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.

indir (3)
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.

IMG_5384

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

Real Mathematics: Game #1

Circle of Numbers

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Most of the countries in the world use these symbols for the remarkable decimal system. The system was designed so perfect, even though symbols might change in some regions, logic of the system is conserved all around the world. Decimal system is one of those things that are “universal”.

If you are willing to participate in today’s modern world, you’d better know how to use numbers. You could go on and try to live a day without using any numbers: You’d see that it is impossible to finish even one single day. Actually, using numbers is not enough: you should be able to understand what a given number represents.

Understanding Numbers

What do I mean by “understanding numbers”?

If I show you 2125555555, can you associate it with anything?

How about 212 555 55 55?

Now most of you realize that 212 555 55 55 is a telephone number. Using three blank spaces inside the number 2125555555 changed the way you look at it.

Numbers take too much space in our daily lives. I think this is a great reason to work and develop a good understanding of them. Circle of Numbers is a game that helps children cultivate the ability of using numbers.

In order to play Circle of Numbers, all you need is a pencil and a pen. This game can have various numbers of versions.

Circle of Numbers 1.0.0

Draw a circle.

Place four boxes on the circle.

You have to place the numbers 0, 1, 2 and 3 inside the boxes such that difference of two adjacent boxes will be an odd number.

Example

Circle of Numbers 1.0.1

Again draw a circle with four boxes on it.

This time each of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 should be used.

In order to do that, you will need ten boxes which means you will have to add six more boxes on the circle.

Difference of two adjacent boxes should be odd.

Example

Let’s assume 4 and 7 are placed as following.

IMG_5152

Player knows no matter which number is chosen, there will be number pairs that will not have odd difference. For instance if 5 is chosen, even though 5-4=1 is odd, 7-5=2 will be even. Player should add a new box to avoid this problem. Following will show you a working strategy step-by-step.

Circle of Numbers 2.0.0

Assume that there are five boxes on the circle and you are allowed to use the numbers 1, 3, 5, 8, 9. Can you construct a valid circle?

IMG_5176

Check it out

I just gave three different versions for the game Circle of Numbers with three different names: Circle of Numbers 1.0.0, Circle of Numbers 1.0.1 and Circle of Numbers 2.0.0. What do these numbers remind you? What would Circle of Numbers 2.0.1 look like?

M. Serkan Kalaycıoğlu

Real Mathematics: Numbers #6

Achtung! Following article contains advertisement for a chocolate company even though I won’t get paid.

How to get the most Milkinis?

Assume that

                – Class has 8 students.

                – All the students adore Milkinis and would do anything to have some.

                – There are 6 Milkinis bars in total.

Rules of the game:

                – 3 tables in the whole classroom named A, B and C.

                – Tables get 1,2 and 3 Milkinis bars in order.

                – On every table each student will get equal share of Milkinis bars. (eg. If there are 4 students on table A, those 4 students will get ¼=0,25 Milkinis bars each.)

                – Ambition is to select the table that has more Milkinis outcome for the student. (eg. While selecting, if student has a chance to get more chocolate from table A, he/she will choose that table.)

Q: Imagine you are one of the 8 students. In order to get the most Milkinis bars, what kind of strategy should you use? (On which turn you should select your table.)

Playing

As long as we don’t change the rules or anything, we will be starting the game in the same fashion every single time: First student will select table C, because it will give him/her 3 Milkinis bars which is greater than A’s 1 and B’s 2 bars.

Second student will choose B, so he/she will get 2 bars of Milkinis.

IMG_4919

Third one has to choose table C as he/she will get 3/2=1,5 bars of Milkinis. Now we have 1 student sitting in table B, 2 students sitting in table C. A is still empty.

IMG_4920

Forth student has 3 identical choices. In all three tables student will be getting same amount of Milkinis bar. (1 Milkinis bar.) Let’s assume forth student selects table B.

IMG_4921

Fifth one will get;

1/1=1 bar from table A,

2/3=0,66 from B,

3/3=1 from C. Let’s say the student selects table C.

IMG_4922

Sixth student will select table A as it will get him/her a full bar of Milkinis.

IMG_4923

Seventh will be selecting table C. Now table C has 4 students and 3 Milkinis bars. Each student here gets ¾=0,75 Milkinis.

IMG_4926

Eight student must select table B as each student on table B gets 2/3=0,66 Milkinis.

Result

Our game is finished with 1 student sitting at the table A, 3 at B and 4 at C. This would result that students get 1 bar of Milkinis from A, 0,66 from B and 0,75 from C. Student who goes to table A is the clear winners in the situation, who was the sixth choice.

One wonders…

  1. Is there a spot while selecting that guarantees the most chocolate?
  2. Why did I feel the need of using decimal point as using fractions would give me the same amount?
  3. One Milkinis bar has 4 little parts. Which student(s) would get the least Milkinis? How many parts of Milkinis would it be?

History of Decimal Point

In early math education teachers discuss decimal point right after teaching what fractions are. If they both mean the same thing, why do we teach both of them?

At the first sight it looks like a waste of time to show the same thing with two different notations. But in truth fractions and decimal point are both very useful and critical in math. Using decimal point might seem confusing, although when it comes to comparing two or more numbers, decimal point notation is easier to the eye than fractions are. (eg. Comparing 0,66 with 0,60 is easier and faster than comparing 2/3 and 3/5.) Also it takes less time when you write down huge numbers as decimals.

Fractions have a history of at least 4000 years. Decimal point notation is relatively a baby next to fractions. In his book History of Mathematics David E. Smith mentions a priest named Christopher Clavius (1537-1612). According to Smith, Clavius is the first known person who used decimal point systematically. In a book his book Clavius made a table called “Tabula Sinuum” where he wrote down his astronomical calculations in decimal point notation.

pellos

In 1492, Francesco Pellos wrote in his arithmetic book that 1/10th of 5836943 makes 583694.3 as shown. Although Pellos wrote the first known decimal point notation in his book, historians of mathematics claim that Clavius should be considered as the inventor of decimal point notation.

M. Serkan Kalaycıoğlu