- 2’s Complement Definition: 2’s complement is a binary number system technique used to represent positive and negative numbers, allowing easier binary subtraction.
- Calculation Methods: Methods to find 2’s complement include subtracting each bit from 1 and adding one to the result, providing consistent results across methods.
- Subtraction Technique: Using 2’s complement for subtraction involves converting the subtrahend to 2’s complement and adding it to the minuend, simplifying binary arithmetic.
- Negative Representation: In binary systems, negative numbers are represented using 2’s complement, where the sign bit ‘1’ indicates a negative value.
- 2’s Complement Examples: Demonstrates practical application of 2’s complement in various binary operations, ensuring clarity on its uses and benefits.
Complement Number System
2’s complement is the usual signed-binary encoding. For a number whose base is N, the (N-1)’s complement is the difference from the highest same-width digit string, and adding one gives the N’s complement. In decimal (base 10) that (N-1)’s form is 9’s complement. In binary, 2’s complement is the 1’s complement plus one.
Method 1: Subtract the number from the highest same-width value. Subtracting 25 from 99 in decimal gives 74, the 9’s complement.
Method 2: Subtract each digit from 9, the highest decimal digit. 9 – 2 and 9 – 5 again give 74. Both methods match, so either may be used.
Binary Number System: A Binary number uses only 0s and 1s and has base 2. Subtracting each binary digit from 1 yields the 1’s complement. Adding 1 to that result gives the 2’s complement. The short way to form the 1’s complement is to invert each digit: replace 0s with 1s and 1s with 0s.
The same two methods apply in binary, plus a bit-flip shortcut, to find 1’s and 2’s complement. Example: 2’s complement of 0100. A leading zero makes this a four-bit word, a common power-of-two width.
Method – 1 Subtract it from 1111, the highest four-bit value, to find the 1’s complement. 1111 – 0100 is 1011. 2’s complement is 1011 + 1, which is 1100.
Method – 2 Subtract every individual digit from 1 to get the 1’s complement. For 0100 that is 1 – 0, 1 – 1, 1 – 0, 1 – 0, which is 1011.
2’s complement will be 1011 + 1 = 1100.
Method – 3 Replace each 1 by 0 and each 0 by 1 to find the 1’s complement. Adding one to that result gives the 2’s complement.
For 0100, replacing 1 by 0 and 0 by 1 gives 1011. Adding 1 gives (1011 + 1), which is 1100.
All three methods give the same 4-bit result, so any one of them may be used.
2’s complement representation of a positive number and its negative.
Why We Do Need 2’s Complement?
The main use of 2’s complement is subtraction of two binary values with the same adder. A computer stores only binary, and the binary number system has no minus sign, so a sign bit is used. If the sign bit is 1 the value is negative. If it is 0 the value is positive.
For subtraction of binary numbers, use the following steps.
Subtracting a Smaller Number from a Larger Number
- Find 2’s complement of the smaller number.
- Add the larger and 2’s complement of the smaller number.
- Discard the carry.
- After discarding the carry, keep the remaining bits. That value is the positive difference.
Subtracting Larger Number from Smaller Number
- Find 2’s complement of larger number.
- Add 2’s complement of larger number to the smaller number.
- If no carry is generated, find the 2’s complement of the result and the result will be negative.
- If a carry is generated here, the assumed larger/smaller order or the chosen bit width is wrong. Do not treat that carry-discarded sum as a negative result.
Subtraction of larger number from smaller number.
Advantages of 2’s Complement
- Subtraction can be done with the help of 2’s complement method.
- The same adder circuit can add or subtract.
- End around carry need not be performed as in the case of 1’s complement.
- Negative number can be represented using 2’s complement.
