Fixed Length Egyptian Fraction Calculator

Understanding Fixed Length Egyptian Fraction Conversion

What are Fixed Length Egyptian Fractions?

Fixed length Egyptian fractions are a way of expressing fractions as the sum of a specified number of distinct unit fractions (fractions with 1 as the numerator). This system is based on the ancient Egyptian method of representing fractions, but with a modern twist of limiting the number of terms.

Formula for Fixed Length Egyptian Fraction Conversion

The conversion process uses an iterative algorithm, which can be represented as:

\[f = \frac{n}{d} \approx \sum_{i=1}^{k} \frac{1}{\lceil \frac{1}{r_i} \rceil}\]

Where:

• \(f\) is the original fraction
• \(n\) is the numerator
• \(d\) is the denominator
• \(k\) is the fixed length
• \(r_i\) is the remainder at step \(i\)
• \(\lceil x \rceil\) denotes the ceiling function (smallest integer greater than or equal to x)

Converting a Fraction to Fixed Length Egyptian Fractions

To convert a fraction to its fixed length Egyptian fraction representation, follow these steps:

1. Initialize the remainder as the original fraction.
2. Take the reciprocal of the ceiling of the reciprocal of the current remainder.
3. Subtract this unit fraction from the current remainder.
4. Repeat steps 2 and 3 until the desired length is reached or the remainder becomes 0.

Calculation Steps

Let's convert 5/6 to a fixed length (3) Egyptian fraction:

1. Initialize: remainder = 5/6 ≈ 0.8333
2. \(\frac{1}{\lceil \frac{1}{0.8333} \rceil} = \frac{1}{2}\)

   New remainder: 0.8333 - 0.5 = 0.3333

3. \(\frac{1}{\lceil \frac{1}{0.3333} \rceil} = \frac{1}{3}\)

   New remainder: 0.3333 - 0.3333 = 0

4. Remainder is 0, so we stop here.

Example with Visual Representation

Let's visualize 5/6 as a fixed length (3) Egyptian fraction:

This diagram shows 5/6 expressed as the sum of unit fractions: 1/2 + 1/3. Note that we only needed two terms to represent 5/6 exactly, so the third term is not necessary.