Digital Arithmetic Digital Arithmetic Operations and Circuits Binary Arithmetic Digital circuits are frequently used for arithmetic operations Fundamental arithmetic operations on binary numbers and digital circuits which perform arithmetic operations will be examined Binary numbers are added like decimal numbers In decimal when numbers sum more than 9 a carry results In binary when numbers sum more than 1 a carry takes place Addition is the basic arithmetic operation used by digital devices to perform subtraction multiplication and division If the numbers are unsigned and positive add them as follow 1 0 1 1 1 0 1 1 1 1 with carry of 0 with carry of 1 with carry of 1 For example 011 011 010 110 110 001 6 125 base 10 Adding Unsigned Numbers Examples X 190 10111110 Y 141 10001101 X Y 331 101001011 X 127 1111111 Y 63 111111 190 10111110 X Y X 170 10101010 Y 85 1010101 255 11111111 X Y Adding Unsigned Numbers Examples X 190 10111110 Y 141 10001101 X Y 331 101001011 X 127 1111111 Y 63 111111 190 10111110 X Y X 170 10101010 Y 85 1010101 255 11111111 X Y Representing Signed Numbers Since it is only possible to show magnitude with a binary number the sign or is shown by adding an extra sign bit A sign bit of 0 indicates a positive number A sign bit of 1 indicates a negative number The 2 s complement system is the most commonly used way to represent signed numbers Magnitude bits 7 0 1 1 1 0 0 0 0 0 1 1 1 1 1 7 1 0 0 1 8 1 0 0 0 Sign bit Examples of 2 s complement representation Representing Signed Numbers Converting to 2 s Complement In order to change a binary number to 2 s complement it must first be changed to 1 s complement To convert to 1 s complement simply change each bit to its complement opposite To convert 1 s complement to 2 s complement add 1 to the 1 s complement A positive number is true binary with 0 in the sign bit A negative number is in 2 s complement form with 1 in the sign bit A binary number can be negated by taking the 2 s complement of it For example 9 01001 sign bit 0 indicating 2 s complement of 9 01001 10110 10110 1 10111 9 This is The sign BIT When the sign bit is zero Positive number If the sign bit is set Negative number Remember 2 s complement is just a conventional way of representing signed numbers in digital arithmetic Don t ask why 2 s Complement Representation Assuming N 1 bits representing a 2 s Complement that is representing the number with N bits and one bit is dedicated to indicate the sign Largest positive number will be 2N 1 Smallest signed number largest negative number will be 2N Total numbers including zero that can be represented will be 2N 1 For example Assume 3 1 bits Largest pos number will be 0111 7 Smallest number will be 1000 8 Remember 1111 1 Zero is represented by 0000 Zero For example Assume 6 1 bits Largest pos number will be Smallest signed number will be Zero is represented by 0 111 111 63 1 000 000 64 0 000 000 0 More Examples Integer Integer 2 s Complement Signed 2 s Complement Signed Unsigned 5 5 0000 0101 4 4 0000 0100 3 3 0000 0011 7 0111 6 0110 5 0101 4 0100 3 0011 2 2 0000 0010 2 0010 1 1 0000 0001 1 0001 0 0 0000 0000 0 0000 1 255 1111 1111 1 1111 2 254 1111 1110 2 1110 3 253 1111 1101 3 1101 4 1100 4 252 1111 1100 5 1011 5 251 1111 1011 6 1010 7 1001 8 1000 Remember Always know how many bits are provided 2 s Complement Representation Integer 2 s Complement Signed 7 0111 6 0110 5 0101 4 0100 3 0011 2 0010 1 0001 0 0000 1 1111 2 1110 3 1101 4 1100 5 1011 6 1010 7 1001 8 1000 Arithmetic Operations using 2 s Complement Inverting A positive number to a negative number A negative number to a positive number Either case just take the 2 s complement Adding A and B A B assuming N bits represent the magnitude and one bit is dedicated as the sign bit Magnitude Overflow max unsigned number that can be represented using 8 bits is 255 The carry will be ignored X 180 10110100 Y 85 01010101 X Y 265 100001001 X 6 0110 Y 3 1101 X Y 3 1 0011 X 4 0100 Y 7 1001 X Y 3 1101 X 3 1101 Y 6 1010 X Y 9 1 0111 X 7 0 111 Y 7 0 111 X Y 14 1 110 Subtracting B from A A B Just take the 2 s Complement of B Add A and B A B Note the value is negative Overflow max signed number that can be represented using 4 bits is 8 Overflow max signed number that can be represented using 4 bits is 7 Arithmetic Operations using 2 s Complement Inverting A positive number to a negative number A negative number to a positive number Adding A and B A B assuming N bits represent the magnitude and one bit is dedicated as the sign bit The carry will be ignored X 180 10110100 Y 85 01010101 X Y 265 100001001 X 6 0110 Y 3 1101 X Y 3 1 0011 X 4 0100 Y 7 1001 X Y 3 1101 X 3 1101 Y 6 1010 X Y 9 1 0111 X 7 0 111 Y 7 0 111 X Y 14 1 110 Subtracting B from A A B Either case just take the 2 s complement Magnitude Overflow max unsigned number that can be represented using 8 bits is 255 Just take the 2 s Complement of B Add A and B A B NOTE When a Positive and a Negative number are added together the overflow will be a signed overflow and it is ok The overflow can be discarded when it is sign overflow max 16 X 4 11100 Y 9 10111 X Y 13 1 10011 Note the value is negative Overflow max signed number that can be represented using 4 bits is 8 Overflow max signed number that can be represented using 4 bits is 7 BCD Addition BDC numbers They are between 0 and 9 Hence each decimal number is represented by 4 bits Add each number between 0 9 individually 0 1 1 0 BCD for 6 0 1 1 1 BCD for 7 1 1 0 1 13 but invalid 0 1 1 0 Add 6 to correct 0 0 0 1 0 0 1 1 BDC for 13 BCD Addition Another Example BDC numbers They are between 0 and 9 Hence each decimal number is represented by 4 bits Add each number between 0 9 individually 1 0 1 0 1 1 0 0 1 BCD …