In order to use an adder to accomplish subtraction, there must be some kind of transformation or complement that produces the negative of a number. Mathematicians sometimes call this an additive inverse. If this is done, the negative of one number can be added to the other number to accomplish the subtraction. Such a transformation has been found. It is the operation NOT N + 1. In machine subtraction parlance, the NOT N operation is called the 1's complement of N, and NOT N + 1 is called the 2's complement of N. Machine arithmetic that makes use of a 2's compliment as the negative of a number is called 2's complement arithmetic.
The expression NOT N + 1 works as the negative of N if and only if N + (NOT N + 1) = 0. Here is a list of 4-bit numbers paired with their 2's complements and added together in an expression that equals zero. The method requires ignoring any carry output beyond the most significant bit.
0000 + (1111 + 1) = 0000 + 0000 = 0000 = 0 + 0 0001 + (1110 + 1) = 0001 + 1111 = 0000 = 1 + -1 0010 + (1101 + 1) = 0010 + 1110 = 0000 = 2 + -2 0011 + (1100 + 1) = 0011 + 1101 = 0000 = 3 + -3 0100 + (1011 + 1) = 0100 + 1100 = 0000 = 4 + -4 0101 + (1010 + 1) = 0101 + 1011 = 0000 = 5 + -5 0110 + (1001 + 1) = 0110 + 1010 = 0000 = 6 + -6 0111 + (1000 + 1) = 0111 + 1001 = 0000 = 7 + -7 1000 + (0111 + 1) = 1000 + 1000 = 0000 = -0 + -0 1001 + (0110 + 1) = 1001 + 0111 = 0000 = -7 + 7 1010 + (0101 + 1) = 1010 + 0110 = 0000 = -6 + 6 1011 + (0100 + 1) = 1011 + 0101 = 0000 = -5 + 5 1100 + (0011 + 1) = 1100 + 0100 = 0000 = -4 + 4 1101 + (0010 + 1) = 1101 + 0011 = 0000 = -3 + 3 1110 + (0001 + 1) = 1110 + 0010 = 0000 = -2 + 2 1111 + (0000 + 1) = 1111 + 0001 = 0000 = -1 + 1
As suggested by the table above, one could decide that the numbers having a most significant bit equal to 1 are the numbers that we can regard as the negative numbers. The others are positive numbers. This is now a widely-adopted convention that has given rise to a distinction between instructions that perform arithmetic on signed numbers or unsigned signed numbers, as the instruction may require.
Each of the pairs of numbers above adding up to zero are of opposite sign except 0 + 0 = 0 and -0 + -0 = 0. In each case, including these two exceptional ones, the addition works out correctly in signed addition whenever we interpret the most significant bit as the place indicating the minus when it is 1.
This system of signed numbers and signed arithmetic has a cost which has proved to be quite affordable. When instructions work on signed numbers, they have only about half as many magnitudes represented by a digital word. In the 4-bit table above, the numbers range from -7 to +7. (There is no explicitly machine indication of the "+" sign.) The unsigned 4-bit numbers range from 0 to 15. There are 15 magnitudes in signed arithmetic and 16 magnitudes in unsigned arithmetic. The same sort of thing happens in wider words, but the system of recognizing negative numbers by the 1 in the most significant bit is the same.
Notice also that the order of the code bits for the magnitudes in negative numbers is the reverse of the order of bits for the magnitudes of positive numbers. Minus 7 in 4-bit numbers is not 1111 (a 111 = 7 magnitude with a 1 in the most significant bit). It is instead 1001.
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