1's Complement vs. 2's Complement: Binary Number Representation
Understand the differences between 1's complement and 2's complement representations of numbers in digital systems. This guide explains how each complement is calculated, their advantages and disadvantages, and their use in simplifying binary arithmetic and logical operations.
1's Complement vs. 2's Complement Representation of Numbers
Complements in Binary Representation
In digital systems, complements are used to simplify subtraction and perform logical operations on binary numbers. Two common types are 1's complement and 2's complement.
1's Complement
The 1's complement of a binary number is obtained by inverting (flipping) each bit: 0s become 1s, and 1s become 0s. We can denote the 1's complement of N as N'.
Calculating 1's Complement
For an n-bit binary number N, its 1's complement N' can be calculated using:
N' = (2n - 1) - N
(Examples of calculating the 1's complement of binary numbers are shown in the original text and should be included here.)
2's Complement
The 2's complement of a binary number is found by first taking the 1's complement and then adding 1 to the result. We can denote the 2's complement of N as N*.
Calculating 2's Complement
For an n-bit binary number N, its 2's complement N* can be calculated using:
N* = 2n - N
(Examples of calculating the 2's complement of binary numbers are shown in the original text and should be included here.)
Comparing 1's and 2's Complement
| Feature | 1's Complement | 2's Complement |
|---|---|---|
| Generation | Invert all bits | Invert all bits, then add 1 |
| Example (for 9) | 1001 becomes 0110 (6) | 1001 becomes 0111 (7) |
| Logic Gates | Uses NOT gates | Uses NOT gates and a full adder |
| Number Representation | Ambiguous representation of 0 | Unambiguous representation of 0 |
| Range (k-bit register) | -(2k-1 - 1) to 2k-1 - 1 | -2k-1 to 2k-1 - 1 |
| Representation of 0 | Two representations (+0 and -0) | One representation (+0) |
| Sign Extension | Same as 2's complement | Same as 1's complement |
| End-Around Carry | Required in addition | Not required |
| Ease of Operation | More complex | Simpler |
Conclusion
Both 1's and 2's complement are used to represent negative numbers in binary. The 2's complement system is generally preferred due to its simpler arithmetic and unambiguous representation of zero.