Arithmetic & Logic Unit

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. Binary Numerals • 01101101b • Bits are numbered from the right b7 b6b5b4b3b2b1b0 • Subscripts represent the place value • bi has place value 2i • Convert to decimal • b7 * 27+b6*26 + b5*25 +b4*24 +b3*23 +b2*22 +b1*21 + b0*20

5. Decimal number is converted to binary by repeated divisions by 2 record the remainders Stop when you reach 0 Concatenate the remainders The order: first remainder is least significant bit 55d = 110111b Decimal to Binary

6. Binary • 3 bits • 000 ;0 • 001 ;1 • 010 ;2 • 011 ;3 • 100 ;4 • 101 ;5 • 110 ;6 • 111 ;7 • 1 bit • 0 ; 0 • 1 ; 1 • 2 bits • 00 ; 0 • 01 ; 1 • 10 ; 2 • 11 ;3

7. 0000 ;0 0001 ;1 0010 ;2 0011 ;3 0100 ;4 0101 ;5 0110 ;6 0111 ;7 1000 ;8 1001 ;9 1010 ;10 (Ah) 1011 ;11 (Bh) 1100 ;12 (Ch) 1101 ;13 (Dh) 1110 ;14 (Eh) 1111 ;15 (Fh) 4 bits and Hex

8. Group bits by fours (from right to left, starting with least significant bits) Add zeros as necessary to complete the last group Convert each group to the equivalent hex digit (look up the previous table) Ex: 0101 1110b = 5Eh Binary to Hex

9. Hex to Binary • Expand each hex digit to the equivalent 4-bit binary form • The leading zeros of leftmost digit can be omitted • 36h = 0011 0110b • (or 110110b)

10. Binary Storage • A pre-arranged storage size is used • typically byte, word, doubleword, or quadword • at least 8 bits will be required • encode the bits Ex: 65d is 0100 0001b

11. all 8 bits used to represent the magnitude of the number Minimum 0 (zero) is coded as 00000000b Maximum 255 is coded as 11111111b Unsigned Byte

12. Singed Byte • two’s complement code is most common • only 7 bits are used for the magnitude • Minimum -128 is coded as 10000000b • Maximum +127 is coded as 01111111b • Zero is 00000000b • Note that we can not read 10000001 as -1.

13. Remember -x’s 2 complement code is the same as 2n-x. Change Sign Rule Flip all the bits Add 1 Negative Numbers

14. Geometric Depiction of Twos Complement Integers

15. The 2’s Complement codes for x and -x add to a power of 2 8-bit code: c+(-c)=28 16-bit code: c+(-c)=216 (-c)=2n-c=[(2n-1)-c]+1 =[(2n-1) + (-c)]+1 Note that (2n-1) is 1111..1b Flip and Plus 1 How Does Change Sign Rule Come From

16. 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

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

18. Can We Read 2’s Complement Code Easily? • Positive ones: yes 00110111b(37h) --->55d • Negative ones: No! 11001001b (C9h) ---> not -49h • by Change Signed rule -55d • double check, the code should be the same as 201d ‘s code 201d = 11001001b since -55+ 256 = 201

19. Conversion Between Lengths • Positive number pack with leading zeros • +18 = 00010010 • +18 = 00000000 00010010 • Negative numbers pack with leading ones • -18 = 11101110 • -18 = 11111111 11101110 • i.e. pack with MSB (sign bit)

20. Numbers can be added in any number base Use the same algorithm you practiced in second grade! Binary Example: c c c 10101 + 0111 11100 Unsigned Addition

21. Hex Addition • Hex Example: c c 2CF03 +435C8 704CB • A carry occurs when the sum is sixteen or greater

22. Unsigned Subtraction • Borrows are worth two • sometimes you need to borrow from afar! b b b b 11000011 - 1101001 01011010

23. A borrow is worth of 16 Example: b b FCF03 -435C9 B993A Hex Subtraction