# Egyptian Fractions Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Last updated: Feb 02, 2023

Our Egyptian fractions calculator will teach you how to write fractions in the Egyptian style. It's not hard — well, the math isn't, at least!

Here you will learn:

• How Egyptians did math;
• What an Egyptian fraction is;
• How to use Egyptian fractions;
• How to calculate Egyptian fractions — the greedy algorithm; and
• How to use our Egyptian fractions calculator.

Discover the ancient Egyptian fractions with us. $1/3$, $1/2$, $1$, go!

## Egyptian math

Egyptians were good at math! Their millennial civilization saw the development of many innovative concepts, spanning from arithmetic to geometry. Look at the pyramids — we even call them pyramids!

The first evidence of the use of mathematics dates back to 3200 BC. Let this sink in: we are closer to Rome than Romans were to those first additions! In many papyruses, historians found traces of mathematical problems and solutions. Thanks to them, we know that Egyptians knew how to calculate surfaces and volumes of many shapes and had knowledge of mathematical constants like pi.

Egyptians had a particular way to write fractions (today, we call them Egyptian fractions) and they made extensive use of it. Let's take a look!

## What are Egyptian fractions?

Our knowledge of the Egyptian fraction expansion comes mainly from a papyrus called the Rhind papyrus, where, out of 87 mathematical problems, as many as 81 involve fractions. Historians think that the Egyptians' interest in fractions stemmed from the particular labor force they used in their construction projects. When managing hundreds of workers, you don't want conflict over an unevenly split piece of bread! An Egyptian fraction is a way to represent a proper fraction (a fraction where the numerator is smaller than the denominator) as a sum of distinct unit fractions.

Like this!

$\frac{11}{14}=\frac{1}{2}+\frac{1}{4}+\frac{1}{28}$

It looks impractical, and it is so. Egyptian fractions were still used shortly after the Egyptian civilization, but quickly gave space to vulgar fractions, which we use today.

Egyptians held certain fractions in high regard. $2/3$ and $3/4$ didn't have a unit fraction expansion, and together with $1/2$, they each had a special symbol.

Every real number smaller than 1 can be represented using Egyptian fractions. One of the most prominent mathematicians in history proved this result: Fibonacci. Egyptian fractions are nothing but an addition to the list of discoveries by the man: we already created the Fibonacci sequence calculator and the fields where his intuitions are used, like the one we introduce in the Fibonacci retracement calculator. We can find Fibonacci even in recent creations, as you can learn at the linear feedback shift registers calculator.

Using a simple recursion, he also laid down the basis for developing the algorithm we will use in our calculator.

## Applications of the Egyptian fractions

When Egyptians developed their fraction system, they mainly applied it in trade and division problems. One of those problems involved dividing loaves of bread among varying numbers of people — it sounds like the type of exercise we'd meet in primary school!

Why use Egyptian fractions today? When you need to divide a certain number of objects into equal parts, and the partition is not straightforward, put it down in terms of those ancient Egyptian fractions. Are you at a party with other 7 people, and there are only 5 pizzas? Let's ask the ancient Egyptians! Well, it looks like they don't know what a pizza is.

Anyway: 8 people, 5 pizzas. This means everyone gets one pizza each and two are left to divide equally:

$\frac{5}{8} = \frac{1}{2}+\frac{1}{8}$

Each person gets half a pizza and a small slice. The reasoning also applies (and in particular) for counterintuitive divisions. Think of 12 people and 13 pizzas. Instead of giving a whole pizza to everyone and then slicing the remaining one in twelfths or worse, try this expansion:

$\frac{13}{12}=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$

Where the fractions represent the amount of pizza each partygoer gets.

## Greedy algorithms: how to calculate an Egyptian fraction expansion

In order to calculate an Egyptian fraction, you can apply many algorithms. The one we judged easiest to learn is the greedy algorithm:

$\frac{x}{y}=\frac{1}{\left\lceil\frac{y}{x}\right\rceil}+\frac{(-y)\mod{x}}{y\left\lceil\frac{y}{x}\right\rceil}$

Let's check what do those operations mean.

The weird "half" square brackets indicate the ceiling operator, defined as the smallest integer greater than or equal to its argument. In other words, we round up toward the integer. Take a look at this:

$\left\lceil\frac{3}{8}\right\rceil=\left\lceil 0.375 \right \rceil = 1$

The ceiling function is often taught in association with the floor function, $\lfloor x \rfloor$. In the latter, the result is the closest integer smaller than (or equal to) the argument. We are not talking about the floors and ceilings of construction and carpentry!

The modulus is the other "uncommon" operator, which returns the remainder of the division between the two arguments. Now that you know how the ceiling operation works, it's going to be even easier to understand:

$\footnotesize y \operatorname{mod}\ {x} = y - \left\lfloor \frac{y}{x} \right\rfloor\cdot x$

The same holds for negative numbers; remember to calculate the remainder in the same "direction"! Here's an example:

$\footnotesize \begin{split} &\ (-11)\operatorname{mod}\ {5}\\ =&\ -11 -\left\lfloor\frac{-11}{5}\right\rfloor\cdot 5\\ =&\ \ 4 \end{split}$

If the second argument has a numerator $>1$, we feed the fraction back to the algorithm until all fractions have denominator $1$. It will happen sooner or later — the finiteness of this algorithm for Egyptian fractions was proved by the man, the myth, the mathematician Fibonacci.

## The greedy algorithm in action

Let's try the greedy algorithm with an example: $6/7$.

First step: apply the algorithm!

$\footnotesize \begin{split} \frac{6}{7} &= \frac{1}{\left\lceil\frac{7}{6}\right\rceil} + \frac{(-7)\operatorname{mod}\ 6}{7\left\lceil\frac{7}{6}\right\rceil} \\[1em] &= \frac{1}{2}+\frac{5}{14} \end{split}$

The first fraction is the first unitary fraction of our result. Now, feed the second fraction to the algorithm again:

$\footnotesize \begin{split} \frac{5}{14} &= \frac{1}{\left\lceil\frac{14}{5}\right\rceil} + \frac{(-14)\operatorname{mod}\ 5}{14\left\lceil\frac{14}{5}\right\rceil}\\[1em] &=\frac{1}{3}+\frac{1}{42} \end{split}$

The numerator of the second fraction is $1$: our greedy algorithm has done its job. We can now write the Egyptian fraction expansion in its entirety:

$\frac{6}{7}=\frac{1}{2}+\frac{1}{3}+\frac{1}{42}$

## Other Egyptian fractions algorithms

The greedy algorithm is not the only way to calculate an Egyptian fraction expansion. Here we will teach you two other algorithms you can use. They are both based on conflict resolution and share the first step: decomposing the fraction $x/y$ in $x$ copies of the fraction $1/y$:

$\frac{x}{y}=\frac{1}{y}+\frac{1}{y}+...+\frac{1}{y}$

We need to solve those conflicts since there can't be identical terms in an Egyptian fraction. We have a choice: pair or split.

#### Solving conflicts by splitting

We apply a simple decomposition of a unity fraction — one you can try on a piece of paper!

$\frac{1}{y}=\frac{1}{y+1}+\frac{1}{y\cdot (y+1)}$

Every time you find a conflict, one of the two identical fractions "surrender" and yield the two "daughter" fractions.

❗ Careful! This method gives an exponentially big number of terms: they follow the value of the numerator $n$ with: $n_{\text{terms}}=2^n-1$.

#### Solving conflicts by pairing

Instead of a single unit fraction of a pair disappearing (giving three fractions as a result), why not pair the two sides of the conflict?

Consider a pair of identical unit fractions, two identities hold, depending on the parity of the denominator.

If the denominator is even, we can do:

$\frac{1}{a}+\frac{1}{a}=\frac{2}{a}=\frac{1}{a/2}$

And if the denominator is odd, we can do:

$\frac{1}{a}+\frac{1}{a}=\frac{2}{a+1}+\frac{2}{a\cdot(a+1)}$

This "peaceful" solution of the conflict gives shorter expansions than the splitting algorithm.

## How to use our Egyptian fraction calculator

Using our Egyptian fraction calculator is straightforward. Insert the fraction that you want to convert to Egyptian fractions, select the algorithm you prefer, and you are good to go!

⚠️ We limited the number of iterations that our Egyptian fraction calculator can take when using the splitting algorithm. The value of the numerator can be at most $5$.

## Ancient math

You discovered how Egyptians wrote their fractions, and with it, a piece of the history of math. Numbers and calculations have always accompanied humanity: discover them with Omni, from the Mayan numerals converter to our comprehensive Mayan calendar converter, from the Babylonian numbers calculator to many more. Discover all our math calculators at the !

## FAQ

### What is an Egyptian fraction?

An Egyptian fraction is the representation of any real number smaller than 1 as a sum of non-repeating unit fractions (fractions with numerator equal to 1).

The expansion was used especially in equal division problems and is of little use today; however, it remains an interesting mathematical topic.

### Why use Egyptian fractions today?

You can still use Egyptian fractions to solve mathematical problems involving the divisions of a certain number of items in equal parts. The result of such problems can be rather interesting — for example, preferring more divisions rather than smaller fractions.

### What is the Egyptian fraction of 4/5?

Using the greedy algorithm to find the Egyptian fraction expansion, we can convert 4/5 in Egyptian fraction: 4/5 = 1/2 + 1/4 + 1/120. As you can see, the denominator grew pretty quickly!

### Are Egyptian fractions unique?

For many fractions, the Egyptian fraction expansion is not unique, and it is possible to find different decompositions. However, every number has an expansion: this was already proved centuries ago by the Italian mathematician Fibonacci!

Davide Borchia
Fraction
Algorithm
Greedy
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