10's Complement of a decimal number?

The 10's complement of a decimal number is a mathematical operation used in digital systems to simplify arithmetic operations, particularly subtraction. It is calculated by subtracting the given number from 10ⁿ, where n is the total number of digits in the number.

The 10's complement can be found in two ways −

• Method 1: Calculate 9's complement first, then add 1
• Method 2: Direct calculation using formula: 10ⁿ - number

Syntax

int tensComplement = pow(10, digitCount) - number;

Method 1: Using 9's Complement

First find the 9's complement by subtracting each digit from 9, then add 1 −

#include <stdio.h>
#include <math.h>

int countDigits(int num) {
    int count = 0;
    while (num != 0) {
        count++;
        num = num / 10;
    }
    return count;
}

int ninesComplement(int num, int digits) {
    int maxNum = pow(10, digits) - 1;
    return maxNum - num;
}

int main() {
    int number = 456;
    int digits = countDigits(number);
    
    printf("Original number: %d
", number);
    
    int ninesComp = ninesComplement(number, digits);
    printf("9's complement: %d
", ninesComp);
    
    int tensComp = ninesComp + 1;
    printf("10's complement: %d
", tensComp);
    
    return 0;
}

Original number: 456
9's complement: 543
10's complement: 544

Method 2: Direct Calculation

Using the direct formula: 10ⁿ - number −

#include <stdio.h>
#include <math.h>

int main() {
    int number = 456;
    int temp = number;
    int digitCount = 0;
    
    /* Count digits */
    while (temp != 0) {
        digitCount++;
        temp = temp / 10;
    }
    
    /* Calculate 10's complement directly */
    int tensComplement = pow(10, digitCount) - number;
    
    printf("Number: %d
", number);
    printf("Number of digits: %d
", digitCount);
    printf("10's complement: %d
", tensComplement);
    
    return 0;
}

Number: 456
Number of digits: 3
10's complement: 544

How It Works

For number 456 (3 digits) −

• 10³ = 1000
• 10's complement = 1000 - 456 = 544

Key Points

• 10's complement = 9's complement + 1
• Direct formula: 10ⁿ - number
• Used in binary arithmetic for efficient subtraction
• Always results in a number with the same number of digits

Conclusion

The 10's complement is a fundamental concept in digital arithmetic that simplifies subtraction operations. Both methods yield the same result, with the direct approach being more efficient for programming implementation.