Real Mathematics – Killer Numbers #2

In the previous article I was talking about the numbers which put an end to Hippasus’ life. These numbers are not only fatal; they are also incommensurable as well. On top of these, it is impossible to write these killer numbers as ratios of two other numbers.

I believe that there are more than enough reasons to choose a name such as “irrational” for these numbers. For me, it is astonishing to accept that there are some lengths which we can’t measure although they are just in front of us.

√2: One of the most famous irrational numbers.

Whether we realize it or not we can easily spot these lengths in everything that has square shape. Just divide a square diagonally into two equal parts and you will get two right-angled equilateral triangles.

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Assume that the square had side lengths 12. This gave a right-angled equilateral triangle with perpendicular sides with length 12. If we apply the Pythagorean Theorem:

This is an irrational number.

In case you’d like to measure this length, you will see a number that has infinite decimals: 16,97056…

I wonder what would happen if I call this number 17.

√2 is Finally Rational

If 12√2=17, we would get:

pisag5

We did it! √2 can be written as a ratio of two other numbers. It means √2 is rational. From now on we can write 17/12 wherever we see √2.

Although let’s stick to geometry a little bit more and see if we really got something or not.

Proof by Contradiction

First we divide the triangle as follows:

We can see that there are two identical right-angled triangles (A and B) that have perpendicular sides with length 5 and 12, and another right-angled triangle (C) that is equilateral.

Let’s analyze the triangle C from close. It has perpendicular sides with length 5 and a hypotenuse that has length 7. Using Pythagorean Theorem we can conclude:

img_45541

25 + 25 = 49.

50 = 49.

This is a contradiction.

√2 is not rational.

One Wonders…

Check and see what would happen if we used a square that has side lengths 10.

Real Mathematics – Killer Numbers #1

Hippasus: First Victim of the Science Mob

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Pythagoras is a very well known historic figure. Even though most of the people know him through the geometry theorem attributed to him, he had accomplished more than just a theorem. He was also the head of the first known science mob in the history.

Pythagorean Theorem: In a right-angled triangle square of the perpendicular sides add up to the square of the third side of the triangle that is also known as the hypotenuse.

Pythagoras was born in the island of Samos. He had an enormous reputation as a mathematician throughout the ancient Greece. His followers (Pythagoreans) chose to live as their leader. They were a tight and closed group that ate neither meat nor beans and isolated themselves from having any kind of possession.

According to Pythagoras universe was built on the numbers. Every number had a character and everything that is happening around us could be explained with numbers. He believed that numbers have categories such as beautiful, ugly, masculine, feminine, perfect and such. For instance 10 was the best number because it contained the summation of the first four numbers: 1+2+3+4=10.

Pythagoreans also believe that every number is rational: Meaning that each number can be represented as a division of two other numbers. (E.g. 10/2 = 5)

Oath Breaker

One day one of Pythagoras’ followers broke his oath and asked the forbidden question: What is the length of the hypotenuse of an equilateral right-angled triangle?

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Geogebra shows that the hypotenuse is around 1,41 units. This is not the exact value of the length as this length can never be measured.

Hippasus was a devoted Pythagorean. One day he sailed away with his brothers. When he was in the open sea, he started thinking about the problem of the right-angled equilateral triangle. In the end he claimed that he found irrational numbers. This was an oath breaker as it was forbidden to question Pythagoras’ words. Hippasus never came back from that trip, and Pythagoreans continued to keep the existence of the irrational numbers as secret.

Incommensurables: Do they exist?

According to the Pythagorean Theorem: Length of hypotenuse on a right-angled equilateral triangle.

kc3b6k2

If Hippasus was wrong, √2 was a rational number which means √2 can be written as the division of two other numbers. Let’s say that this is true and a/b is equal to √2.

Ps: a and b are relatively prime. This means that a/b can’t be simplified; they are the smallest numbers for that ratio.

kc3b6k21

Let’s square both sides so that we are free from the square root.

kc3b6k22

Now send the denominator to the left side of the equality.

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This actually means that two squares that have side b add up to another square that has side a.

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Hence, we just need to show that when we add two identical squares, we can get another square.

karekc3b6k1

Since the little squares add up to the large square, let’s try to put them inside the large one.

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As seen above, little squares intersect in the middle and leave gaps on the corners. If we stick to our initial assertion, this intersection must have same area as the gaps. But there is something absurd here, because this intersection is a square. Also the gaps are identical squares that add up to the intersection.

If I call sides of the little squares d, and the big square c:

kc3b6k25

This result is the same as our starting point. We just found ourselves in a loop which means that our initial assertion was wrong. √2 can’t be shows as a ratio of a/b. Hence, √2 is not a rational number.

One Wonders…

  1. Try to prove that √2 is an irrational number, using Euclid’s tools which are compass and an unmarked ruler.
  2. How can we understand if √3 is rational or not? (Hint: Try to prove geometrically like I did in the article.)

M. Serkan Kalaycıoğlu

Real Mathematics – Killer Numbers#6

Socrates’ Lesson

In the previous articles I have talked about Plato and his effect on science; particularly geometry. Thanks to his book named Meno, we know about one of the most influential philosophers of all times: Socrates.

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Meno was another book of Plato that was written as dialogues. In this book there were two main characters: Meno and Socrates.

In the beginning of the book Meno asks Socrates if virtue is teachable or not. Even though Meno is crucial for understanding Socrates’ philosophy, there is one part of the book that interests me the most.

Problem

The book gets interesting when Socrates starts asking “the boy” who was raised near Meno. At first, Socrates is asking the boy to describe shape of a square and its properties. After a series of questions Socrates asks his main problem: How can one double the area of a given square?

This is an ancient problem that is also known as “doubling the square”. The boy answers Socrates’ questions and eventually finds the area of a square with side length of 2 units. The boy also concludes that since this area has 4 units, double of such square should have 8 units. But when asked to find one side of such square, the boy gives the answer of 4 units. However after his answer the boy realizes that a square with sides of 4 units has 16 units of area, not 8.

Classical Greek Mathematics

After this point the boy follows Socrates’ descriptions in order to draw a square that has 8 units of area. At first Socrates commands the boy to draw a square that has sides 2:

1kare

This square’s area is 4 units. Then Socrates tells him to draw three identical squares:

4kare

Now Socrates tells the boy to unite these squares as follows:

tekkare

Socrates asks the boy to draw the diagonals in each square. They both know the fact that a diagonal divides a square into two equal areas:

kosekare

It is easy to see that the inner square has a total area of 8 units:

kosekare2

One side of the inner square is the diagonal from small squares. In order to find that diagonal the boy uses Pythagorean Theorem:

karepis

Conclusion

Even though he only uses a compass and an unmarked ruler, the boy found a length that is irrational thanks to Socrates’ instructions. Back in ancient Greece numbers were imagined as lengths/magnitudes. This is why as long as they constructed it neither Socrates nor the boy cared about irrationality of a length.

Pythagoras and his cult claimed that all numbers are rational and they tried to hide the facts that irrational numbers exist. But in the end philosophers like Socrates won the debate and helped mathematics to flourish into many branches.

M. Serkan Kalaycıoğlu

Real Mathematics – Killer Numbers #5

Equilateral Triangle and Irrational Number

In the previous article I asked you to prove whether it is possible or not to draw and equilateral triangle on the system of lattice points. Here is one of the possible proofs for that.

Proof by Contradiction

Let’s assume that we have an equilateral triangle that has sides of length 2 units:

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Here, we will make a critical assumption: Corners of this triangle sits on the system of lattice points. Because of that the triangle must have an area that is rational. Why?

Because Pick’s theorem says that whenever we are inside the system of lattice points the numbers of points and area of the polygon are directly related with each other. And since we can’t count irrational number of points (eg. we can’t have √3 points, can we?!), area of the polygon must be rational too.

euake3

We already know that we can find a triangle’s area: ½(height x base). Then let me draw the height of the base and find its length using Pythagoras’ theorem:

h2 + 12 = 22

h2 = 4 – 1

h2 = 3

h = √3.

We just found the height of our equilateral triangle an irrational number. From here we will find the area

1/2(2*√3) = √3 units.

CONTRADICTION

This result is a contradiction. Despite what Pick’s theorem says (polygons inside the system of lattice points must have rational areas) this result shows an irrational number. Then we can conclude that this or any equilateral triangle can never be drawn on the system of lattice points.

One wonders…

Can you find another polygon which you can’t draw on the system of lattice points?

M. Serkan Kalaycıoğlu

Real Mathematics – Killer Numbers #4

Lattice Points

Imagine a page from a squared notebook. Take all the edges of little squares and leave the vertices. If you assume that horizontally and vertically distance between two consecutive points is exactly 1 unit. I name this plane as the system of lattice points.

In the system of lattice points every point is represented by a number duo:

latis3

Pick’s Theorem: An alternative method in order to find the area of a polygon.

In the system of lattice points one can construct as many polygons as wanted with joining points together. Pick’s theorem is handy for finding the areas of these polygons.

According to the theorem only two things are needed to find the area of a given polygon: Number of points polygon has on its edges (let’s call it e) and the number of points staying inside the polygon (let’s call it i). Pick’s theorem gives the following formula for the area of a polygon sitting on the system of lattice points:

Area = i + (e/2) – 1

Example 1: Triangle.

üçge

Assume that there is a triangle as shown above. Number of points on its edges e is equal to 4 as the number of points inside the triangle i is equal to 0. Hence Pick’s theorem says the area of this triangle is:

Area = 0 + (4/2) – 1

          = 0 + 2 – 1

          = 1 unit.

We know from basic geometry that the area of a triangle is the half of height times base: (2*1)/2 = 1 unit.

Example 2: Square.

karre

In the picture we can see that e=12 and i=4. Thus the area is:

4 + (12/2) – 1 = 4 + 6 – 1 = 9 units.

We know that area of a square is the square of the lenght of its edge which makes 3*3=9 units.

Square and triangle are easy examples and perhaps you are thinking that Pick’s theorem is redundant. Then let’s continue drawing a more complex polygon and find its area.

Example 3: Polygon.

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Now Pick’s theorem shows its strength. Without the theorem we’d have to divide this polygon into various polygons and then calculate areas one by one. But with Pick’s theorem it is just counting points:

e=12 and i=72. Hence the area of the polygon can be found with:

Area = 72 + (12/2) – 1 = 72 + 6 – 1 = 77 units.

Equilateral Triangle

Up to this point it looks like we are dealing with geometry but the headlines said that it is an article about numbers. Let me change the course of the article with a question.

Q: Is it possible to draw an equilateral triangle on our points system while the corners of the triangle sits on the lattice points?

For instance let me draw the base of an equilateral triangle that has 2 units of length:

ekui
Third corner (Q) sits between lattice points.

As seen above third corner of the triangle won’t be on a lattice point.

One wonders…

Can you prove that this is the case for each and every equilateral triangle on the system of lattice points?

To be continued…

M. Serkan Kalaycıoğlu

Real Mathematics: Killer Numbers #3

Theodorus of Cyrene

City of Cyrene was one of the ancient Greek cities that were located in the Northern Africa at around 5th century BC. We are certain that Theodorus (465 BC – 398 BC) was born in Cyrene and he was a tutor of Plato. Theodorus probably met Socrates and lived in Athens for some time. Unfortunately we don’t know much about his life.

Although we know through his student Plato that Theodorus had made significant works on irrational numbers. During ancient times, even though they had number symbols called Attic (also  known as Herodianic symbols) Greek philosophers did not care about number symbols. According to their thinking numbers were just magnitudes of lines. Hence they used line segments to represent numbers.

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Plato

Pythagoreans thought that every number can be shown as a ratio of any other two rational numbers, and that is why they claimed that numbers are rational. Oddly enough a geometric property what we call Pythagoras Theorem contradicted Pythagoreans’ claims as an isosceles right triangle with 1 unit length sides would have a hypotenuse that is √2 unit length. Unfortunately for them √2 is not a rational number. In other words, it can’t be shown as a ratio of two rational numbers.

Theodorus was also a Pythagorean, but he proved that √2, √3, √5, … are all irrational. For some reason he stopped at √17.

In this article I’ll be talking about a work of Theodorus now called Spiral of Theodorus.

Construction of the Spiral of Theodorus

This lovely geometric shape, which is also called “Einstein Spiral”, “π Spiral” and “Square Root Spiral”, could be constructed by anyone who knows Pythagoras Theorem.

At first construct an isosceles right triangle with 1 unit length sides. Through Pythagoras Theorem we can calculate the length of the hypotenuse as √2.

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In the next step, continue drawing a base with length 1 unit that is perpendicular to the hypotenuse of the previous triangle and construct a second triangle which would have hypotenuse with length √3.

As long as you continue the same process, you will be ending up with hypotenuses with lengths √4, √5, √6, √7 … Theodorus stopped at √17. Probably he ended his structure at 17 because that is the final triangle before overlapping starts.

IMG_5379

It is obvious to the naked eyes that these triangles form a beautiful spiral, which is called Spiral of Theodorus.

Nothing like a spiral

What is so special about the Spiral of Theodorus?

  • All the lengths of the hypotenuses of the spiral, except the perfect square number lengths, are irrational.

    IMG_5390
    √1, √4, √9, √16, √25… are the only numbers that are rational.
  • If one continues to add triangles which would mean that the spiral is going to the infinity, no two hypotenuses overlap.

    IMG_5380
    Even if it looks too close, no two hypotenuses overlap in the Spiral of Theodorus.
  • The windings of the spiral would have length π between themselves as one adds infinitely many triangles.

    IMG_5383
    I have found 3,1 in the first 30 triangles. If I kept going for infinity, this number would approach to π.
  • The angle between two consecutive perfect square number hypotenuses would approach to 360/π as the spiral goes to infinity.

    360/π = 114,591559026… which is very close to what I draw. I was certainly lucky but if you draw this perfectly, as you approach to infinity, you would find exactly. 360/π.

Killer Curve

This is my favorite property of the Spiral of Theodorus: Cut off every single triangle from the spiral and align them on the x-y coordinate system.

If you connect the tipping points of every triangle, you would end up with the y=√x curve. I like calling irrational numbers as “killer numbers” because of the story of poor Hippasus. I think it would be suitable to call this curve The Killer Curve.

I also used a program called GeoGebra and find the following result.

teodorusgeo

One might wonder…

What would happen if you take the first triangle like the following?

IMG_5394

What kinds of changes do you observe?

M. Serkan Kalaycıoğlu