Unsigned numbers involve only non-negative numbers, like 0 and 3. Signed numbers involve both positive and negative numbers, like 3 and -3.
In binary, a signed-magnitude representation uses the left bit for the sign: 0 means positive, 1 means negative. Ex: For 4-bit numbers, 0011 is 3, and 1011 is -3. Signed-magnitude representation is rarely used, because calculations involving negative numbers, such as 5 - 3, would require special circuits beyond an adder.
A more clever negative number representation exists that can use an adder for both positive and negative numbers. A complement of an N-digit number is another number that yields a sum of 100...00 (with N 0's), and can be used to represent the negative of that number.
Base 10:
5
- 3
----
2
Replace by:
5
+ 7
----
12 ignore cary
7 + 3 = 10; 7 is the complement of 3. Thus 5 + 7 is 10 too much, so the carry can be ignored.
0011 (3) has complement: (0011)' + 1
1100 + 1
So -3 is: 1101
The above is called the two's complement representation, which inverts every bit and adds 1.
Example:
In base ten, the complement of 33 (two digits) is: 67
In base two, the complement of 0010 (four bits) is: 1101 + 1 = 1110
What is -2 in four-bit two's complement representation:
210 = 00102 , -210 = 11012 +12 = 11102
Assuming two's complement representation, what base ten number does 1001 represent?
Left bit is 1, so negative. Complement to find positive. 0110 + 1 = 0111. Magnitude is 7. Negate to yield -7.
Method2: 1001 - 1 = 1000, invert bits to 01111, 011112 = 710, then -7.
2. Subtracting by adding
Two's complement representation has the benefit of allowing an adder to be used even when dealing with negative numbers.z
Ex: 5 + -3 is just 0101(5) + 1101(-3) = 10010, or 0010(2) after ignoring the carry.
No extensive special circuitry for negative numbers is needed.
Example:
6 + 2 is 0110 + ?
0010 (2 is 0010 in binary. The leftmost 0 means positive.)
6 + -2 is 0110 + ?
1101+1=1110 (2 is 0010. The negative is the complement, so 1101 + 1 or 1110.)
3. Overflow
The largest positive four-bit two's complement number is 0111, or 7.
The smallest negative is 1000, or -8 (0111 + 1 = 1000, so magnitude is 8).
Adding two positives, or adding two negatives, may yield a value that can't be represented in the given number of bits, a situation known as overflow.
Ex: 0101 (5) + 0011 (3) incorrectly yields 1000, which is -8 in two's complement.
Adding two positives may overflow:
0010 + 0011 = 0101 (ok)
2 + 3 = 5
0111 + 0001 = 1000 (overflow)
7 + 1 != -8
(The leftmost 0 means positive, so 01112 = 710 , 00012 = 110 . 1000 is -8 in four-bit two's complement representation)
Adding two negatives may overflow:
1111 + 1111 = (1)1110 (ok)
-1 + -1 = -2
(1110 is -2 in four-bit two's complement representation)
1000 + 1111 = (1)0111 (overflow)
-8 + -1 != 7
(The leftmost 0 means positive, so 01112 = 710 )
Adding positive and negative cannot overflow:
1000 + 0111 = 1111 (ok)
-8 + 7 = -1
0010 + 1000 = 1010 (ok)
2 + -8 = -6
As seen above, overflow occurs if the numbers being added have the same sign bit but the sum's sign bit differs. In other words, overflow occurs if two positives sum to a negative (clearly wrong), or two negatives sum to a positive (clearly wrong).
Adding a positive number and negative number (or vice-versa) cannot result in overflow. The sum always has a smaller magnitude than one or both of the numbers, so clearly can fit in the same number of bits. Ex: 7 + -2 = 5, and 5's magnitude is smaller than 7.
1. Two's complement
Unsigned numbers involve only non-negative numbers, like 0 and 3. Signed numbers involve both positive and negative numbers, like 3 and -3.
In binary, a signed-magnitude representation uses the left bit for the sign: 0 means positive, 1 means negative. Ex: For 4-bit numbers, 0011 is 3, and 1011 is -3. Signed-magnitude representation is rarely used, because calculations involving negative numbers, such as 5 - 3, would require special circuits beyond an adder.
A more clever negative number representation exists that can use an adder for both positive and negative numbers. A complement of an N-digit number is another number that yields a sum of 100...00 (with N 0's), and can be used to represent the negative of that number.
Base 10:
Replace by:
7 + 3 = 10; 7 is the complement of 3. Thus 5 + 7 is 10 too much, so the carry can be ignored.
Base 2:
Replace by:
Complement: invert bits, add 1
The above is called the two's complement representation, which inverts every bit and adds 1.
Example:
2. Subtracting by adding
Two's complement representation has the benefit of allowing an adder to be used even when dealing with negative numbers.z Ex: 5 + -3 is just 0101(5) + 1101(-3) = 10010, or 0010(2) after ignoring the carry. No extensive special circuitry for negative numbers is needed.
Example:
3. Overflow
The largest positive four-bit two's complement number is 0111, or 7. The smallest negative is 1000, or -8 (0111 + 1 = 1000, so magnitude is 8). Adding two positives, or adding two negatives, may yield a value that can't be represented in the given number of bits, a situation known as overflow. Ex: 0101 (5) + 0011 (3) incorrectly yields 1000, which is -8 in two's complement.
Adding two positives may overflow:
(The leftmost 0 means positive, so 01112 = 710 , 00012 = 110 . 1000 is -8 in four-bit two's complement representation)
Adding two negatives may overflow:
(1110 is -2 in four-bit two's complement representation)
(The leftmost 0 means positive, so 01112 = 710 )
Adding positive and negative cannot overflow:
As seen above, overflow occurs if the numbers being added have the same sign bit but the sum's sign bit differs. In other words, overflow occurs if two positives sum to a negative (clearly wrong), or two negatives sum to a positive (clearly wrong).
Adding a positive number and negative number (or vice-versa) cannot result in overflow. The sum always has a smaller magnitude than one or both of the numbers, so clearly can fit in the same number of bits. Ex: 7 + -2 = 5, and 5's magnitude is smaller than 7.