# Egyptian Fraction Calculator

You can use this Egyptian fraction calculator to employ the greedy algorithm to express a given fraction (x/y) as the finite sum of unit fractions (1/a + 1/b + 1/c + ...).

How to use the calculator:

Simply input the numerator and denominator of the fraction in the associated fields and click on the "Calculate" button to generate the results.

## Egyptian Fractions

Ancient Egyptians used numbers and basic calculators as early as 2700 BCE. However, they expressed fractions in a very different way to the methods we employ today. Specifically, as opposed to recording three quarters as 3/4, they opted to record the ratio as the sum total of the fractions that had a numerator of one. As such, 3/4 was expressed as 1/2 + 1/4.

An Egyptian fraction representation is available for every rational number between 0 and 1, and every number in this continuum can be expressed as the finite sum of the unit fractions. Each representation is not unique. For instance, 2/3 can be expressed as follows:

1/3 + 1/3

1/3 + 1/4 + 1/12

1/3 + 1/5 + 1/20 + 1/12

1/3 + 1/6 + 1/30 + 1/20 + 1/12

...

## The Greedy Algorithm

The most basic approach by which we can express a vulgar fraction in the form of an Egyptian fraction (i.e., the sum of the unit fractions) is to employ the greedy algorithm that was first proposed by Fibonacci in 1202.

This method involves subtracting the largest potential unit fraction from the fraction of interest and then repeating the process by subtracting the largest possible unit fraction from the remainder until the point at which the maximum number of summands required matches the numerator value. However, the algorithm typically terminates in advance of that point.

It is important to note that the greedy algorithm may not always generate the simplest unit fraction expression. For instance, using the greedy Egyptian fraction algorithm on the vulgar fraction 5/121 produces the following:

5/121 = 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225

However, 5/121 can be expressed in much simpler forms:

5/121 = 1/25 + 1/759 + 1/208725

5/121 = 1/26 + 1/350 + 1/275275

5/121 = 1/27 + 1/234 + 1/84942

5/121 = 1/30 + 1/132 + 1/2420

5/121 = 1/33 + 1/91 + 1/33033

...

You may also be interested in our Fraction Calculator or/and Repeating Decimal to Fraction Converter

Egyptian Fraction (EF) Calculator

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Results

You can use this Egyptian fraction calculator to employ the greedy algorithm to express a given fraction (x/y) as the finite sum of unit fractions (1/a + 1/b + 1/c + ...).

How to use the calculator:

Simply input the numerator and denominator of the fraction in the associated fields and click on the "Calculate" button to generate the results.

## Egyptian Fractions

Ancient Egyptians used numbers and basic calculators as early as 2700 BCE. However, they expressed fractions in a very different way to the methods we employ today. Specifically, as opposed to recording three quarters as 3/4, they opted to record the ratio as the sum total of the fractions that had a numerator of one. As such, 3/4 was expressed as 1/2 + 1/4.

An Egyptian fraction representation is available for every rational number between 0 and 1, and every number in this continuum can be expressed as the finite sum of the unit fractions. Each representation is not unique. For instance, 2/3 can be expressed as follows:

1/3 + 1/3

1/3 + 1/4 + 1/12

1/3 + 1/5 + 1/20 + 1/12

1/3 + 1/6 + 1/30 + 1/20 + 1/12

...

## The Greedy Algorithm

The most basic approach by which we can express a vulgar fraction in the form of an Egyptian fraction (i.e., the sum of the unit fractions) is to employ the greedy algorithm that was first proposed by Fibonacci in 1202.

This method involves subtracting the largest potential unit fraction from the fraction of interest and then repeating the process by subtracting the largest possible unit fraction from the remainder until the point at which the maximum number of summands required matches the numerator value. However, the algorithm typically terminates in advance of that point.

It is important to note that the greedy algorithm may not always generate the simplest unit fraction expression. For instance, using the greedy Egyptian fraction algorithm on the vulgar fraction 5/121 produces the following:

5/121 = 1/25 + 1/757 + 1/763309 + 1/873960180913 + 1/1527612795642093418846225

However, 5/121 can be expressed in much simpler forms:

5/121 = 1/25 + 1/759 + 1/208725

5/121 = 1/26 + 1/350 + 1/275275

5/121 = 1/27 + 1/234 + 1/84942

5/121 = 1/30 + 1/132 + 1/2420

5/121 = 1/33 + 1/91 + 1/33033

...

You may also be interested in our Fraction Calculator or/and Repeating Decimal to Fraction Converter

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