Lecture 01
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The document discusses various methods for representing numeric data in a computer system, including binary, decimal, fixed-point, and floating-point representations. It describes word length in bits and bytes and how numbers are stored in memory in big-endian and little-endian formats. Signed number representations like sign-magnitude, one's complement, and two's complement are also summarized. Various decimal coding schemes such as BCD, ASCII, excess-three, and two-out-of-five are defined.
Lecture 01
- 1. Start where you are. Use what you have. Do what you can. -Arthur Ashe
- 3. Reference Books Computer Architecture and Organization Hayes J.P., McGraw-Hill. Computer organization and design: The hardware/software interface Patterson D.A., Hennessy J.L., Morgan Kaufmann.
- 4. Lecture - 01
- 5. The Basic Information Types Instruction Data Nonnumerical data Numbers Fixed-point Binary Decimal Floating-point Binary Decimal Information
- 6. Word Length Bit / Bits Name Descriptions 1 Bit Logic variable, flag. 8 Byte Smallest addressable memory item, Binary-coded decimal digit pair. 16 Halfword Short fixed-point number. Short address, Short instruction. 32 Word Fixed or floating point number, Memory address, Instruction. 64 Double word Long instruction, Double-precision floating point number. Some Instruction formats of the Motorola 680X0 microprocessor series
- 7. Byte Storage Byte 0Byte 1Byte 2Byte 3 07152331 Least significant bit Most significant bit Indexing convention for the bits and bytes of a word
- 8. Big-Endian Storage Method Suppose a word Wi is represented as Bi, 3 Bi, 2 Bi, 1 Bi, 0 Most significant byte Bi, 3 of Wi is assigned the lowest address and the least significant byte Bi, 0 is assigned the highest address. It is so named because it assigns highest address to byte 0 of a word. Byte 0,3 Byte 0,2 Byte 0,1 Byte 0,0 Byte 1,3 Byte 1,2 Byte 1,1 Byte 1,0 00 01 …000 …001 …002 …003 …004 …005 …006 …007
- 9. Little-Endian Storage Method Suppose a word Wi is represented as Bi, 0 Bi, 1 Bi, 2 Bi, 3 Most significant byte Bi, 3 of Wi is assigned the highest address and the least significant byte Bi, 0 is assigned the lowest address. It is so named because it assigns lowest address to byte 0 of a word. Byte 0,0 Byte 0,1 Byte 0,2 Byte 0,3 Byte 1,0 Byte 1,1 Byte 1,2 Byte 1,3 00 01 …000 …001 …002 …003 …004 …005 …006 …007
- 10. Tags It is a technique of determining the type of a word. This is done by associating with each information word a group of bits, called a tag, that identifies the word’s type. Advantages: It simplifies instruction specifications. Tag inspection permits the hardware to check for software errors. Disadvantages: They increase memory size. Add system hardware costs without increasing computing performance.
- 11. Tags 04751 Parity Check bit Tagged-word format of the B6500/7500 Series Information BitsTags
- 12. Factors to select a number representation The number types to be represented; for example, integers or real numbers. The range of values (number magnitudes) likely to be encountered. The precision of numbers, which refers to the maximum accuracy of the representation. The cost of the hardware required to store and process the numbers.
- 13. Signed Numbers 3 types of representations of a signed number: Sign magnitude +75=01001011 -75=11001011 1’s complement +75= 01001011 -75 = 10110100 2’s complement +75= 01001011 -75 = 10110101
- 14. Signed Numbers Sign magnitude number employ the positional notation for magnitude and simply change the sign bit to represent + or -. In both complement code xn-1 retains the role as sign bit, but remaining bits no longer form a simple positional code when the number is negative. The advantage of the 2’scomplement code is that subtraction can be performed by logical complementation and addition only. 2’s complement addition and subtraction can be implemented by a simple adder for unsigned numbers. Multiplication and division are more difficult to implement of 2’s complement code.
- 15. Signed Numbers Decimal Representation Sign Magnitude 1’s Complement 2’s Complement +7 0111 0111 0111 +6 0110 0110 0110 +5 0101 0101 0101 +4 0100 0100 0100 +3 0011 0011 0011 +2 0010 0010 0010 +1 0001 0001 0001 +0 0000 0000 0000 -0 1000 1111 0000 -1 1001 1110 1111 -2 1010 1101 1110 -3 1011 1100 1101 -4 1100 1011 1100 -5 1101 1010 1011 -6 1110 1001 1010 -7 1111 1000 1001
- 16. Decimal Codes BCD (Binary coded decimal) In BCD format each digit di of a decimal number is denoted by a 4-bit equivalent bi,3bi,2bi,1bi,0. BCD is a weighted (positional) number code where each bi,j has the weight 10i 2j . ASCII The 8-bit ASCII code represents the 10 decimal digits by a 4-bit BCD field and the remaining 4-bits of the ASCII word have no numerical significance.
- 17. Decimal Codes Excess-Three Code The excess-three code can be formed by adding 00112 to the corresponding BCD number. The advantage of the excess-three code is that it may be processed using the same logic used for binary codes. If two excess-three numbers are added like binary numbers, the required decimal carry is automatically generated from the high-order bits. The sum must be corrected by adding +3. 1000 = 5 + 1100 = 9 Carry 1 0100 Binary sum + 0011 Correction 0111 = 4 Excess-three sum
- 18. Decimal Codes Two-out-of-Five Each decimal digit is represented by a 5-bit sequence containing two 1s and three 0s. There are exactly 10 distinct sequence of this type. It is single-error detecting code. It uses 5 rather than 4 bits per decimal digit.
- 19. Decimal Codes Decimal Digit BCD ASCII Excess-three Two-out-of-five 0 0000 00110000 0011 11000 1 0001 00110001 0100 00011 2 0010 00110010 0101 00101 3 0011 00110011 0110 00110 4 0100 00110101 0111 01001 5 0101 00110101 1000 01010 6 0110 00110110 1001 01100 7 0111 00110111 1010 10001 8 1000 00111000 1011 10010 9 1001 00111001 1100 10100