How do Egyptian fractions work?

Answer 1

The idea behind Egyptian fractions is to write any fraction as the sum of unit fractions which are fractions with the number 1 in the numerator, like 1/2 or 1/3. The catch is all the fractions have to be different. This means no two fractions with the same denominator can be added.

So we write 2/3 but that is not a unit fraction. You cannot write it as 1/3+1/3 using Egyptian fractions because the violates the repeating the fraction rule.

Saying 3/4 = 1/2 + 1/4 is totally OK.

The reason they are worth understanding and studying, other than their pure beauty, is they allow you to compare fractions easier than our current system. They also allow you to divide things up into parts more easily than our current system.

So since we cannot write 2/3 as 1/3 + 1/3 how do we write it?

We write it as 1/6 +1/2.

One common notation for this Egyptian fraction is [2,6]. Using this notation, here are a few others: 2/3= [2,6]2/5= [3,15]2/7= [4,28]

Now that you see what they are, let me explain what I meant about dividing and comparing. If I write 5/8 as 1/2+1/8 and I want to divide 5 things among 8 people, each would get 1/2 and 1//8. That is 5/8 and 8 ( 5/8)=5 . It is as simple as that.

In general if I have m things to divide among n people, I write m/n as an Egyptian fraction and each person gets that fractions worth of the thing I am dividing.

When we compare fractions we usually have to either convert them to decimals or create fractions with a common denominator.

With Egyptian fractions, this is not necessary. You write the numbers as Egyptian fractions and then keep doing that with the fractions you have until you can compare the two. You get the added advantage of seeing just how much bigger or smaller one number is from the other.