Tag Archives: fractions

A New World of Thought in Fractions

I never thought about math as a fluid subject that would be approached differently depending on the needs of a culture.  We are taught from a young age how to do math and what it applies to, but I never thought that in different cultures, in a different time, people’s application of math could be so different from ours.

Let me explore my thought this way.  When I think of a math problem I need to solve on a daily basis, it is usually based around money, or computer memory.  I have five dollars and I want to buy a sandwich and fries.  Do I have enough money?  Or I have 3 GB of data for my phone.  Can I watch my favorite TV show on 4G or do I need to wait until I get home?  Many of the math problems we run into on a day-to-day basis support a monetary society where I have money, so I use that money to get more things.

This is starkly different from what we learned of different cultures in class.  The culture in particular I think of is Egyptian culture.  From the book, Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics, I learned that in ancient Egypt they had a very different fractional system than we do.  When we think of fractions we have a denominator and a numerator.  I like to think of the denominator as a bucket with n places for an object. The numerator is how many objects we have in the bucket, or how full it is.  In Egypt they had a very different system.  They represent fractions as a single number with a line over it.  The number represented an amount and the line indicated that it was a fraction.  It is a little foreign to us, but if the Egyptians wrote 6, with a line over the six, then we could think of it as 1/6.  It is important to note, and I will discuss later that this system can represent any fraction.  It would just take a bit more work in order to do that.

Why did they do it that way and why do we do things the way we do?  I think in our society our number system fulfills our desire to fill and make sure we have enough.  Our fractional system is very supportive of the money we use, because money is how we support ourselves. To elaborate on this I will use my bucket analogy for fractions. If you recall I said the denominator in our fractional system is like a bucket with room for n amount of objects.  The numerator tells us how many objects are in the bucket.  If the numerator and denominator are the same then we have filled one bucket. We use this all the time when we go shopping food.  When we shop the denominator is the price of what we want to buy.  The numerator is how much money we have.  If we have enough money, which represents the numerator, to fill the price, which represents the denominator, then we can get that item.  Or using my analogy you could say we filled the bucket.

In ancient Egypt their fraction system also could have supported their means of commerce.  They could have had more of a barter system were you worked for food and your master or boss gave you a portion of what you helped him produce.  For example if ten workers made 12 loaves of bread the boss possibly would have taken four for himself, leaving eight for the workers.  Using the Egyptian fractional system he could quickly and efficiently think, I have eight loaves for ten workers.  I will divide 5 loaves into halves giving me 2, with a line over the 2(1/2), which is, half a loaf for each worker.  Now I still have 3 loaves left.  If I divide those into fourths or 4, with a line over the 4(1/4), then I will have given each worker ½ a loaf and ¼ a loaf and there will be ½ a loaf left.  Know all I need to do is break that into tenths, but since it was a half loaf that would actually be twentieths.  That gives me 2 4 20, all with a with a line over them, which to us would be ½+1/4+1/20.  Now the boss had divided his loaves evenly and easily.

Learning this got me wondering.  How was it that other cultures did math?  Could I find another unique way fractions were used?  In the article, History of Fractions, Liz Pumfry talks about a Babylonian method of writing fractions.  Their method was also very simple and a little confusing.  To explain the Babylonian fraction system I first have to explain that the Ancient Babylonian number system was base 60.  The Babylonians wrote their fractions as demonstrated in the image below.

Image from nrich.maths.org.

The first grouping of symbols is for twelve and the second is for fifteen.  In Babylonian fractions this would come out to be 12 + (15/60).  That is, if this was a fraction.  Because the ancient Babylonians did not have a symbol for zero or a decimal point this grouping of numbers could have other meanings.

x60 Units Sixtieths Number
12 15 12+15/60=12
12 15 720+15

But why did the Babylonians use this method for fractions?  It seems to me that this method kept their number system simple, but in my quick analysis of this study I have found Babylonian fractions to be limited. In the article Babylonian Mathematics, it says, “Irregular fractions such as 1/7, 1/ 11, etc were not normally not used. There are some tablets that remark, ‘7 does not divide’, or ‘11 does not divide’, etc.” From this it seems we can deduce that the Babylonians disregarded some fractions.  It seems to me this could have limited them in some regards. For example, if they only used sixty as a denominator then the smallest number they could represent would be 1/60.  Let’s say they could increase this, though, using 602 or 60n.  If this is the case they could represent much smaller numbers, but it would still be difficult to represent values such as 1/7 or 1/13.  They might be able to do this by adding different values together, but that sounds very difficult and sloppy.   From this I conclude that they would lose accuracy with their fractional system.

Despite losing accuracy, addition and subtraction of fractions would be made much easier if the denominator was always sixty or a power of sixty.  This would make it easier for them to teach fractions as well as learn fractions.  They would not have to worry so much about finding a common denominator as we do.

It is very interesting to note that the Babylonian representation of fractions is very similar to our representation of decimals.  It is, in fact, so similar that I feel it is important to point out.  Our decimal system is base 10.  If we write 0.1 we are essentially writing 1/10.  This is essentially the exact same as the Babylonian system except they are using base sixty.  This is in fact so similar that it causes us to have the same problem representing some fractions.  If you want to write 1/7 in our decimal system you will quickly realize that this is impossible.  There is no concrete way to represent 1/7 in a base ten system.

It is very interesting to think about these different methods for representing fractions.  The Egyptians, if you think about it, actually had a very strong system that, while confusing, could represent all fractions very accurately.  It also suited their needs very well.  The Babylonians had a system very similar to our decimal system.  It seems foreign to us to use base sixty, but base ten is all we know and if they were confronted with our decimal system they would probably find it similarly confusing.  Our fractional system is, I believe, very strong, but in some ways I look at the Egyptian method and think it could have been very helpful when I wanted to share candy with friends or at the dinner table when everyone is fighting to make sure they get their equal portion of food.

In my studies I could not find any other Ancient Babylonian use of fractions.  I am also left to speculate on their reasons for their fractional system.  It is interesting to consider what a different cultures mathematical systems might have been and speculate as to why they used the method they did.



Count Like an Egyptian: A hand-on Introduction to Ancient Mathematics by David Reimer


Egyptian Numbers: Counting By Ten

When most people think of early mathematics, the first thing that comes to mind is probably Archimedes or Pythagoras of ancient Greece; or perhaps ancient Mesopotamia and its thousands of clay tablets, many of them containing math problems.  Few people think of ancient Egypt. This is mainly due to the fact that little is known about ancient Egyptian math in comparison to these other civilizations. However, by looking at what we do know about Egyptian math and the feats of engineering they accomplished with it, we can recognize how complex and sophisticated it was.

Rhind Mathematical Papyrus

A fragment from the Rhind Mathematical Papyrus. Image: Public domain, via Wikimedia Commons.

The majority of our knowledge of ancient Egyptian math comes from two papyri: the Rhind Mathematical Papyrus, which was originally written about 1985-1975 BCE; and the Moscow Mathematical papyrus, dating from around 1850 BCE. Math problems were not written on the walls of temples or great monuments in Egypt, so all math was probably done on papyrus or other perishable media, meaning that many mathematical works have not come down to us. This in in contrast to the mathematical documents from Mesopotamia, which were primarily done on clay tablets that are not as susceptible to the elements and ravages of time.

The ancient Egyptians probably developed the first base ten numerical system in human history. It was fully in use around 2700 BCE and perhaps even earlier. They had different symbols representing the powers of ten. A straight vertical line represented one, 10 was shown by a drawing of a hobble for cattle, a picture of a coil of rope served as a symbol for 100, a lotus plant delineated 1,000, a bent finger for 10,000, a tadpole or a frog for 100,000 and the picture of a god, perhaps Heh, represented 1 million. Although the Egyptian numerals were in a decimal system, it was not a positional place value system like the decimal system we use, or the sexagesimal (base 60) system developed by the Mesopotamians. Multiples of the powers of ten were written by repeating the symbol as many times as needed, and although they had a symbol for 1 million, other large numbers would have been very tedious to write. For instance, the number 987,654 would have required 39 characters. This Egyptian system possibly had an influence on the later Greek numerical system, but the Greeks improved upon it, creating a different symbol for each number 1-9, and other symbols for 20, 30 and so on.

Parts of the Eye of Horus were used to write fractions. Image: BenduKiwi, via Wikimedia Commons.

Parts of the Eye of Horus were used to write fractions. Image: BenduKiwi, via Wikimedia Commons.

The Egyptians also had notation for fractions, although all Egyptian fractions were unit fractions (meaning they always had one in the numerator), with only two exceptions, 2/3 and 3/4. Fractions were marked by the hieroglyph for ‘R’ which is a long skinny oval, very similar in shape to the Eye of Horus. In fact, in an ancient Egyptian myth, the evil god Seth attacked his brother the god Horus and in the fight Seth gouged out the eye of Horus and tore it into pieces, fortunately the god Throth was able to put Horus’s eye back together and heal it. Thus, to honor the gods and this myth, whenever the Egyptians used fractions in relation to their measurement of volume, which is the hekat, the commonly used fractional parts were transcribed by using the corresponding parts of the Eye of Horus.

The great feats of engineering that the ancient Egyptians accomplished would have required an enormous amount of sophisticated math. The pyramids, for instance, are considered a marvel of mathematics and engineering. The base of the pyramids are almost perfect squares which the Egyptians would have achieved by using trigonometry, like the 3-4-5 trick. It was known in Egypt that a triangle with 3, 4 and 5 unit sides would always be a perfect right triangle. This Pythagorean triple, observed by the Egyptians long before Pythagoras, is sometimes called an “Egyptian triangle.” The Egyptians would have utilized this rule of geometry when laying out the base of a pyramid by tying knots in rope at 3, 4 and 5 unit intervals. Other tricks of trigonometry were known as well. The Rhind Mathematical Papyrus contains an equation for calculating the slope of a pyramid’s face, which is the same thing as finding the cotangent. It is also likely that they knew how to find a pyramid’s volume.

It is unfortunate that more is not known about the mathematics that were used by the ancient Egyptians, because they were obviously very skillful and innovative, and they must have been among the first people to develop important mathematical principals.