# How did egyptian mathematical fractions evolve in antiquity?

• Hamilton080
In summary, the evolution of fractions in ancient Egypt can be traced back to the old kingdom. The earliest use of fractions was in metrological systems, where they were written using specific symbols. These fractions were later used in more abstract contexts. The symbols used for fractions have their origin in the eye parts of Horus. The reason for the development of fractions was to solve problems with agricultural field boundaries after the annual Nile flood. However, the exact reason for the use of fractions in other contexts, such as in mathematical calculations, is not known.
Hamilton080
There are certain explanations on how integers might have evolved, like for example "the wings of a bird to symbolize the number two, clover--leaves three, the legs of an animal four, the fingers on his own hand five."1 Seeing all these, and making experience short--abstract, can be said to have given the numbers2. Egyptian integer number notation can itself be traced back to their notion of marks on tally (for the sign of one), hobble for cattle (for ten), measuring rope (for hundred).3

Similarly what are the explanations given for the evolution of fractions in Egypt, by any of the found evidence? Did division discovery gave raise to fractions or fractions discovery gave raise to division, in Egypt?

Book and journal reading suggestions would be helpful to a greater extent.

I had already asked this question in other websites, but didn't get the help, so I have posted it here again.
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1Tobias Dnatzig, Number: The Langauge of Science
2This expression can be found in Hamilton's letter of Sep.16, 1828, in the book Life of Sir William Rowan Hamilton.
3Annette Imhausen, Mathematics in Ancient Egypt: A Contextual History Page No.18-21.

Egyptian fractions were really the sum of reciprocals, in modern terms. The main reason for their development was to reestablish agricultural field boundaries after the annual Nile flood. As part of grad school requirement I had to pass tests from the Chicago Field Museum - hieratics (Middle Egyptian), hieroglyphs, translation English<-> middle Egyptian and also some math translations. That was 50 years ago. I do not think I want to do that again. I cannot do it most of that a priori now. What are you doing? Coding Egyptian fractions? https://rosettacode.org/wiki/Egyptian_fractions

I don't know what you want to find, but start with this: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

Yours truly not a pharaoh:
james

Hamilton080 and jedishrfu
I don't think we will ever know a definitive answer to this question. Number systems come about when we have a problem that can't be solved by the best number system we have and so after awhile someone invents/discovers an extension that allows us to make the intractable problem more tractable.

Interestingly our english unit-based rulers have a notion of egyptian fractions embedded in the markings if you consider how you read the length 7/8" by reading the 1/2" mark + 1/4" mark +1/8" mark and then mentally say 7/8"

Here's a discussion on Egyptian fractions that shows some of the advantages:

https://nrich.maths.org/1173

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html#section2

Also there's a book by Jan Gulberg that goes into much of the history of many number systems:

https://www.amazon.com/dp/039304002X/?tag=pfamazon01-20

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Hamilton080
"Rhind papyrus" and "Moscow papyrus" and read about their interpretations and implications. IMO, practical math to solve problems about crop field boundaries that were obliterated every year by the flood.

Hamilton080 and jedishrfu
I was trying to think of a more definitive example like imaginary numbers came when we couldn't compute the square root of negative numbers. So in the case of Egyptian surveying was it that they had to write down some fraction of a cubit?

The wiki article shows a cubit rod with the fractional measurements on it so that might be the reason:

https://en.wikipedia.org/wiki/Cubit

Hamilton080
Here is what I have found:

Below in quote 1, the usual requirement of using parts of an object can be seen, like 1/4 of st3t (which is usually a symbol for a particular area of land)--as like @jedishrfu seems to suggest the origin from surveying. We can see a possibility that the Egyptians might have used words of half, quarter, etc, in the usual sense, without any attachment to numbers. But later they might have come to next level of abstraction by seeing the relation of half being one part of two formed by original.
Quote 2 is added to notice that the signs of fractions have their origin in the eye "parts" of Horus.

(1) From page 53 of Annette Imhausen:
The evolution of fractions in ancient Egypt can be traced back at least as far as the old kingdom. The beginnings consisted of a small group of specific fractions written by special signs [see the symbols here: notation]. These are first attested within the context of metrological systems, that is, 3/4 in 3/4 of a finger and 1/4 in 1/4 of ----[a symbol which is similar to st3t], but they retain their notation in later times as abstract fractions

(2) From wiki-the Eye of Horus:
In one myth, when Set and Horus were fighting for the throne after Osiris's death, Set gouged out Horus's left eye. The majority of the eye was restored by either Hathor or Thoth (with the last portion possibly being supplied magically).

Image: Eye of Horus

The reason why I wanted to know the evolution of fractions was to know on how 1/3 (=0.333..), 2/3 (=0.66..), etc came to be accepted. But, now I feel to have been a fool to think it to help me.

Thank you very much for the help till now. Better I should leave Pharaohs and move ahead to meet the inventors of decimal notation.

## 1. How did ancient Egyptians use fractions in their daily lives?

Ancient Egyptians used fractions primarily in measuring and dividing quantities of food and other goods. They also used fractions in construction and land surveying.

## 2. What is the earliest evidence of Egyptian fractions?

The earliest evidence of Egyptian fractions can be found in the Rhind Mathematical Papyrus, which dates back to around 1650 BCE. This papyrus contains a variety of mathematical problems and solutions, including fractions.

## 3. How did Egyptian fractions differ from modern fractions?

Egyptian fractions were written as sums of unit fractions, where the numerator is always 1. For example, 3/4 would be written as 1/2 + 1/4. They also did not have a symbol for zero and did not use decimals or percentages.

## 4. How did Egyptian mathematical fractions evolve over time?

Egyptian fractions remained largely unchanged for thousands of years until the introduction of Greek mathematics. This led to the development of new methods for working with fractions, such as the use of common denominators and the concept of equivalent fractions.

## 5. Did other ancient civilizations use fractions?

Yes, many other ancient civilizations, such as the Babylonians, Greeks, and Romans, also used fractions in their mathematics. However, their methods and notation were different from those used by the ancient Egyptians.