An Introduction to Egyptian Mathematics

Transcription

1 An Introduction to Mathematics Some of the oldest writing in the world is on a form of paper made from papyrus reeds that grew all along the Nile river in Egypt. The reeds were squashed and pressed into long sheets like a roll of wall-paper and left to dry in the sun. When dry, these scrolls could be rolled up and easily carried or stored. Some of the papyrus scrolls date back to about 2000 BC, around the time of the construction of the larger pyramids. Because there are deserts on either side of the Nile, papyrus scrolls have been well preserved in the dry conditions. So what was on them do you think? How to preserve a body as a mummy? Maybe it was how to construct the extensive system of canals used for irrigation across Egypt or on storage of grain in their great storage granaries? Perhaps they tell how to build boats out of papyrus reeds which float very well because pictures of these boats have been found in many tombs? The surprising answer is that the oldest ones are about mathematics! Henry Rhind and his Papyrus scroll One of the papyrus scrolls, discovered in a tomb in Thebes, was bought by a 25 year old Scotsman, Henry Rhind at a market in Luxor, Egypt, in After his death at the age of 30, the scroll found its way to the British Museum in London in 1864 and remained there ever since, being referred to as the Rhind

2 Mathematical Papyrus (or RMP for short). So what did it say? The hieroglyphs (picture-writing) on the papyrus were only deciphered in 1842 (and the Babylonian clay-tablet cuneiform writing was deciphered later that century). It starts off by saying that the scribe "Ahmes" is writing it about 1600 BC but that he had copied it from "ancient writings" which, from his description of the Pharoah of that time dates it to 2000 BC or earlier. The picture is also a link so click on it to go to the St Andrews MacTutor biography of Ahmes. Since early civilisations would need to predict the start of spring accurately in order to sow seeds, then a large part of such mathematical writing has applications in astronomy. Also, calculations were needed for surveying (geometry) and for building and for accounting and for sharing bread and beer (given as wages) amongst several workers. On this page we will look at how the s of 4000 years ago worked with fractions. Fractions The s of 3000 BC had an interesting way to represent fractions. Although they had a notation for 1 / 2 and 1 / 3 and 1 / 4 and so on (these are called reciprocals or unit fractions since they are 1 / n for some number n), their notation did not allow them to write 2 / 5 or 3 / 4 or 4 / 7 as we would today. Instead, they were able to write any fraction as a sum of unit fractions where all the unit fractions were different. For example, 3 / 4 = 1 / / 4

3 6 / 7 = 1 / / / 42 A fraction written as a sum of distinct unit fractions is called an Fraction. Why use fractions today? Suppose you and 7 other friends go for a pizza. You all like the same kind but you don't want a whole one each. The 8 of you decide to buy 5 identical pizzas. How do you divide them up between you? Decimals don't help and that you each get 5/8 only tells you what the problem is (split 5 things into 8 parts) not the solution! It is easier with beer or sacks of grain. This was an everyday problem in ancient Egypt when barley loaves may have to be divided amongst workers. Math in Action: A practical use of Fractions So suppose Fatima has 5 loaves of bread to share among the 8 workers who have helped dig her fields today and clear the irrigation channels. Pause for a minute and decide how YOU would solve this problem before reading on... HINT: What if there were only 4 loaves not 5 to be split amongst 8 people? First Fatima sees that they all get at least half a loaf, so she uses 4 of the loaves to give all 8 of them half a loaf each. She has one whole loaf left. Now it is easy to divide one loaf into 8, so they get an extra eighth of a loaf each and all the loaves are divided equally between the 5 workers. On the picture here they each receive one red part ( 1 / 2 a loaf) and one green part ( 1 / 8 of a loaf): and 5 / 8 = 1 / / 8 The solution has the added benefits of fewer crumbs/slices than dividing each loaf into 8 and giving 5 slices to each person.

4 it is easy to see everyone has an identical number of pieces of the same sizes. each portion received is a fraction 1/n of a loaf. All the fractions are different and unit fractions Try the following using the style of thinking: Things to do 1. Suppose Fatima had 3 loaves to share between 4 people. How would she do it? 2....and what if it was 2 loaves amongst 5 people? 3....or 4 loaves between 5 people? 4. What about 13 loaves to share among 12 people? We could give them one loaf each and divide the 13 th into 13 parts for the final portion to give to everyone. Try representing 13 / 12 as 1 / / / *. What does this mean - that is, how would you divide the loaves using this representation? Was this easier? It turns out that fractions are not only a very practical solution to some everyday problems today but are interesting in their own right. They had practical uses in the ancient method of multiplying and dividing, and every fraction t / b can always be written as an fraction, which we will show further down on this page. There are also many unsolved problems concerning them, which are still a puzzle to mathematicians today.

5 Name Pg # Lotus Diagram: Math : Read the different sections about math. In the top boxes take notes with important ideas about Math. In the bottom boxes review the facts placed in the top boxes and pick out the aspects you listed that reflect the way math transformed their society and we can see it in today s society. Complete the problem Intro to Math Henry Rhind & Scroll Fractions Explain how using math can solve everyday problems. Math transformed society We see math today in this way