Every positive fraction can be represented as sum of unique unit fractions. A fraction is unit fraction if numerator is 1 and denominator is a positive integer, for example 1/3 is a unit fraction. Such a representation is called Egyptial Fraction as it was used by ancient Egyptians.

Following are few examples:

Egyptian Fraction Representation of 2/3 is 1/2 + 1/6 Egyptian Fraction Representation of 6/14 is 1/3 + 1/11 + 1/231 Egyptian Fraction Representation of 12/13 is 1/2 + 1/3 + 1/12 + 1/156

We can generate Egyptian Fractions using Greedy Algorithm. For a given number of the form ‘nr/dr’ where dr > nr, first find the greatest possible unit fraction, then recur for the remaining part. For example, consider 6/14, we first find ceiling of 14/6, i.e., 3. So the first unit fraction becomes 1/3, then recur for (6/14 – 1/3) i.e., 4/42.

Below is C++ implementation of above idea.

// C++ program to print a fraction in Egyptian Form using Greedy // Algorithm #include <iostream> using namespace std; void printEgyptian(int nr, int dr) { // If either numerator or denominator is 0 if (dr == 0 || nr == 0) return; // If numerator divides denominator, then simple division // makes the fraction in 1/n form if (dr%nr == 0) { cout << "1/" << dr/nr; return; } // If denominator divides numerator, then the given number // is not fraction if (nr%dr == 0) { cout << nr/dr; return; } // If numerator is more than denominator if (nr > dr) { cout << nr/dr << " + "; printEgyptian(nr%dr, dr); return; } // We reach here dr > nr and dr%nr is non-zero // Find ceiling of dr/nr and print it as first // fraction int n = dr/nr + 1; cout << "1/" << n << " + "; // Recur for remaining part printEgyptian(nr*n-dr, dr*n); } // Driver Program int main() { int nr = 6, dr = 14; cout << "Egyptian Fraction Representation of " << nr << "/" << dr << " is\n "; printEgyptian(nr, dr); return 0; }

Output:

Egyptian Fraction Representation of 6/14 is 1/3 + 1/11 + 1/231

The Greedy algorithm works because a fraction is always reduced to a form where denominator is greater than numerator and numerator doesn’t divide denominator. For such reduced forms, the highlighted recursive call is made for reduced numerator. So the recursive calls keep on reducing the numerator till it reaches 1.

References:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

This article is contributed by **Shubham**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.