Integer represention
- 1. Computer Arithmetic KAMRAN RIASAT … 18 MUHAMMAD HASNAIN … 29 MUHAMMAD SAIF ULLAH … 30 USAMA ALI … 38 Chapter 09
- 2. Arithmetic & Logic Unit (ALU) Part of the computer that actually performs arithmetic and logical operations on data All of the other elements of the computer system are there mainly to bring data into the ALU for it to process and then to take the results back out Based on the use of simple digital logic devices that can store binary digits and perform simple Boolean logic operations
- 5. Integer Representation In the binary number system arbitrary numbers can be represented with: ▪ The digits zero and one ▪ The minus sign (for negative numbers) ▪ The period, or radix point (for numbers with a fractional component) For purposes of computer storage and processing we do not have the benefit of special symbols for the minus sign and radix point Only binary digits (0,1) may be used to represent numbers
- 6. Sign-Magnitude Representation Left most bit is sign bit 0 means positive 1 means negative Drawbacks: ▪ Need to consider both sign and magnitude in arithmetic ▪ Two representations of zero (+0 and -0) Rarely used in implementing the integer portion of the ALU
- 7. Twos Complement Representation Uses the most significant bit as a sign bit Differs from sign-magnitude representation in the way that the other bits are interpreted
- 8. Table 9.2 Alternative Representations for 4-Bit Integers
- 9. Benefits One representation of zero Arithmetic works easily (see later) Negating is fairly easy ▪ 3 = 00000011 ▪ Boolean complement gives 11111100 ▪ Add 1 to LSB 11111101
- 10. Range Extension Range of numbers that can be expressed is extended by increasing the bit length In sign-magnitude notation this is accomplished by moving the sign bit to the new leftmost position and fill in with zeros This procedure will not work for twos complement negative integers ▪Rule is to move the sign bit to the new leftmost position and fill in with copies of the sign bit ▪For positive numbers, fill in with zeros, and for negative numbers, fill in with ones ▪This is called sign extension
- 11. Fixed-Point Representation The radix point (binary point) is fixed and assumed to be to the right of the rightmost digit Programmer can use the same representation for binary fractions by scaling the numbers so that the binary point is implicitly positioned at some other location
- 13. Negation(Twos Complement) Twos complement operation ▪ Take the Boolean complement of each bit of the integer (including the sign bit) ▪ Treating the result as an unsigned binary integer, add 1 The negative of the negative of that number is itself: +18 = 00010010 (twos complement) bitwise complement = 11101101 + 1 11101110 = -18 -18 = 11101110 (twos complement) bitwise complement = 00010001 + 1 00010010 = +18
- 14. Addition
- 15. OVERFLOW RULE: If two numbers are added, and they are both positive or both negative, then overflow occurs if and only if the result has the opposite sign.
- 16. Negation Special Case 1 0 = 00000000 (twos complement) Bitwise complement = 11111111 Add 1 to LSB + 1 Result 100000000 Overflow is ignored, so: - 0 = 0
- 17. Negation Special Case 2 -128 = 10000000 (twos complement) Bitwise complement = 01111111 Add 1 to LSB + 1 Result 10000000 So: -(-128) = -128 X Monitor MSB (sign bit) It should change during negation
- 18. SUBTRACTION RULE: Rules: 1. To subtract one number (subtrahend) from another (minuend) 2. Take the twos complement (negation) of the subtrahend 3. And add it to the minuend.
- 19. Subtraction
- 20. Hardware for Addition and Subtraction Figure 9.6 Block diagram of Hardware for Addition and Subtraction
- 21. MULTIPLICATION
- 22. Multiplication Complex Work out partial product for each digit Take care with place value (column) Add partial product
- 23. Multiplication Example 1011 Multiplicand (11 dec) x 1101 Multiplier (13 dec) 1011 Partial products 0000x Note: if multiplier bit is 1 copy 1011xx Multiplicand (place value) 1011xxx Otherwise zero 10001111 Product (143 dec) Note: need double length result`
- 25. Flowchart for Unsigned Binary Multiplication
- 26. Multiplying Negative Numbers This does not work! Solution 1 ▪ Convert to positive if required ▪ Multiply as above ▪ If signs were different, negate answer Solution 2 ▪ Booth’s algorithm
- 28. Example of Booth’s Algorithm
- 29. Examples Using Booth’s Algorithm
- 30. Division
- 32. Example of Restoring Twos Complement Division
- 33. Thank You