Understanding Two's Complement
Two's complement is a mathematical operation on binary numbers, and is the standard method for representing signed (positive or negative) integers in binary in virtually all modern computer architectures. This representation simplifies arithmetic operations like addition and subtraction for digital circuits, as it allows subtraction to be treated as addition of a negative number.
Key Principles and Formation
To find the two's complement of a binary number, typically two steps are followed: first, invert all the bits (change 0s to 1s and 1s to 0s), which is known as the one's complement. Second, add 1 to the result of the one's complement. For example, to represent -5 in an 8-bit system, start with +5 (00000101). Its one's complement is 11111010. Adding 1 yields 11111011, the two's complement of -5.
A Practical Example: Addition with Two's Complement
Consider adding 7 and -3 using 8-bit two's complement. +7 is 00000111. To get -3, we take +3 (00000011), find its one's complement (11111100), and add 1 (11111101). Now, add them: 00000111 (+7) + 11111101 (-3) = (1)00000100. The leftmost '1' (overflow bit) is discarded, resulting in 00000100, which is +4. This shows how subtraction is handled via addition.
Importance in Computing
Two's complement is crucial because it allows a single addition circuit to perform both addition and subtraction, eliminating the need for separate, more complex hardware. Furthermore, it provides a unique representation for the number zero (all zeros), unlike other signed binary representations that can have both positive and negative zero. This inherent efficiency and simplicity are why it is universally adopted in computer processors.