Egyptian fractions

The Egyptian fraction is a sum of unique fractions with a unit numerator (unit fractions). There is an infinite number of ways to represent a fraction as a sum of unit fractions. Several methods have been developed to convert a fraction to this form. This calculator can be used to expand a fractional number to an Egyptian fraction using Splitting, Golomb, Fibonacci/Sylvester, Binary, or Bleicher/Erdős methods¹. Enter any number between 0 and 1 in decimal or simple fraction form, and the calculator will expand it to a sum of distinct unit fractions. The calculator also can try to find the best method among listed above, minimizing either denominators sum or maximal denominator (see below for more details on best method criteria).

Egyptian fraction expansion

Common fraction

Expansion methodBest method : Denominator sum Best method : Maximal denominator Binary remainder Bleicher/Erdös Fibonacci /Sylvester Golomb Splitting

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Egyptian fractions

$\displaystyle{\frac{1}{7}+\frac{1}{91}+\frac{1}{247}+\frac{1}{475}+\frac{1}{775}}$

Method

Golomb

Denominators

7

91

247

475

775

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See Egyptian Fraction to Rational Number Converter for the inverse transformation.
Ancient Egyptians did not use the fraction expansion methods mentioned above to represent a fraction as a unit fraction sum. We can consider that analyzing ancient documents surviving to this day. The calculator below uses the algorithms mentioned above to expand fractions with the numerator 2 and the odd denominator in the range from 5 to 101 and compare the results with the Rhind papyrus (1650 B.C.E). The Golomb method does not participate in comparison since it gives the same results as the Fibonacci/Sylvester method for the Rhind papyrus data (and all fractions with numerator = 2 in the general case).

Rhind papyrus and fraction expansion algorithms

Best algorithm criteriaExact match : Binary remainder Exact match : Bleicher/Erdos Exact match : Fibonacci/Sylvester Exact match : Rhind papyrus Exact match : Splitting Minimise : Denominator sum Minimise : Maximal denominator Minimise : Number of hieroglyphs Minimise : Number of hieroglyphs 2 Minimise : Terms count

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Calculation precision

Digits after the decimal point: 2

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Summary

Method name	The best results count	One of the best
Rhind papyrus	43	6
Fibonacci /Sylvester	0	4
Binary remainder	0	2
Bleicher/Erdös	0	1
Items per page: 5102050100200500100010K100KAll 1 - 4 of 4

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Details

Fraction	Denominators	Best methods	Terms count	Hieroglyph count
2/5	3 15	Rhind papyrus,Fibonacci /Sylvester	2	11
2/7	4 28	Rhind papyrus,Fibonacci /Sylvester,Binary remainder	2	16
2/9	6 18	Rhind papyrus	2	17
2/11	6 66	Rhind papyrus,Fibonacci /Sylvester	2	20
2/13	8 52 104	Rhind papyrus,Binary remainder	3	23
2/15	10 30	Rhind papyrus,Bleicher/Erdös	2	6
2/17	12 51 68	Rhind papyrus	3	26
2/19	12 76 114	Rhind papyrus	3	25
2/21	14 42	Rhind papyrus	2	13
2/23	12 276	Rhind papyrus,Fibonacci /Sylvester	2	20
2/25	15 75	Rhind papyrus	2	20
2/27	18 54	Rhind papyrus	2	20
2/29	24 58 174 232	Rhind papyrus	4	42
2/31	20 124 155	Rhind papyrus	3	23
2/33	22 66	Rhind papyrus	2	18
2/35	30 42	Rhind papyrus	2	11
2/37	24 111 296	Rhind papyrus	3	29
2/39	26 78	Rhind papyrus	2	25
2/41	24 246 328	Rhind papyrus	3	34
2/43	42 86 129 301	Rhind papyrus	4	40
2/45	30 90	Rhind papyrus	2	14
2/47	30 141 470	Rhind papyrus	3	23
2/49	28 196	Rhind papyrus	2	28
2/51	34 102	Rhind papyrus	2	12
2/53	30 318 795	Rhind papyrus	3	39
2/55	30 330	Rhind papyrus	2	11
2/57	38 114	Rhind papyrus	2	19
2/59	36 236 531	Rhind papyrus	3	32
2/61	40 244 488 610	Rhind papyrus	4	45
2/63	42 126	Rhind papyrus	2	17
2/65	39 195	Rhind papyrus	2	29
2/67	40 335 536	Rhind papyrus	3	32
2/69	46 138	Rhind papyrus	2	24
2/71	40 568 710	Rhind papyrus	3	34
2/73	60 219 292 365	Rhind papyrus	4	49
2/75	50 150	Rhind papyrus	2	13
2/77	44 308	Rhind papyrus	2	21
2/79	60 237 316 790	Rhind papyrus	4	48
2/81	54 162	Rhind papyrus	2	20
2/83	60 332 415 498	Rhind papyrus	4	49
2/85	51 255	Rhind papyrus	2	20
2/87	58 174	Rhind papyrus	2	27
2/89	60 356 534 890	Rhind papyrus	4	53
2/91	70 130	Rhind papyrus	2	13
2/93	62 186	Rhind papyrus	2	25
2/95	60 380 570	Rhind papyrus	3	32
2/97	56 679 776	Rhind papyrus	3	56
2/99	66 198	Rhind papyrus	2	32
2/101	101 202 303 606	Rhind papyrus	4	28
Items per page: 5102050100200500100010K100KAll 1 - 49 of 49

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The following comparison criteria give the best results for original data from Rhind papyrus comparing to all method results:

• Minimise : Maximal denominator
• Minimise : Denominator sum

Both comparison criteria chooses the Rhind papyrus fraction expansions as the best in 46 of 49 cases. The Fibonacci/Sylvester methods wins for Minimise : Hieroglyph count and Minimise : Terms count criteria. But if you slightly change the method of counting hieroglyphs (if you count as one any dash set, denoting numbers from 2 to 9), the Rhind papyrus expansion will only lose slightly to the Fibonacci/Sylvester method.