How to convert decimal fractions to hexadecimal fractions?

Question

So I was thinking, how do you convert a decimal fraction into a hexadecimal fraction? What are some methods for converting and are there short cuts?

Answer 1

You can use this algorithm:

1. Take a fractional part of the number (i.e. integer part equals to zero)
2. Multiply by 16
3. Convert integer part to hexadecimal and put it down
4. Go to step 1

For instance, let's find out hexadecimal representation for pi = 3.141592653589793...

integer part is evident - 0x3; as for fractional part (0.141592653589793) we have

  0.14159265358979 * 16 =  2.26548245743664; int part  2 (0x2); frac 0.26548245743664
  0.26548245743664 * 16 =  4.24771931898624; int part  4 (0x4); frac 0.24771931898624
  0.24771931898624 * 16 =  3.96350910377984; int part  3 (0x3); frac 0.96350910377984
  0.96350910377984 * 16 = 15.41614566047744; int part 15 (0xF); frac 0.41614566047744
  0.41614566047744 * 16 =  6.65833056763904; int part  6 (0x6); frac 0.65833056763904
  0.65833056763904 * 16 = 10.53328908222464; int part 10 (0xA); ...

So pi (hexadecimal) = 3.243F6A...

Possible (C#) implementation

public static String ToHex(Double value) {
  StringBuilder Sb = new StringBuilder();

  if (value < 0) {
    Sb.Append('-');

    value = -value;
  }

  // I'm sure you know how to convert decimal integer to its hexadecimal representation
  BigInteger bi = (BigInteger) value;
  Sb.Append(bi.ToString("X"));

  value = value - (Double)bi;

  // We have integer value in fact (e.g. 5.0)
  if (value == 0)
    return Sb.ToString();

  Sb.Append('.');

  // Double is 8 byte and so has at most 16 hexadecimal values
  for (int i = 0; i < 16; ++i) {
    value = value * 16;
    int digit = (int) value;

    Sb.Append(digit.ToString("X"));

    value = value - digit;

    if (value == 0)
      break;
  }

  return Sb.ToString();
}

Test

   Console.Write(ToHex(Math.PI)); // <- returns "3.243F6A8885A3"

Answer 2

You can get the fractional part by multiplying the input number by a whole number of hex digits. Then you can use regular integer-to-hex conversion. For example, to get 6 characters after the (hexa)decimal point, multiply the fractional part by 0x1000000.

Here is some Java code that will do it.

String toHexFraction(double x, int digits) {
    // Get fractional part.
    if (x < 0.0)
        x = 0.0 - x;
    x = x % 1.0;

    // Shift left by n digits
    long multiplier = (1L << (digits * 4));
    long fraction = (long)(x * multiplier);

    // Convert integer to hex string.
    // String should have at least n digits; prefix with zeros if not.
    String hex = Long.toHexString(fraction);
    String padding = "000000000000000";
    hex = padding.substring(0, digits - hex.length()) + hex;

    return hex;
}

String toHexInteger(double x) {
    long whole = (long) x;
    String prefix;
    if (whole < 0) {
        // Long.toHexString treats the number as an unsigned integer.
        whole = 0 - whole;
        prefix = "-";
    } else {
        prefix = "";
    }
    return Long.toHexString(whole);
}

String toHex (double x, int digits) {
    return toHexInteger(x) + "." + toHexFraction(x, digits);
}

The number of digits will be limited by the largest integer you can represent in a double.

This should work for other square bases too, e.g. for octal change digits * 4 to digits * 3, and use Long.toOctalString.