Base Conversions

What is a base?

In mathematics, a base is the number of different digits or combination of digits and letters that a system of counting uses to represent numbers. For example, the decimal system uses ten digits (0-9), the binary system uses two digits (0-1), and the hexadecimal system uses sixteen digits (0-9 and A-F).

• Decimal: 0123456789
• Binary: 01
• Hexadecimal: 0123456789ABCDEF

How to convert a number to a base?

To convert a number to a different base, you need to divide the number by the base and keep track of the remainders. The remainders will be the digits of the new number in the desired base. You can repeat this process until the quotient is zero.

Converting Decimal to Binary

For example, to convert the decimal number 42 to binary, you can divide 42 by 2 and keep track of the remainders:

42 / 2 = 21 // remainder 0
21 / 2 = 10 // remainder 1
10 / 2 = 5  // remainder 0
5 / 2 = 2   // remainder 1
2 / 2 = 1   // remainder 0
1 / 2 = 0   // remainder 1

The remainders in reverse order are 101010, which is the binary representation of 42.

How does this look like in a C?

#include <stdio.h>
 
void to_binary(int n) {
    if (n > 1) {
        to_binary(n / 2);
    }
    printf("%d", n % 2);
}
// flow:
// a. 42 > 1, spawn to_binary(21)
// b. 21 > 1, spawn to_binary(10)
// c. 10 > 1, spawn to_binary(5)
// d. 5 > 1, spawn to_binary(2)
// e. 2 > 1, spawn to_binary(1)
// f. 1 !> 1, print 1 % 2 = 1
// e. print 2 % 2 = 0
// d. print 5 % 2 = 1
// c. print 10 % 2 = 0
// b. print 21 % 2 = 1
// a. print 42 % 2 = 0
 
int main(void) {
    int decimal = 42;
    printf("Binary: ");
    to_binary(decimal);
    printf("\n");
    return 0;
}

After running the above code, you should see Binary: 101010 printed to the console. to_binary is a recursive function that divides the number by 2 and prints the remainder while the number is greater than 1.

Converting Decimal to Hexadecimal

To convert a decimal number to hexadecimal, you can follow a similar process as above, but this time divide by 16.

#include <stdio.h>
 
void to_hex(int n) {
    if (n > 15) {
        to_hex(n / 16);
    }
    int remainder = n % 16;
    if (remainder < 10) {
        printf("%d", remainder);
    } else {
        printf("%c", 'A' + remainder - 10);
    }
}
 
int main(void) {
    int decimal = 42;
    printf("Hexadecimal: ");
    to_hex(decimal);
    printf("\n");
    return 0;
}

Info

Why 'A' + remainder - 10? Because the ASCII value of A is 65, and we need to start from 10 to 15 for the hexadecimal digits A-F. By adding A (65) to the remainder, we get the correct ASCII value for the hexadecimal digit.