Two’s Two’s ComplementComplement
Two’s ComplementTwo’s Complement
A system used to represent a A system used to represent a negative number in binarynegative number in binary
Positive numbers start with a 0Positive numbers start with a 0 Negative numbers start with a 1Negative numbers start with a 1
On a cars Mileometer if you On a cars Mileometer if you went backwards this would went backwards this would
happen:happen:
0 0 0 0 0 0 0 32109 9 9 9 9 9 9 99 9 9 9 9 9 9 89 9 9 9 9 9 9 79 9 9 9 9 9 9 6
Two’s Complement is like a Two’s Complement is like a Speedometer…….Speedometer…….
0 0 0 0 0 0 1 1
Positive MSB (Most Significant Bit) indicated by 0
= 3
128 64 32 16 8 4 2 1
Two’s Complement is like a Two’s Complement is like a Speedometer…….Speedometer…….
0 0 0 0 0 0 1 0
Positive MSB indicated by 0
= 2
128 64 32 16 8 4 2 1
Two’s Complement is like a Two’s Complement is like a Speedometer…….Speedometer…….
0 0 0 0 0 0 0 1
Positive MSB indicated by 0
= 1
128 64 32 16 8 4 2 1
Two’s Complement is like a Two’s Complement is like a Speedometer…….Speedometer…….
0 0 0 0 0 0 0 0
Positive MSB indicated by 0
= 0
128 64 32 16 8 4 2 1
Two’s Complement is like a Two’s Complement is like a Speedometer…….Speedometer…….
1 1 1 1 1 1 1 1
Negative MSB indicated by 1
= -1
-128 64 32 16 8 4 2 1
Two’s Complement is like a Two’s Complement is like a Speedometer…….Speedometer…….
1 1 1 1 1 1 1 0
Negative MSB indicated by 1
= -2
-128 64 32 16 8 4 2 1
Two’s Complement is like a Two’s Complement is like a Speedometer…….Speedometer…….
1 1 1 1 1 1 0 1
Negative MSB indicated by 1
= -3
-128 64 32 16 8 4 2 1
How can you convert a How can you convert a negative two’s complement negative two’s complement
number to denary?number to denary?
Example: Convert two’s complement Example: Convert two’s complement 1001011010010110
-128-128 64 64 32 16 32 16 8 8 4 4 22 11
11 0 0 1 0 1 1 0 0 0 1 0 1 1 0
This would be:This would be:
-128 + 16 + 4 + 2 = -106-128 + 16 + 4 + 2 = -106
Example 2Example 2
Convert two’s complement 11011011Convert two’s complement 11011011
-128-128 64 64 32 16 32 16 8 8 4 4 2211
11 1 0 1 1 0 1 1 1 0 1 1 0 1 1
This would be:This would be:-128 + 64 + 16 + 8 + 2 + 1 = -37-128 + 64 + 16 + 8 + 2 + 1 = -37
ActivityActivity
Convert the following numbers to Convert the following numbers to denary:denary:
1.1. Two’s complement 00110011Two’s complement 00110011
2.2. Two’s complement 11001100Two’s complement 11001100
3.3. Two’s complement 10111101Two’s complement 10111101
4.4. Binary (no +\- MSB at the start) Binary (no +\- MSB at the start) 1111010111110101
How can you convert a negative How can you convert a negative denary number into two’s denary number into two’s
complement?complement?Step 1Step 1
pretend negative denary number is pretend negative denary number is positive and convert it into binary e.g. positive and convert it into binary e.g. -52 becomes 52 -52 becomes 52
128128 64 64 32 16 32 16 8 8 4 4 2211
0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0
How can you convert a negative How can you convert a negative denary number into two’s denary number into two’s
complement?complement?Step 2Step 2
Starting from the right, leave all Starting from the right, leave all digits alone, up to and including the digits alone, up to and including the first 1, then change all the other first 1, then change all the other digits around (0 to 1) or (1 to 0)digits around (0 to 1) or (1 to 0)
126 64126 64 32 16 32 16 8 8 4 4 2211
0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0
00110011
Alternative Method:Alternative Method:Create positive binary number as before:Create positive binary number as before:
128128 6464 32 16 32 16 8 4 8 4 2 2 11
0 0 1 1 0 1 0 0 0 0 1 1 0 1 0 0 Next change all 1’s and 0’s around:Next change all 1’s and 0’s around:128128 64 64 32 16 32 16 8 4 2 8 4 2 1 1 1 1 0 0 1 0 1 11 1 0 0 1 0 1 1Then add 1 to the number:Then add 1 to the number:128128 64 64 32 16 8 32 16 8 4 2 4 2 1 1 1 1 0 0 1 0 1 11 1 0 0 1 0 1 1+ + 0 0 0 0 0 0 0 10 0 0 0 0 0 0 1= = 1 1 0 0 1 1 0 01 1 0 0 1 1 0 0
Addition in BinaryAddition in Binary
0 1 0 0 1 1 0 10 1 0 0 1 1 0 1
+ + 0 0 1 1 1 0 1 1 0 0 1 1 1 0 1 1 00010001
111111
1248163264128
77
59
136
Denary
Addition ExerciseAddition Exercise
Add 47 to 159 in binaryAdd 47 to 159 in binary Add 12 to 47 in binaryAdd 12 to 47 in binary
Subtraction in Binary Subtraction in Binary
Rather than subtracting, convert the Rather than subtracting, convert the number that is to be subtracted into number that is to be subtracted into a negative number then add them a negative number then add them together as beforetogether as before
Subtraction in Binary Subtraction in Binary
e.g. 15-12e.g. 15-12
12 = 0000110012 = 00001100
-12 = 11110100-12 = 11110100
15 = 0000111115 = 00001111
0000111100001111
+ + 1111010011110100
(1) 00000011(1) 00000011
The carry of 1 is ignored
Subtraction ExercisesSubtraction Exercises
1.1. Subtract 23 from 123 in binarySubtract 23 from 123 in binary
2.2. Subtract 14 from 32 in binarySubtract 14 from 32 in binary
Rules:Rules:
Positive numbers have a 0 as the MSBPositive numbers have a 0 as the MSB Negative numbers have a 1 as the MSBNegative numbers have a 1 as the MSB Even numbers have 0 as the LSB (Least Even numbers have 0 as the LSB (Least
Significant Bit)Significant Bit) Odd numbers have 1 as the LSBOdd numbers have 1 as the LSB -1 is always represented by a 1 in every -1 is always represented by a 1 in every
bit regardless of number of bitsbit regardless of number of bits
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