What Are Egyptian Fractions?

It's a real shame that the Library of Alexandria burned down, because the Egpytians were coming up with some pretty cool stuff 4000 years ago! An Egyptian fraction is simply a fraction that can be written as the sum of numerous different unit fractions. A unit fraction is a fraction of the form $1/n$, where n is a positive integer number. For example, $3/4$ can be written as $1/2 + 1/4$, and $6/7$ can be written as $1/2 + 1/3 + 1/42$. Mathematicians have further proved that any and all fractions can be written as Egyptian fractions!

First, let's show how every Egyptian fraction can be written in an infinite number of ways.

Take the fraction $3/4$. This fraction can be expressed as $1/2 + 1/4$. Now take $1/4$. This can be expressed as $1/6 + 1/12$. $1/6$ can be expressed as $1/8 + 1/24$ - $1/8$ can be expressed as $1/12 + 1/24$ - and so on! Though this may seem like a rather non-rigorous proof, it makes sense.

More rigorously, a chain of infinite fractions can be constructed using the identity:

$1/a = {1}/{(a+1)} + {1}/{(a(a+1))}$

If you don't believe it, try it out for yourself!

Now let's show how any fraction can be written as an Egyptian fraction.

Say you're given the funky fraction $401/900$. How would you write this as an Egyptian fraction? Well, we know that $401/900$ is bigger than $1/3$ and less than $1/2$ (respectively, those two fractions are $300/900$ and $450/900$. So, we take out the biggest chunk we can out of the fraction - in this case, the chunk happens to be $1/3$. Now we are left with $401/900 - 300/900 = 101/900$. Repeat the same process; take the biggest chunk you can out of it. We can see that $101/900$ is very close to $100/900$, or $1/9$. Now we are left with $101/900 - 100/900$ = 1/900. Yay! Now we have all unit fractions. $401/900$ can be written as $1/3 + 1/9 + 1/900$. This process of taking the biggest chunk possible out of a fraction is called the Greedy algorithm - it subtracts the largest unit fraction it can and then repeats the same process on the remainder.

If you are skeptical about the fact that all fractions can be written as a sum of unit fractions, think about it like this. Before you subtracted anything from $401/900$, the numerator was $401$. After you subtracted $1/3$, the remainder was $101/900$ (numerator is 101). After you subtracted $1/9$, the remainder was $1/900$ (numerator of 1 - this is what we want!) The numerators of the remainders continually get smaller and smaller - it is realistic to expect that they always will eventually converge to 1.