Number systems: binary, octal and hexadecimal

We write numbers in base 10: each digit is worth ten times the digit to its right. Computers use base 2 (binary). Programmers also use base 16 (hexadecimal, "hex"), because one hex digit is exactly 4 bits, so a byte is two hex digits.

Converting by repeated division

#include <stdio.h>

int main(void) {
    int n = 45;
    int base = 2;
    int digits[32];
    int count = 0;
    int x = n;
    while (x > 0) {
        digits[count++] = x % base;   // the remainder is the next digit
        x /= base;
    }
    printf("%d in base %d: ", n, base);
    for (int i = count - 1; i >= 0; i--) printf("%d", digits[i]);
    printf("\n");
    printf("hex %x, octal %o\n", n, n);
    return 0;
}

To write n in base b, divide by b again and again: the remainders, read from the last to the first, are the digits. 45 ÷ 2 gives remainders 1, 0, 1, 1, 0, 1, so 45 is 101101 in binary. The program stores the digits in an array, then prints them backwards.

Back to decimal, and in C

#include <stdio.h>

int main(void) {
    char digits[] = "101101";
    int base = 2;
    int value = 0;
    for (int i = 0; digits[i] != '\0'; i++) {
        value = value * base + (digits[i] - '0');
    }
    printf("%s in base %d is %d\n", digits, base, value);
    printf("0x2D = %d, 055 = %d\n", 0x2D, 055);
    return 0;
}

To read digits in base b, start from 0 and, for each digit, multiply by b and add the digit: 101101 gives ((((1×2+0)×2+1)×2+1)×2+0)×2+1 = 45.

C understands these bases directly:

• write 0x2D for hex, 055 for octal (a leading 0!), and 0b101101 for binary in newer compilers;
• print with %x (hex) and %o (octal).

Mark lesson complete