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.

440px-rhind_mathematical_papyr

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 #5

pythagoras1

I was wondering; if there was a list of hall of fame for famous ancient Greeks Pythagoras would find himself in the top ten for sure. What is striking about his fame is that it comes directly from a geometry property. Although mathematicians know that so called Pythagorean Theorem was known to other cultures at least 1000 years before he “discovered” it.

Pythagorean Theorem: In a right-angled triangle sum of the squares of the perpendicular sides gives the square of the hypotenuse that is the longest side of the triangle.

It is being told that there are 367 different proofs for this theorem. Some of them are so similar, even mathematicians have trouble seeing the difference among these proofs.

Let’s check a few of the proofs.

Proof 1

Elisha Loomis talks about a proof for the Pythagorean Theorem in his book “The Pythagorean Proposition”. This proof is special because it came from a high school student named Maurice Laisnez.

I decided to use cutting papers for the explanation. First of all I cut a random right-angled triangle and then made 3 more copies of it.

I lined these four triangles up such that it gave me a square inside a square:

pissa2

Since sides of the inner square are c, it has area c2.

Now let’s line the triangle as follows:

pissa3

Marked areas 1 and 2 are squares and their area is equal to the area of the inner square from the previous alignment. Now let’s find the areas of 1 and 2: They make a2 and b2.

Their addition will make c2. Hence:

a2 + b2 = c2

Proof 2

For the second proof I decided to go to the ancient China.

Zhoubi Suanjing is believed to be written around 500 BC to 200 BC. In the Loomis’ book you can find this proof in the page 253.

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Pythagorean Theorem’s proof in the Suanjing.

Again I will cut four right-angled triangles for the explanation of the proof. But this time I will cut the triangles such that their perpendicular sides will have length 3 and 4 units. Chinese mathematicians tried to find the third side of the triangle as follows.

pisasaads

In order to start the proof I lined the triangles up like below and a tiny square formed in the middle:

pisasa

Tiny square A has sides that have 1 unit each. This is why area of A is 1 unit as well.

We know that the area of one triangle is (3*4)/2 = 6 units. There are four of such triangles and that gives us 6*4 = 24 units of area. When I add the area of A to this result, I can find area of the whole square as 25 units.

If area of a square is 25 units, its one side is square root of the area: √25 = 5 units.

From here we found length of the third side from the triangles:

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This proof shows us that 3-4-5 triangle and Pythagorean Theorem were both known in ancient China.

One wonders…

A farmer dad wants to retire. He would like to divide three of his lands to his two sons equally. But he wants to do that without dividing the lands from each other. What should he do?

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X-Y-Z are squares as DCG is a right triangle.

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #9

Finding Pi

Humans discovered a connection between a circle’s circumference and its diameter around 4000 years ago. This connection, what we call the number pi now, affected everyone who was involved in mathematics and/or engineering in the early times. Countless people, including historical figures, worked in order to find what this number is, but all of them failed to find such answer.

Some of them tried to find an approximation and got really close. When it was their time to shine for geometry, one ancient Greek whom we are familiar with had taken approximation methods to a different level. This famous Greek dude was Archimedes.

Archimedes is believed to live between 287 BCE and 211 BCE. If you take a look at the history of mathematics it was common to see approximations for the number pi, especially in Babylon, Sumer, ancient Egypt and China civilizations.

Although, Archimedes was different in an important way: Before him, method to find the number pi was not that complicated. A certain circle would be inspected and compared to a similar regular polygon. Their areas or perimeters were thought to be same which would give an approximation for pi.

Let’s try to find an approximation for the number pi using this method.

Draw a circle and divide it into 12 equal parts, like a pizza.

Cut the 12th slice into two equal parts and lay down all the slices side to side.

Slices will seem like a rectangle which is a regular polygon. At this point circle and rectangle are believed to have same shapes.

IMG_5838

Circumference of circle is found with pi*R, where R is the diameter of the circle. It is obvious to the naked eye that the long sides of the rectangle would give the circumference of the circle. Hence, we can find an approximation for the number pi.

Try this for another circle that has different diameter length.

IMG_5841

And take the arithmetic mean of both results. There is the number pi… Well, it is just an approximation.

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Archimedes’ Method

Even though there were so many engineering marvels as well as important geometry knowledge, Archimedes was not happy with the methods used to find the number pi. He developed his own method that required inscribing and circumscribing the unit circle with regular polygons. Archimedes did something different than the methods which were used before him: He continued using polygons with more sides and compared those results until he found the best polygon.

His method today is known as “the method of exhaustion”. It is the earliest known version of calculus.

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You could draw a unit circle yourself. Try to inscribe and circumscribe a square. You’d get pi between 2,8 and 4, which is not bad for the first try. If you continue with polygons that have more sides, you’d get even better intervals.

The thing is, Archimedes didn’t try to find an exact value. Instead he found an interval which was 99,9% accurate!

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Archimedes found this interval with starting a hexagon, and he finished his method at a 96-sided polygon. You could also see that 22/7, which is given to students for pi in school, is the upper limit that Archimedes found nearly 2200 years ago!

M. Serkan Kalaycıoğlu