Computer Arithmetic_Computer_Architecture.ppt
- 1. William Stallings Computer Organization and Architecture Chapter 9 Computer Arithmetic
- 2. Key Points Two principle concerns for computer arithmetic • the way in which numbers are represented • the algorithms used for the basic arithmetic operations These two considerations apply both to integer and floating point arithmetic
- 3. part of the computer that actually performs arithmetic and logical operations on data. Handles integers May handle floating point (real) numbers Arithmetic & Logic Unit (ALU)
- 4. ALU Inputs and Outputs
- 5. 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 complement
- 6. Sign-Magnitude Left most bit is sign bit 0 means positive 1 means negative +18 = 00010010 -18 = 10010010 Problems Need to consider both sign and magnitude in arithmetic Two representations of zero (+0 and -0)
- 7. Two’s Complement +3 = 00000011 +2 = 00000010 +1 = 00000001 +0 = 00000000 -1 = 11111111 -2 = 11111110 -3 = 11111101
- 8. Benefits of 2’s complement One representation of zero Arithmetic works easily (see later) Negating is fairly easy 3 = 00000011 Boolean complement gives 11111100 Add 1 to LSB 11111101
- 9. Geometric Depiction of Twos Complement Integers
- 10. Negation Special Case 1 0 = 00000000 Bitwise not 11111111 Add 1 to LSB +1 Result 1 00000000 Overflow is ignored, so: - 0 = 0
- 11. Negation Special Case 2 -128 = 10000000 bitwise not 01111111 Add 1 to LSB +1 Result 10000000 So: -(-128) = -128 X Monitor MSB (sign bit) It should change during negation
- 12. Range of Numbers 8 bit 2’s complement +127 = 01111111 = 2 -1 -128 = 10000000 = -2 16 bit 2’s complement +32767 = 011111111 11111111 = 2 - 1 -32768 = 100000000 00000000 = - 2
- 13. Conversion Between Lengths Positive number pack with leading zeros +18 = 00010010(sign magnitude, 8bits) +18 = 00000000 00010010(sign magnitude,16bits) Negative numbers pack with leading ones -18 = 10010010(sign magnitude, 8bits) -18 = 10000000 10010010(sign magnitude,16bits) i.e. pack with MSB (sign bit)
- 14. Addition and Subtraction Normal binary addition Monitor sign bit for overflow Take two’s complement of subtrahend and add to minuend i.e. a - b = a + (-b) So we only need addition and complement circuits
- 15. Hardware for Addition and Subtraction
- 16. Multiplication Complex Work out partial product for each digit Take care with place value (column) Add partial products
- 17. Multiplication Example 1011 Multiplicand (11 dec) x 1101 Multiplier (13 dec) 1011 Partial products 0000 1011 1011 10001111 Product (143 dec) Note: need double length result if multiplier bit is 1 copy otherwise zero
- 18. Flowchart for Unsigned Binary Multiplication
- 21. Multiplying Negative Numbers Solution 1 Convert to positive if required Multiply as above If signs were different, negate answer Solution 2 Booth’s algorithm
- 22. Division More complex than multiplication Negative numbers are really bad! Based on long division