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Computer Organization and ASSEMBLY LANGUAGE

Understand integer representation and twos complement in computer arithmetic. Learn about sign-magnitude and twos complement representations, calculations, and hardware requirements for addition and subtraction operations.

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Computer Organization and ASSEMBLY LANGUAGE

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  1. Computer Organization and ASSEMBLY LANGUAGE Lecture 5 & 6 Computer Arithmetic Integer Representation, Integer Arithmetic Course Instructor: Engr. Aisha Danish

  2. Arithmetic & Logic Unit • Does the calculations • Everything else in the computer is there to service this unit • Handles integers • May handle floating point (real) numbers • May be separate FPU (maths co-processor) • May be on chip separate FPU (486DX +)

  3. ALU Inputs and Outputs

  4. Integer Representation • Only have 0 & 1 to represent everything • Positive numbers stored in binary • e.g. 41=00101001 • No minus sign • No period • Sign-Magnitude • Two’s compliment

  5. Sign-Magnitude • The simplest form of representation that employs a sign bit is the sign-magnitude representation • In an n-bit word, the rightmost n-1 bits hold the magnitude of the integer • +18 = 00010010 • -18 = 10010010

  6. Sign-Magnitude • Problems • Need to consider both sign and magnitude in arithmetic • Two representations of zero (+0 and -0) • Inconvenient because it is slightly more difficult to test for 0 (an operation performed frequently on computers) than if there were a single representation • Sign-magnitude representation is rarely used in implementing the integer portion of the ALU • Instead, the most common scheme is twos complement representation

  7. Two’s Compliment • Like sign magnitude, twos complement representation uses the most significant bit as a sign bit, making it easy to test whether an integer is positive or negative • It differs from the use of the sign-magnitude representation in the way that the other bits are interpreted

  8. Characteristics of Twos Complement Representation and Arithmetic

  9. Two’s Compliment • +3 = 00000011 • +2 = 00000010 • +1 = 00000001 • +0 = 00000000 • -1 = 11111111 • -2 = 11111110 • -3 = 11111101

  10. Alternative Representations for 4-Bit Integers

  11. Benefits • One representation of zero • Arithmetic works easily • Negating is fairly easy • 3 = 00000011 • Boolean complement gives 11111100 • Add 1 to LSB 11111101

  12. Negation Special Case 1 • 0 = 00000000 • Bitwise not 11111111 • Add 1 to LSB +1 • Result 1 00000000 • Overflow is ignored, so: • - 0 = 0

  13. Negation Special Case 2 • 128 = 10000000 • bitwise not 01111111 • Add 1 to LSB +1 • Result 10000000 • So: • -128 = 128 • Monitor MSB (sign bit) • It should change during negation

  14. Range of Numbers • 8 bit 2s compliment • +127 = 01111111 = 27 -1 • -128 = 10000000 = -27 • 16 bit 2s compliment • +32767 = 011111111 11111111 = 215 - 1 • -32768 = 100000000 00000000 = -215

  15. Conversion Between Lengths • Positive number pack with leading zeros • +18 = 00010010 • +18 = 00000000 00010010 • Negative numbers pack with leading ones • -18 = 10010010 • -18 = 11111111 10010010 • i.e. pack with MSB (sign bit)

  16. Addition and Subtraction • Normal binary addition • Monitor sign bit for overflow • Take twos compliment of substahend and add to minuend • i.e. a - b = a + (-b) • So we only need addition and complement circuits

  17. Hardware for Addition and Subtraction

  18. Problems • Represent the following decimal numbers in both binary sign/magnitude and twos complement using 16 bits: • +512 • -29 • Represent the following twos complement values in decimal: • 1101011 • 0101101.

  19. Problem • Assume numbers are represented in 8-bit twos complement representation. Show the calculation of the following: • 6+13 • -6+13 • 6-13 • -6-13

  20. Problem • Find the following differences using twos complement arithmetic:

  21. Multiplication • Complex • Work out partial product for each digit • Take care with place value (column) • Add partial products

  22. Multiplication Example • 1011 Multiplicand (11 dec) • x 1101 Multiplier (13 dec) • 1011 Partial products • 0000 Note: if multiplier bit is 1 copy • 1011 multiplicand (place value) • 1011 otherwise zero • 10001111 Product (143 dec) • Note: need double length result

  23. Division • More complex than multiplication • Negative numbers are really bad! • Based on long division

  24. Division of Unsigned Binary Integers Quotient 00001101 Divisor 1011 10010011 Dividend 1011 001110 Partial Remainders 1011 001111 1011 Remainder 100

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