Storage Sizes (^) Binary and Hexadecimal Addition (^) Binary and Hexadecimal subtraction (^) Carry and Overflow (^) Character Storage
Data Representation Computer Organization and Assembly Language
(^) Computers only deal with binary data (0s and 1s), hence all data manipulated by computers must be represented in binary format. (^) Machine instructions manipulate many different forms of data: (^) Numbers: (^) Integers: 33, +128, - (^) Real numbers: 1.33, +9.55609, -6.76E12, +4.33E- (^) Alphanumeric characters (letters, numbers, signs, control characters): examples: A, a, c, 1 ,3, ", +, Ctrl, Shift, etc. (^) Images (still or moving): Usually represented by numbers representing the Red, Green and Blue (RGB) colors of each pixel in an image, (^) Sounds: Numbers representing sound amplitudes sampled at a certain rate (usually 20kHz). (^) So in general we have two major data types that need to be represented in computers; numbers and characters.
Data Representation Computer Organization and Assembly Language
(^) Each digit (bit) is either 1 or 0 (^) Each bit represents a power of 2 (^) Every binary number is a sum of powers of 2
Data Representation Computer Organization and Assembly Language
(^) Weighted positional notation shows how to calculate the decimal value of each binary bit: Decimal = ( d n-
n - ) ( d n-
n - ) ... ( d 1
1 ) ( d 0
0 ) d = binary digit (^) binary 10101001 = decimal 169: (1 2 7 ) + (1 2 5 ) + (1 2 3 ) + (1 2 0 ) = 128+32+8+1=
Data Representation Computer Organization and Assembly Language
(^) Start with a binary representation of all 0’s (^) Determine the highest possible power of two that is less or equal to the number. (^) Put a 1 in the bit position corresponding to the highest power of two found above. (^) Subtract the highest power of two found above from the number. (^) Repeat the process for the remaining number
Data Representation Computer Organization and Assembly Language
(^) Example: Converting 76d to Binary (^) The highest power of 2 less or equal to 76 is 64, hence the seventh (MSB) bit is 1 (^) Subtracting 64 from 76 we get 12. (^) The highest power of 2 less or equal to 12 is 8, hence the fourth bit position is 1 (^) We subtract 8 from 12 and get 4. (^) The highest power of 2 less or equal to 4 is 4, hence the third bit position is 1 (^) Subtracting 4 from 4 yield a zero, hence all the left bits are set to 0 to yield the final answer
Data Representation Computer Organization and Assembly Language
(^) Each hexadecimal digit corresponds to 4 binary bits. (^) Example: Translate the binary integer 000101101010011110010100 to hexadecimal
Data Representation Computer Organization and Assembly Language
(^) Each Hexadecimal digit can be replaced by its 4-bit binary number to form the binary equivalent. M1021.swf
Data Representation Computer Organization and Assembly Language
Decimal 422 = 1A6 hexadecimal stop when quotient is zero least significant digit most significant digit (^) Repeatedly divide the decimal integer by 16. Each remainder is a hex digit in the translated value:
Data Representation Computer Organization and Assembly Language
What is the largest unsigned integer that may be stored in 20 bits? Standard sizes:
Data Representation Computer Organization and Assembly Language
(^) Divide the sum of two digits by the number base (16). The quotient becomes the carry value, and the remainder is the sum digit. 36 28 28 6A 42 45 58 4B 78 6D 80 B 1 1 21 / 16 = 1, remainder 5 Important skill: Programmers frequently add and subtract the addresses of variables and instructions.
Data Representation Computer Organization and Assembly Language
(^) Several ways to represent a signed number (^) Sign-Magnitude (^) 1's complement (^) 2's complement (^) Divide the range of values into 2 equal parts (^) First part corresponds to the positive numbers (≥ 0) (^) Second part correspond to the negative numbers (< 0) (^) Focus will be on the 2's complement representation (^) Has many advantages over other representations (^) Used widely in processors to represent signed integers
Data Representation Computer Organization and Assembly Language
Sum of an integer and its 2's complement must be zero: 00100100 + 11011100 = 00000000 (8-bit sum) Ignore Carry The easiest way to obtain the 2's complement of a binary number is by starting at the LSB, leaving all the 0s unchanged, look for the first occurrence of a 1. Leave this 1 unchanged and complement all the bits after it. starting value 00100100 = + step1: reverse the bits (1's complement) 11011011 step 2: add 1 to the value from step 1 + 1 sum = 2's complement representation 11011100 = -
Data Representation Computer Organization and Assembly Language
Highest bit indicates the sign. 1 = negative, 0 = positive If highest digit of a hexadecimal is > 7, the value is negative Examples: 8A and C5 are negative bytes A21F and 9D03 are negative words B1C42A00 is a negative double-word
