Real Mathematics – Numbers #4

In schools we start learning mathematics with learning what numbers are. Unfortunately numbers are taken for granted and being overlooked just because it starts in the elementary school. The truth is this part of mathematics is a joint work of countless civilizations that lasted thousands of years. Although, categorizing and defining all those information were done only in the near past. This means that things we learn in the first few years of school have so much more depth than we think they have.

Especially fractions (or rational numbers) weren’t used in Europe in the sense we understand them today until the 17th century. In fact for a long time people thought of fractions not as numbers but as two numbers being divided to one another.

Rhind

Ancient Egyptians were one of the first known civilizations that used fractions. They created one of the most important and oldest documents in the history of civilizations using papyrus trees. Around 4000 years ago they started writing valuable information on papyrus leafs. Rhind papyrus is one of those documents. It is believed to be written around 1800 BC. Thanks to Rhind, we can understand how ancient Egyptians used fractions.

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It is uncanny how commonly they used fractions in Rhind. Although they were obsessed with unit fractions as they found ways to describe every fraction with them.

Unit Fraction: Fractions that have 1 on their numerators.

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In the ancient Egypt they used a shaped that looks like an open mouth (or an eye). This shape was the notation for the unit fraction. Denominator of the fraction would be placed under the mouth.

Table of 2/n

Inside Rhind there is a method for describing fractions in the form of 2/n (when n is odd) with two unit fractions. Table starts with 2/3 and ends with 2/101. In the papyrus it says that 2/3 is equal to ½ + 1/6. For the rest of the papyrus a formula was given in order to describe fractions in the form of 2/3k: It is 1/2k + 1/6k.

Let’s try it for 2/9. 9 is 3k, hence k=3. This gives 2/9 = 1/6 + 1/18. Ingenious, isn’t it?

Next number on the table is 2/5 = 1/3 + 1/15. This is also a general formula just like the previous one. Any fraction in the form of 2/5k can be shown as 1/3k + 1/15k.

Fraction Line

Besides ancient Egyptians, Babylonians used fractions too. But their choice of symbols was so confusing, it is impossible to understand which number is written. You could only check the rest of the calculation (if there is any) and guess which number is being used.

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In the Babylon civilization number system has base 60. The one on the left is 12, other one is 15. But they can also mean 12+(15/60) too. Lack of symbol for fractions caused a lot of problems in this civilization.

Around 1500 years ago Indian mathematicians were shining. They found the number system we use today and even the number zero was invented (or discovered). Their brilliance was key for fractions too as they showed fractions one under the other. Muslims were the ones who thought of putting the fraction line between numbers.

In the end we owe our modern notation to Indian and Muslim mathematicians.

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The way 7/15 was shown in the old Indian symbols.

One wonders…

Try to find answers to following questions about the Rhind:

  1. Why did they only consider odd numbers in the denominator of the fractions?
  2. Find 2/7 and 2/11 using their methods.
  3. What happens after 11?
  4. Try to come up with a general formula for 3, 5, 7 and 11.

M. Serkan Kalaycıoğlu

Real Mathematics: Geometry – #7

Ancient Greek philosophers used only compass and an unmarked ruler to come up with incredible results in geometry. In this article, I’ll be talking about what kinds of methods were used in geometry’s holy book Euclid’s Elements in order to cut a segment into as many equal parts as we’d like.

So far, I’ve showed in my previous articles how to bisect any segment and any angle with just using a compass and an unmarked ruler.

How about trisecting a given segment?

Trisecting a segment is shown in Euclid’s Elements, Book 6, Proposition 9. Let me try to show you Euclid’s method in a nutshell.

Define any two points A and B on a plane and construct a line segment between them. Choose a point C which is not on the line segment AB.

Draw a line from the point A passing through C.

Use compass to mark the points D, E and F on the line AC such that AD, DE and EF are equal line segments.

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Connect F and B. Then draw line segments from points D and E which should be parallel to BF.

Thus, line segment AB is cut into three equal parts. Actually, using Euclid’s method, we can divide AB into as many equal parts as we’d like.

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It might seem easy at first sight but when you think about it, there are a few information we need in order to use Euclid’s method for trisecting a line segment. If you are careful enough, you might have realized that one should know how to draw a parallel to any given line. In order to do that, we should know the method from Elements Book I, Proposition 31, which requires how to create a specific angle on a random point (Elements Book I, Proposition 23). And it doesn’t end here: Proposition 23 requires knowledge of proposition 22 from the same book.

Suddenly Euclid’s ingenious methods seem a bit complicated.

Muslim Ingenious

Elements were written more than 2400 years ago and humanity should thank a few Muslim scholars for its existence today!

Al-Nayrizi (865-922) was one of the first Muslim philosophers who read and commented on Elements. Actually you could still buy his commentary on Elements even today on amazon.com even though it costs a little over 200 USD.

Al-Nayrizi has come up with a truly magnificent method for cutting a segment into equal parts.

Let’s start with drawing a straight line AB. Al-Nayrizi’s method requires only the following knowledge: Drawing lines through A and B respectively which are perpendicular to the line AB.

If we’d like to cut AB into n equal parts, we should mark n-1 equal segments on these perpendicular lines. We could easily do that using a compass. Let’s assume that we’d like to cut AB into four equal parts. That means we need 4-1=3 equal segments on the perpendicular lines.

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Now all we have to do is to connect the dots as shown below.

In the end we managed to cut AB into four equal parts.

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M. Serkan Kalaycıoğlu