1's and 2's complement of a Binary Number?

Binary numbers are expressed in base 2, using only the digits '0' and '1'. Each digit in a binary number is called a bit. Understanding 1's and 2's complement is essential for representing signed numbers in computer systems.

1's Complement

One's complement of a binary number is obtained by inverting all bits − changing every '0' to '1' and every '1' to '0'.

Syntax

1's complement = invert all bits
Example: 101100 ? 010011

2's Complement

Two's complement of a binary number is obtained by adding 1 to the 1's complement of the binary number.

2's complement = 1's complement + 1
Example: 101100 ? 010011 + 1 = 010100

Example: Finding 1's and 2's Complement

Here's a C program to calculate both complements of a binary number −

#include <stdio.h>
#include <string.h>

int main() {
    char binary[10] = "01001011";
    char ones_complement[10];
    char twos_complement[10];
    int length = strlen(binary);
    
    printf("Binary number: %s
", binary);
    
    // Calculate 1's complement
    for(int i = 0; i < length; i++) {
        if(binary[i] == '0') {
            ones_complement[i] = '1';
        } else {
            ones_complement[i] = '0';
        }
    }
    ones_complement[length] = '\0';
    printf("1's complement: %s
", ones_complement);
    
    // Calculate 2's complement (1's complement + 1)
    strcpy(twos_complement, ones_complement);
    int carry = 1;
    for(int i = length - 1; i >= 0 && carry; i--) {
        if(twos_complement[i] == '0') {
            twos_complement[i] = '1';
            carry = 0;
        } else {
            twos_complement[i] = '0';
        }
    }
    printf("2's complement: %s
", twos_complement);
    
    return 0;
}

Binary number: 01001011
1's complement: 10110100
2's complement: 10110101

Key Points

• 1's complement: Simply flip all bits (0?1, 1?0)
• 2's complement: Add 1 to the 1's complement
• 2's complement is widely used to represent negative numbers in computers
• The range for n-bit 2's complement is -2^(n-1) to 2^(n-1)-1

Conclusion

1's and 2's complement are fundamental concepts in digital systems. While 1's complement inverts all bits, 2's complement adds 1 to the result, making it the standard method for representing signed integers in modern computers.