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

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

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

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

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

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

Binary Systems7 Analog Signal

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

Binary Systems9 Binary Signal

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

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

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 X -(m-1) x -m

Binary Systems13 Radix Number System  Base – 2 (binary numbers)  0 1  Base – 8 (octal numbers)   Base – 16 (hexadecimal numbers)  A B C D E F

Binary Systems14 Radix Operations  The same as for decimal numbers *

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

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 x 2 = x 2 = x 2 = x 2 =

Binary Systems17 Different Bases

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

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

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

Binary Systems21 9’s and 10’s Complements  9’s complement of  =  9’s complement of  =  10’s complement of  =  10’s complement of  =976579

Binary Systems22 1’s and 2’s Complements  1’s complement of  – =  Simply replace 1’s and 0’s  1’s complement of   2’s complement of  =  Add 1 to 1’s complement  2’s complement of  =

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)

Binary Systems24 Subtraction with Complements of Unsigned  ’s complement of Discard end carry Answer 59815

Binary Systems25 Subtraction with Complements of Unsigned  ’s complement of There is no end carry => -(10’s complement of 40185)

Binary Systems26 Subtraction with Complements of Unsigned  ’s complement of Discard end carry 2^ Answer

Binary Systems27 Subtraction with Complements of Unsigned  ’s complement of There is no end carry => -(2’s complement of ) Answer =

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

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

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 = = = = For 8 bits, we can represent the signed integers –128 to +127 How about for N bits?

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

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

Binary Systems33 Arithmetic Addition  Usually represented by 2’s complement Discard Discard

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

Binary Systems35 Processor-Memory Registers

Binary Systems36 Operations

Binary Systems37 Logic Gates - 1

Binary Systems38 Logic Gates - 2

Binary Systems39 Ranges The gate inputThe gate output

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

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

Binary Systems42 Questions