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:

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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:

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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.

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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.

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

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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.

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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:

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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 #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.

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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.

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    √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.

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    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.

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    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.

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One might wonder…

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

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What kinds of changes do you observe?

M. Serkan Kalaycıoğlu