1ECE 274 - Digital LogicLecture 12 Chapter 5.3 – 5.4 Signed Numbers Fast Adders2ECE 274 - Digital LogicRepresenting Signed Numbers Decimal Represented by “+” or “-” sign Binary Unsigned binary number All bits determine magnitude of number Signed binary number n-1 bits determine magnitude of number Sign denoted by left most bit 0 indicates positive number 1 indicates negative number523- 100 + 3bn-1b1b0magnitudeUnsigned number...bn-2bn-1b1b0Signed numbermagnitudeSign0 denotes “+”1 denotes “-”...3ECE 274 - Digital LogicSigned Number Representations2423221012120 How do we represent the magnitude? Positive Binary Numbers Represented by position numbering systems previously discussed Negative Binary Numbers Sign-and-Magnitude Representation 1’s Complement 2’s Complement1012= (1 * 22) + (0 * 21) + (1 * 20)= (1 * 4) + (0 * 2) + (1 * 1)= 5104ECE 274 - Digital LogicSign-and-Magnitude Representation of Binary Numbers Sign-and-Magnitude Representation Decimal representation Magnitude of a number is expressed the same way Symbol distinguishes positive or negative Binary can adopt same scheme n-1 bits denote magnitude Leftmost bit denotes positive (0) or negative value (1) Intuitive representation for humans Not well suited for computers+532 -532magnitude is the same (532)symbols distinguish between positive and negative value01012= 51011012= -5105ECE 274 - Digital LogicAddition of Sign-and-Magnitude Numbers Addition of sign-and-magnitude integers If signs are same Add magnitude values Copy sign If signs are different Subtract smaller magnitude from larger magnitude Copy sign of larger magnitude Circuitry required Adder Subtractor Compare +0101 (510)0010 (210)0111 (710)+1011 (-310)1011 (-310)1110 (-610)-0111 (710)1010 (210)0101 (510)6ECE 274 - Digital Logic1’s Complement Representation of Binary Numbers 1’s Complement Representation K = n-bit negative number P = corresponding positive number K = (2n-1) - PWhat we are actually doing is just complementing each of the bits (including sign bit)n = 4K= (24–1) -PConvert +5 (01012) to a negative number, using 1’s complementK = 1510-PK = 11112-PK = 11112- 01012K = 10102K= (24–1) -PConvert +3 (00112) to a negative number,using 1’s complementK = 1510-PK = 11112-PK = 11112- 00112K = 110027ECE 274 - Digital LogicAddition of 1’s Complement Numbers Addition of 1’s complement integers Consider four possible combination of signs Top two are correct Bottom two are incorrect Carry produced by sign bit If carry produced by sign bit, add it to the LSB New result correct Circuitry required Adder Detection of carry from sign bit Drawback - signed addition may require twice as long as unsigned addition+0101 (+510)0010 (+210)0111 (+710)+1010 (-510)0010 (+210)1100 (-310)+0101 (+510)1101 (-210)10010 (-1310)+1010 (-510)1101 (-210)10111 (-810)+10011 (+310)+11000 (-710)xx8ECE 274 - Digital Logic2’s Complement Representation of Binary Numbers 2’s Complement Representation K = n-bit negative number P = corresponding positive number K = 2n–P Notice value plus it’s 2’s complement result in 0 (ignoring carry)n = 4K= 24-PConvert +5 (01012) to a negative number, using 2’s complementK = 1610-PK = 100002-PK = 100002-01012K = 10112K= 24-PConvert +3 (00112) to a negative number, using 2’s complementK = 1610-PK = 100002-PK = 100002-00112K = 11012+0101 (+510)1011 (-510)1 0000+0011 (+310)1101 (-510)1 00009ECE 274 - Digital Logic2’s Complement Representation of Signed Binary Numbers -- Shortcut 2’s Complement conversion shortcut Given signed number, B = bn-1, bn-2, ..., b1, b0 Start from right to left, copy all bits that are 0 and the first bit that is 1 Complement remaining bitsConvert +6 (01102) to a negative number, using 2’s complement shortcut01100101Start from right, copy all 0’s until we hit first 1Copy first bit that is 1Complement remaining bitsResult:Convert +180 (0 1011 01002) to a negative number, using 2’s complement shortcut0 1011 0100Result:Start from right, copy all 0’s until we hit first 1Copy first bit that is 1001Complement remaining bits00101110ECE 274 - Digital Logic Comparison of Signed Number Representations Interpretation of 4-bit signed integersb3b2b1b00111011001010100001100100001000010001001101010111100110111101111Sign and Magnitude+7+6+5+4+3+2+1+0-0-1-2-3-4-5-6-71’s complement2’s complement+7+6+5+4+3+2+1+0-7-6-5-4-3-2-1-0+7+6+5+4+3+2+1+0-8-7-6-5-4-3-2-1• Multiple zero representations• Sign-and-magnitude• 1’s Complement• Single zero representation• 2’s Complement• Represent numbers from -7 to + 7• Sign-and-magnitude• 1’s Complement• Represent numbers from -8 to +7• 2’s Complement11ECE 274 - Digital LogicAddition of 2’s Complement Numbers Addition of 2’s complement integers Consider four possible combination of signs All are correct If carry produced by sign bit, ignore it Circuitry required Adder Highly suitable for implementation of addition operations+0101 (+510)0010 (+210)0111 (+710)+1011 (-510)0010 (+210)1100 (-310)+0101 (+510)1110 (-210)10011 (-310)+1011 (-510)1110 (-210)11001 (-710)ignore carry out bit12ECE 274 - Digital LogicSubtraction Using Addition What about subtraction of 2’s complement integers? We can use an adder to perform subtraction Negate subtrahend, add to the minuend A – B = A + (-B) A – (-B) = A + (+B)D = X – Y minuendsubtrahend13 Consider four possible combination of signs All are correct If carry produced by sign bit, ignore it Using 2’s complement representation of negative numbers Only 1 adder required to perform addition and subtractionECE 274 - Digital LogicSubtraction Using Addition – Does it really work? +0101 (+510)1110 (-210)10011 (+310)= 5 – 2= 5 + (-2)= 3 +1011 (-510)1110 (-210)11001 (-710)= (-5) – 2= (-5) + (-2)= -7+0101 (+510)0010 (+210)0111 (+710)= 5 – (-2)= 5 + 2= 7+1011 (-510)0010 (+210)1100 (-310)= (-5) – (-2)= (-5) + 2= - 3ignoreignore14 1’s Complement Representation K = (2n-1) – P Shortcut: obtained by complementing each bit 2’s Complement Representation K = 2n–P K = 2n–P + 1 -1 K