An Egyptian fraction is a fraction expressed as a sum of unit fractions. For example, you can write 3/7 as 1/3 + 1/11 + 1/231.

You can write any fraction as an Egyptian fraction, although there may be more than one way to do so.

One simple algorithm for finding an Egyptian fraction representation for a fraction is to subtract the largest unit fraction that is smaller than the original fraction and then repeat with the result.

This example uses the Fraction class described in the example Make a Fraction class in C#. See that example for most of the class's details. To make this example, I added some comparison operators. To make comparing fractions to integers easier, I also added an operator to convert from long to Fraction. The following code shows the most interesting pieces of the new code.

// Return a < b.
public static bool operator <(Fraction a, Fraction b)
{
    return (double)a < (double)b;
}

// Return a > b.
public static bool operator >(Fraction a, Fraction b)
{
    return (double)a > (double)b;
}

The following code shows how the program uses the revised Fraction class to find Egyptian fraction representations.

// Return a string representation of the Egyptian fraction.
private List GetEgyptianFraction(Fraction frac)
{
    List result = new List();

    // Remove any whole number part.
    int whole_part = (int)(frac.Numerator / frac.Denominator);
    if (whole_part > 0)
    {
        result.Add(whole_part);
        frac = frac - whole_part;
    }

    // Pull out unit fractions.
    long denom = 2;
    while (frac > 0)
    {
        // Make the unit fraction smaller until it fits.
        Fraction unit_fraction = new Fraction(1, denom);
        while (unit_fraction > frac)
        {
            denom++;
            unit_fraction = new Fraction(1, denom);
        }

        // Remove the unit fraction.
        result.Add(unit_fraction);
        frac -= unit_fraction;
        denom++;
    }

    return result;
}

The code first removes any whole number part from the fraction. For example, it converts 12/5 into 2 + 2/5.

Next, as the fraction is greater than 0, the code loops through successively smaller unit fractions subtracting any that fit. It tries 1/2, 1/3, 1/4, and so forth until it removes the last part of the fraction.