{"slug": "math-of-egyptian-fractions-old-web-pre-ai", "title": "Math of Egyptian fractions (old web pre-AI)", "summary": "Egyptian fractions, a method of expressing rational numbers as sums of distinct unit fractions, were used in Egypt as early as 4000 BC and remained the primary way to write non-integer numbers in Europe until the 18th century. Every rational number can be represented as an Egyptian fraction, a theorem proven in 1880, with algorithms dating back to Fibonacci in the 12th century.", "body_md": "Egyptian Fractions\n\nYou may find it amazing that fractions, as\nwe know them, barely existed, for the european civilization, until\nthe 17'th century. Even in the 19th century, a method called russian\npeasant fractions, was the same used by the Europeans since they\nmet the African, and the Egyptians at least since 4000BC in Egypt.\nAs the method was found on several papyrus, we now call this technique\n**egyptian fractions**. Let us agree to call a number a *unit\nfraction *if it has form 1/n , where n is a positive integer.\n*An egyptian fraction *is an expression of the sum of unit\nfractions\n\n1/a + 1/b + 1/c + ... , where the denominators a,b,c, ... are increasing.\n\nAn *egyptian number *is any number equal\nwhich can be expressed as the sum of an integer plus the sum of\nan Egyptian fraction. Here are some egyptian fractions:1/2 + 1/3\n(so 5/6 is an egyptian number), 1/3 + 1/11 + 1/231 (so 3/7 is\nan egyptian number), 3 + 1/8 + 1/60 + 1/5280 (so 749/5280 is an\negyptian number). The egyptians also made note of the fraction\n2/3.\n\n1/5 + 1/37 + 1/4070 and 1/6 + 1/22 + 1/66 are regarded as different egyptian fractions even though the sum of each is 5/22.\n\n[ Other\nEgyptian fraction pages of note](egyptian-fractions.html).\n\n[The earliest records of\negyptian fractions date to nearly 3900 years ago in the papyrus\ncopied by Ahmes (sometimes called Ahmos - ][ref1](madrefs_ancient.html#rhind1),\n[ref2](madrefs_ancient.html#rhind0)) purportedly from\nrecords at least 300 years earlier. It is conjectured that the\nmysterious, so called, meaningless, egyptian triple 13, 17, 173\nactually means\n\n3 + 1/13 + 1/17 + 1/173 = 3.141527\nwhich [approximates](mad_ancient_egypt_geometry.html#old pi)** to\n4 places**!!!\n\nHowever, to the victor goes the spoils; i.e., prior to the \"discovery\" of the Rhind papyrus, egyptian fractions were thought, by european mathematicians, to come from the Greeks. Even the name pi is Greek.\n\nThe rule of Egyptian fractions requires us\nto write only unit fractions, integers, and their sums. So if\na duke is awarded 3/7'th of the conquered land, the quanity might\nbe represented as (1/4 + 1/7 + 1/28)'th of the conquered land,\nwhich is a bit better than\n\n(1/3 + 1/11 + 1/231)'th of the conquered land, but still awkward.\nUntil the 18'th century, when our present method, from India and\nalso thousands of years old, of writing any integer in the numerator\nbecame popular in Europe, Egyptian fractions were the primary\nmethod of writing non-integer numbers.\n\nTHEOREM. Every rational number is an egyptian number.\n\nThe modern proof of the Theorem was discovered in 1880, but European's have known how to compute Egyptian numbers since Fibonacci in the 12'th century. Before exhibiting the rule we make a convention. Given a non-integer r, let [r] denote the smallest integer > r.\n\nSuppose p/q < 1 is written in lowest terms\nthen there is an egyptian fraction with at most p terms and whose\nsum is p/q (see the [proof](mad_ancient_egypt_arith.html#proof of theorem1)\nbelow). Let r/s = p/q -1/[q/p]. So p/q = 1/[q/p] + r/s. If r=1,\nwe are done; otherwise, repeat the process. Here are some examples:\n\n1. Consider p/q = 4/23. Since 23/4 = 5.65, [23/4] = 6. Compute 4/23 - 1/6 as a fraction, and get 1/138. Thus, 4/23 = 1/6 + 1/138.\n\n2. Consider 5/22. [22/5] = 5. 5/22 - 1/5 =\n3/110. Now [110/3] = 37. But\n\n3/110 - 1/37 = 1/4070. So 5/22 = 1/5 + 1/37 + 1/4070.\n\n3. With a little ingenuity, you can determine other egyptian fractions whose sum is 5/22. For example, let's start, for no special reason, with 1/6 instead of 1/5. 5/22 - 1/6 = 2/33. [33/2] = 17 and 2/33 - 1/17 = 1/561. Thus, 5/22 = 1/6 + 1/17 + 1/561.\n\n4. In the last expression for 5/22, keep 1/6 but exchange 1/17 for 1/22. This means we compute 2/33 - 1/22 = 1/66. Thus, 5/22 = 1/6 + 1/22 + 1/66. This expression is more satisfactory since the denominators are not as large as in the proceeding two cases..\n\nNote: There is significant interest in determining\nwhich expression is \"best\" or what the egyptians would\nhave used. We discuss this on the page \"** The\nBest Egyptian Fraction**.\"\n\n[About the proof\nof the theorem:]\n\nNotice that when 0 < p < q are integers with only 1 as a common divisor (i.e., p/q < 1 is written in lowest terms) the construction in the 2nd paragraph after the theorem gives, via mathematical induction, the proof of the theorem, since for p[q/p]-q < p.\n\nIn spite of the Theorem, there is very little\ninterest in egyptian fractions (or even their modernized version\n- continued fractions) today, and only infinite series, a topic\nof elementary calculus could indicate their passage (recall from\ncalculus\n\ne-2 = 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + ...). There\nare, however, interesting and important problems. We include some\nof these in the [problem\nsection](mad_ancient_egypt_arith.html#egyptianfractionproblems) below.\n\nDivide 1077 by 25:\n\n|\n1 |\n25* |\n|\n|\n2 |\n50* |\n|\n|\n4 |\n100 |\n|\n|\n8 |\n200* |\n|\n|\n16 |\n400 |\n|\n|\n32 |\n800* |\n|\n|\n2/5 |\n2 |\n|\n|\n1+2+8+32+ 1/3 + 1/15 =43 |\n25+50+200+800+2=1077 |\n\nAs the egyptians wrote 1/3 + 1/15 for our 2/5. Infact, the Ahmos scroll contains a table of decompositions of each odd fractions of the form 2/n where n ranges from 3 to 101. Here are a few:\n\n|\n2/5 |\n1/3 + 1/5 |\n|\n2/7 |\n1/4 + 1/28 |\n|\n2/9 |\n1/6 + 1/18 |\n|\n2/15 |\n1/10 + 1/30 |\n|\n2/17 |\n1/12 + 1/51 + 1/68 |\n|\n2/101 |\n1/101 + 1/202 + 1/303 + 1/606 |\n| for more see\n|\n\nMultiplication of egyptian fractions\n\nMultiply 383/15 by 130/3 or 25 + 8/15 = 25 + 1/3 +1/5 by 43 + 1/3\n\n|\n1* |\n25 + 1/3 +1/5 |\n|\n2* |\n50 + 2/3 + 2/5 = 50 + 2/3 + (1/3 + 1/15) = 51 + 1/15 |\n|\n4 |\n102 + 2/15 = 102 + 1/10 + 1/30 |\n|\n8* |\n204 + 1/5 + 1/15 |\n|\n16 |\n408 + 2/5 + 2/15 = 408 + (1/3 + 1/15) + (1/10 + 1/30) |\n|\n32* |\n816 + (2/3 + 2/15 )+ (1/5 + 1/15) = 816 + 2/3 + 2/5 = 816 + 2/3 +1/10 + 1/30 |\n|\n2/3 |\n544 + 4/9 + 1/15 + 1/45 |\n|\n1/3* |\n272 + 2/9 + 1/30 + 1/90 |\n|\n|\n25 + 1/3 +1/5 + 51 + 1/15 + 204 +\n1/5 + 1/15 + 816 + 2/3 +1/10 + 1/30 + 272+ 2/9 + 1/30 + 1/90= 1369 + 1/3 + 1/5 + 1/6 + 1/18 + 1/30 + 1/90 |\n\n[PROBLEMS\nUSING EGYPTIAN FRACTIONS]\n\n1. **The Mullah's horse**: The former Grand Wizier, Mullah\nNasrudin was approached by three men with 19 horses. The men asked\nhim to adjudicate the will of their recently dead father which\nrequired that his horses be divided among his three sons so that\nthe oldest son receives 1/2, the middle son gets 1/3, and the\nyoungest son would get 1/7. With little hesitation Nasrudin added\nhis own horse to the herd and said, \"What is half of 20,\n1/4 of 20, and 1/5 of 20\" After some time the men replied,\n\"10, 5, and 4\". The eldest son then took 10 of the horses,\nthe middle son took 5 of the horses, and the youngest son took\n4 of the horses. The Mullah Nasrudin, then took the remaining\nhorse and rode home. Can you explain what occured?\n\n2. Find all the solutions (there are less than 10) to the problem\n(n-1)/n = 1/a + 1/b + 1/c, where a < b,\n\nb < c, a, b, and c are positive integers with least common\nmultiple n. Note. a = 2, b = 4, c = 6, and n = 12 gives one solution.\n\n3. How many different egyptian fractions can be used to describe 2/3? Two of them are 1/2 + 1/3 + 1/6 and 1/3 + 1/10 + 1/15.\n\n5. Write a program (in BASIC or any other language) for computing\negyptian fractions representing all fractions p/q, p < q with\nq at most 50 (hint: look at the proof above to check your answer\nsee the ** Rhind Papyrus\n2/n table**.\n\n6. The following is problem 33 from the Ahmes Papyrus. Solve it using egyptian fractions only: The sum of a certain quantity together with its two-thirds and its one-seventh becomes 37. What is the quantity?\n\n7. **Want to solve an unsolved problem?** One of the most\nfamous problems on Egyptian Fractions asks, \"Can every proper\nfraction of the form 4/q be expressed with an egyptian fraction\nwith less than 4 terms?\" Can every proper fraction of the\nform 5/q be expressed with an egyptian fraction with less than\n4 terms?\n\n8. **The sailor, coconut, and monkey problem**: Five sailors\nwere abandoned on an island. To provide food, they collected all\nthe coconuts they could find. During the night one of the sailors\nawoke and decided to take his share of the coconuts. He divided\nthe nuts into five equal piles and discovered that one was left\nover, so he threw the extra coconut to the monkies. He then hid\nhis share and went back to sleep. A little later a second sailor\nawoke and had the same idea as the first. He divided the remainder\nof the nuts into five equal piles, discovered also that one was\nleft over, and through it to the monkies before hiding his share.\nIn turn each of the other three sailors did the same - dividing\nthe observable amount into five equal piles, hiding one, throwing\none left over to the monkies. The next morning the sailors, looking\ninnocent, divided the remaining nuts into five piles with none\nleft over. Find the smallest number of nuts in the original pile.\n\n4. 355/113 approximates to 6 places. (355/113) - 3 = 16/113. Find an egyptian fraction whose sum is 16/113.\n\n9. ** Above we noted the triple 13, 17, 173**\nused to approximate- The numbers 3 and 7 were\nvery important to egyotian mythology.\n\n**URL: http://www.math.buffalo.edu/mad/Ancient-Africa/mad_egyptian-fractions.html**\n\nGOTO [The\nBest Egyptian Fraction](best-egyptian-fraction.html)\n\n**BACK TO **[EGYPTIAN ARITHMETIC](mad_ancient_egypt_arith.html)\n\nThe web pages\n\n[MATHEMATICIANS OF THE AFRICAN DIASPORA](../mad0.html)\n\nare brought to you by\n\nThe Mathematics Department of\n\nThe State University of New York at Buffalo.\n\nThey are created and maintained\nby\n\nScott W. Williams\n\nProfessor of Mathematics\n\n**CONTACT\nDr. Williams**\n\n**5/24/99**", "url": "https://wpnews.pro/news/math-of-egyptian-fractions-old-web-pre-ai", "canonical_source": "https://www.math.buffalo.edu/mad/Ancient-Africa/mad_egyptian-fractions.html", "published_at": "2026-06-25 04:09:21+00:00", "updated_at": "2026-06-25 04:43:40.269118+00:00", "lang": "en", "topics": ["machine-learning"], "entities": ["Ahmes", "Fibonacci", "Rhind papyrus"], "alternates": {"html": "https://wpnews.pro/news/math-of-egyptian-fractions-old-web-pre-ai", "markdown": "https://wpnews.pro/news/math-of-egyptian-fractions-old-web-pre-ai.md", "text": "https://wpnews.pro/news/math-of-egyptian-fractions-old-web-pre-ai.txt", "jsonld": "https://wpnews.pro/news/math-of-egyptian-fractions-old-web-pre-ai.jsonld"}}