1. Binary Systems1 DIGITAL LOGIC DESIGN by Dr. Fenghui Yao Tennessee State University Department of Computer Science Nashville, TN

2. Binary Systems2 Digital Systems  They manipulate discrete information (A finite number of elements)  Example discrete sets 10 decimal digits, the 26 letters of alphabet 10 decimal digits, the 26 letters of alphabet  Information is represented in binary form  Examples  Digital telephones, digital television, and digital cameras  The most commonly used one is DIGITAL COMPUTERS

3. Binary Systems3 CENTRAL PROCESSING UNIT Digital Computers Control Unit Arithmetic Logic Unit (ALU) Registers R1 R2 Rn Main Memory DiskKeyboardPrinter I/O Devices Bus

4. Binary Systems4 Binary Signals  It means two-states  1 and 0  true and false  on and off  A single “on/off”, “true/false”, “1/0” is called a bit  Example: Toggle switch

5. Binary Systems5 Byte  Computer memory is organized into groups of eight bits  Each eight bit group is called a byte

6. Binary Systems6 Why Computers Use Binary  They can be represented with a transistor that is relatively easy to fabricate (in silicon)  Millions of them can be put in a tiny chip  Unambiguous signal (Either 1 or 0)  This provides noise immunity

7. Binary Systems7 Analog Signal

8. Binary Systems8 Binary Signal  A voltage below the threshold  off  A voltage above the threshold  on

9. Binary Systems9 Binary Signal

10. Binary Systems10 Noise on Transmission  When the signal is transferred it will pick up noise from the environment

11. Binary Systems11 Recovery  Even when the noise is present the binary values are transmitted without error

12. Binary Systems12 Binary Numbers  A number in a base-r system x = x n-1 x n-2... x 1 x 0. x -1 x -2... X -(m-1) x -m

13. Binary Systems13 Radix Number System  Base – 2 (binary numbers)  0 1  Base – 8 (octal numbers)  0 1 2 3 4 5 6 7  Base – 16 (hexadecimal numbers)  0 1 2 3 4 5 6 7 8 9 A B C D E F

14. Binary Systems14 Radix Operations  The same as for decimal numbers 11001011 +10011101 101101000 11001011 - 10011101 00101110 101 * 110 000 1010 +10100 11110

15. Binary Systems15 Conversion From one radix to another  From decimal to binary

16. Binary Systems16 Conversion From one radix to another  From decimal to base-r  Separate the number into an integer part and a fraction part  For the integer part Divide the number and all successive quotients by r Divide the number and all successive quotients by r Accumulate the remainders Accumulate the remainders 165 23 3 0 423423 0.6875 x 2 = 1 + 0.3750 0.3750 x 2 = 0 + 0.7500 0.7500 x 2 = 1 + 0.5000 0.5000 x 2 = 1 + 0.0000

17. Binary Systems17 Different Bases

18. Binary Systems18 Conversion From one radix to another  From binary to octal  Divide into groups of 3 bits  Example 11001101001000.1011011 = 31510.554 11001101001000.1011011 = 31510.554  From octal to binary  Replace each octal digit with three bits  Example 75643.5704 = 111101110100011.101111000100 75643.5704 = 111101110100011.101111000100

19. Binary Systems19 Conversion From one radix to another  From binary to hexadecimal  Divide into groups of 4 bits  Example 11001101001000.1011011 = 3348.B6 11001101001000.1011011 = 3348.B6  From hexadecimal to binary  Replace each digit with four bits bits  Example 7BA3.BC4 = 111101110100011.101111000100 7BA3.BC4 = 111101110100011.101111000100

20. Binary Systems20 Complements  They are used to simplify the subtraction operation  Two types (for each base-r system)  Diminishing radix complement (r-1 complement)  Radix complement (r complement) For n-digit number N r-1 complement r complement

21. Binary Systems21 9’s and 10’s Complements  9’s complement of 674653  999999-674653 = 325346  9’s complement of 023421  999999-023421 = 976578  10’s complement of 674653  325346+1 = 325347  10’s complement of 023421  976578+1=976579

22. Binary Systems22 1’s and 2’s Complements  1’s complement of 10111001  11111111 – 10111001 = 01000110  Simply replace 1’s and 0’s  1’s complement of 10100010  01011101  2’s complement of 10111001  01000110 + 1 = 01000111  Add 1 to 1’s complement  2’s complement of 10100010  01011101 + 1 = 01011110

23. Binary Systems23 Subtraction with Complements of Unsigned  M – N  Add M to r’s complement of N Sum = M+(r n – N) = M – N+ r n Sum = M+(r n – N) = M – N+ r n  If M > N, Sum will have an end carry r n, discard it  If M<N, Sum will not have an end carry and Sum = r n – (N – M) (r’s complement of N – M) Sum = r n – (N – M) (r’s complement of N – M) So M – N = – (r’s complement of Sum) So M – N = – (r’s complement of Sum)

24. Binary Systems24 Subtraction with Complements of Unsigned  65438 - 5623 65438 10’s complement of 05623 +94377 159815 Discard end carry 10 5 -100000 Answer 59815

25. Binary Systems25 Subtraction with Complements of Unsigned  5623 - 65438 05623 10’s complement of 65438 +34562 40185 There is no end carry => -(10’s complement of 40185) -59815

26. Binary Systems26 Subtraction with Complements of Unsigned  10110010 - 10011111 10110010 2’s complement of 10011111 +01100001 100010011 Discard end carry 2^8 -100000000 Answer 000010011

27. Binary Systems27 Subtraction with Complements of Unsigned  10011111 -10110010 10011111 2’s complement of 10110010 +01001110 11101101 There is no end carry => -(2’s complement of 11101101) Answer = -00010011

28. Binary Systems28 Signed Binary Numbers  Unsigned representation can be used for positive integers  How about negative integers?  Everything must be represented in binary numbers  Computers cannot use – or + signs

29. Binary Systems29 Negative Binary Numbers  Three different systems have been used  Signed magnitude  One’s complement  Two’s complement NOTE: For negative numbers the sign bit is always 1, and for positive numbers it is 0 in these three systems

30. Binary Systems30 Signed Magnitude  The leftmost bit is the sign bit (0 is + and 1 is - ) and the remaining bits hold the absolute magnitude of the number  Examples -47 = 1 0 1 0 1 1 1 1 -47 = 1 0 1 0 1 1 1 1 47 = 0 0 1 0 1 1 1 1 47 = 0 0 1 0 1 1 1 1 For 8 bits, we can represent the signed integers –128 to +127 How about for N bits?

31. Binary Systems31 One’s complement  Replace each 1 by 0 and each 0 by 1  Example (-6)  First represent 6 in binary format (00000110)  Then replace (11111001)

32. Binary Systems32 Two’s complement  Find one’s complement  Add 1  Example (-6)  First represent 6 in binary format (00000110)  One’s complement (11111001)  Two’s complement (11111010)

33. Binary Systems33 Arithmetic Addition  Usually represented by 2’s complement + 5 00000101 +11 00001011 +16 00010000 - 5 11111011 +11 00001011 +6 100000110 Discard + 5 00000101 -11 11110101 -6 11111010 - 5 11111011 -11 11110101 -16 111110000 Discard

34. Binary Systems34 Registers  They can hold a groups of binary data  Data can be transferred from one register to another

35. Binary Systems35 Processor-Memory Registers

36. Binary Systems36 Operations

37. Binary Systems37 Logic Gates - 1

38. Binary Systems38 Logic Gates - 2

39. Binary Systems39 Ranges The gate inputThe gate output

40. Binary Systems40 Study Problems  Course Book Chapter – 1 Problems  1 – 2  1 – 7  1 – 8  1 – 20  1 – 34  1 – 35  1 – 36

41. Binary Systems41 Sneak Preview  Next time  ASSIGNMENT Will be given Will be given  QUIZ……. Expect a question from each one of the following Expect a question from each one of the following  Convert decimal to any base  Convert between binary, octal, and hexadecimal  Binary add, subtract, and multiply  Negative numbers

42. Binary Systems42 Questions