ECE 2110: Introduction to DigitalSystems
Signed Number Conversions
2
Previous class Summary
Signed-magnitude, two’s complement,one’s complement
Different for negatives numbers
Representations of positive numbers areSAME.
0 may have different representations.
Sign bit: 0 for positive, 1 for negative
3
Signed Decimal to Hex conversion
Step 1: Know what format you are converting to!!! You mustknow if you are converting the signed decimal to SM, 1scomplement, or 2s complement. Convert +34, -20 to all three formats.
Step 2: Ignore the sign, convert the magnitude to binary. 34 = 2 x 16 + 2 = 2216 = 001000102 20 = 1 x 16 + 4 = 1416 = 000101002
Step 3 (positive decimal number): If the decimal number waspositive, then you are finished no matter what the format is! +34 as an 8 bit SM number is 2216 = 001000102 +34 as an 8 bit 1s complement number is 2216 = 001000102 +34 as an 8 bit 2s complement number is 2216 = 001000102
4
Signed Decimal to Hex conversion (cont)
Step 3 (negative decimal number): Need to do more if decimalnumber was negative. If converting to SM format, set Sign bit to One: 20 = 000101002, then -20 => 100101002 = 9416If converting to 1s complement, complement each bit. 20 = 000101002, then -20 => 111010112 = EB16If converting to 2s complement, complement and add one. 20 = 000101002 , then -20=> 111010112 + 1 = 111011002 = EC16
5
Signed Decimal to Hex conversion (cont)
Final results:
Compare! +34 as an 8 bit SM number is 2216 = 001000102 +34 as an 8 bit 1s complement number is 2216 = 001000102 +34 as an 8 bit 2s complement number is 2216 = 001000102
-20 as an 8 bit SM number is 9416 = 100101002 -20 as an 8 bit 1s complement number is EB16 = 111010112 -20 as an 8 bit 2s complement number is EC16 = 111011002
6
Sign extension?
Pad with 0s for positive numbers and 1sfor negative numbers
7
A common Question?
Given a hex number, how do I know if it is in 2’s complement or1’s complement; is it already in 2’s complement or do I have putit in 2’s complement, etc.
8
Answer: We cant!
If I write a HEX number, I will ask for a decimalrepresentation if you INTERPRET the encoding as aparticular method (i.e, either 2’s complement, 1’scomplement, signed magnitude).
A Hex or binary number BY ITSELF can representANYTHING (unsigned number, signed number, charactercode, etc). You MUST HAVE additional information thattells you what the encoding of the bits mean.
For example: FE16 or 111111102
9
Example Conversions
FE16 as an 8 bit unsigned integer = 254FE16 as an 8 bit signed magnitude integer = -126FE16 as an 8 bit ones complement integer = - 1FE16 as an 8 bit twos complement integer = -27F16 as an 8 bit unsigned integer = 1277F16 as an 8 bit signed magnitude integer = +1277F16 as an 8 bit ones complement integer = +1277F16 as an 8 bit twos complement integer = +127To do hex to signed decimal conversion, we need todetermine sign (Step 1), determine Magnitude (step 2),combine sign and magnitude (Step 3)
10
Hex to Signed Decimal Conversion Rules
Given a Hex number, and you are told to convert to a signed integer(either as signed magnitude, 1s complement, 2s complement)
•STEP 1: Determine the sign! If the Most Significant Bit iszero, the sign is positive. If the MSB is one, the sign isnegative. This is true for ALL THREE representations: SM, 1scomplement, 2s complement. F016 = 111100002 (MSB is ‘1’), so sign of result is ‘-’ 6416 = 011001002 (MSB is ‘0’), so sign of result is ‘+’.
If the Most Significant Hex Digit is > 7, then MSB = ‘1’ !!!(e.g., 8,9,A,B,C,D,E,F => MSB = ‘1’ !!!)
11
Hex to Signed Decimal (cont)
•STEP 2 (positive sign): If the sign is POSITIVE, then justconvert the hex value to decimal. The representation is thesame for SM, 1s complement, 2s complement. 6416 is a positive number, decimal value is 6 x 16 + 4 = 100.
•STEP 3 : Just combine the sign and magnitude to get the result. 6416 as an 8 bit signed magnitude integer = +1006416 as an 8 bit ones complement integer = +1006416 as an 8 bit twos complement integer = +100
•Note: Final answer is +100 regardless of whether encoding wasSM, 1s complement, or 2s complement.
12
Hex to Signed Decimal (cont)
•STEP 2 (negative sign): If the sign is Negative, then need tocompute the magnitude of the number.
• We will use the trick that - (-N) = + Ni.e. Take the negative of a negative number will give you the positivenumber. In this case the number will be the magnitude.If the number is SM format, set Sign bit to Zero: F016 = 111100002 => 011100002 = 7016 = 112If the number is 1s complement, complement each bit. F016 = 111100002 => 000011112 = 0F16 = 15If the number is 2s complement, complement and add one. F016 = 111100002 => 000011112 + 1 = 000100002 = 1016 = 16
13
Hex to Signed Decimal (cont)
•STEP 3 (also): Just combine the sign and magnitude to get the result.
F016 as 8 bit Signed magnitude number is -112 F016 as 8 bit ones complement number is -15 F016 as 8 bit twos complement number is -16
14
Next…
Quiz#1
Signed Addition/Subtraction