Real Mathematics – Special Numbers #3

Lynx – Snowshoe Hare War

In North America there is a predator-prey relationship between lynx and snowshoe hare. Snowshoe hare is the primary food of the lynx which effects the populations of both species directly. For instance if lynx population is high in a forest, it is safe to assume that snowshoe hare population is (or at least once was) high in the same forest. And if lynx population decreases, one can conclude that one of its main reasons is the decrease in snowshoe hare population.

Ecologists have been conducting researches to understand the behaviors of this population change between lynx and snowshoe hare. Within time they found interesting results:

  • Whenever lynx population sees its highest point, snowshoe hare population hits its lowest point.
  • As its primary food supply almost runs out, population of lynx starts to decrease.
  • Decrease for lynx population directly effects the population of snowshoe hare as it increases rapidly.
  • Around the time when population of snowshoe hare hits its heights, lynx population bounces back from its lowest point.
  • More snowshoe hare meant more food for lynx and this leads lynx population to rise as snowshoe hare population to fall down once again.
Figure_45_06_01
Comparison of  the lynx (blue) population with the snowshoe hare (red) population between 1845-1935.

This is a cycle. Ecologists came up with a conclusion for the cycle between these two species. Their results showed them that one cycle lasts 8 to 11 years. It is still unknown what derives this time period. Ecology is filled with numerous secrets such as this one. Scientists showed that predator-prey relationship has a direct effect for the changes in the populations of lynx and snowshoe hare. Although this is not the only reason behind the cycle as climate, human factor and other predators also have an effect.

Periodic Living Beings

Population cycles are crucial for all the species living on Earth. Living beings had evolved so that they can maintain their lives and they managed it with evolving defense mechanisms. Some species don’t have enough strength so they had to come up with new defense mechanisms which are not physical. For instance some animals hide for long periods of time and only come out to breed. Hiding is their defense against predators. These animals could be called as “periodic living beings”.

Assume that the animal A comes out only once in every 10 years to breed for a couple of weeks.

Q: If A’s predator B is also a periodic living being, how long its period should be in order to catch A’s coming out?

If A comes out every 10 years, predator’s period should be a factor to A’s period so that B could catch A as soon as possible.

Factors of 10 are 1, 2, 5 and 10.

  • If predator’s period is 1 year, then (10/1=10) predator catches A in its 10th cycle.
  • If predator’s period is 2 years, then (10/2=5) predator catches A in its 5th cycle.
  • If predator’s period is 5 years, then (10/5=2) predator catches A in its 2nd cycle.
  • If predator’s period is 10 years, then (10/10=1) predator catches A in its 1st cycle.

There is still more bad news to come for A. These four numbers are not the only cycles a predator can have in order to catch A. For example if a predator’s cycle is 6 years, then it would catch A in their least common multiple. For 6 and 10, least common multiple is 30. Thus these two species meet once in every 30 years.

In the end, 10 years of cycle is not suitable for a living being.

Q: Is there a periodic living being on nature?

Cicadas

There are different types of cicada which only come out once in every 7, 13 and 17 years just for breeding. This is why cicadas are known as “periodic living beings”.

indir (10)

Their cycles are 7, 13 and 17. These three numbers are not ordinary numbers: They are primes.

Prime numbers are special in their own way as they don’t have factors besides 1 and themselves.

It is still a mystery how these cicadas evolved into these specific primes. But we can speculate over this fact. A physicist named Mario Markus claimed that cicadas emerge in these prime numbers so that they will survive. This is a bold but interesting assumption.

Example: Let’s choose a kind of cicada that has 13 year-cycles. If its predator has a cycle of 5 years, there two creatures could only meet once in every (13*5=65) 65 years. Since 13 is a prime number, predator and prey can meet only at the multiples of 13.

M. Serkan Kalaycıoğlu

 

Real Mathematics – Numbers #1

First thing first; mathematics and philosophy can’t be separated. I can back this idea with a historical fact: Compared to history of science, “scientist” is a relatively new title. For instance, Isaac Newton who is regarded as the father of modern physics was known as a “natural philosopher”. Use of the word scientist had made us forget the fact that a scientist is actually a person who does deep thought sessions in majority of his/her work.

Mathematics Education – “This is that, and hence the answer should be that.”

Unfortunately, in traditional mathematics education there is almost no time for one to think. I think I should explain what I mean with the term “traditional mathematics education”: Imagine that numbers is the subject of a specific mathematics class. Teacher follows the traditional way and gives definitions to number types.

Counting Numbers: These types are also known as whole numbers. They start from 1 and go to infinity one by one:

1, 2, 3, 4 …

Natural Numbers: They start from 0 and continue to the infinity one by one. Their only difference with counting numbers is the mighty 0 (zero).

0, 1, 2, 3, 4 …

Odd Natural Numbers: They are the numbers which has remainder 1 when divided to 2. First odd number is 1. Odd numbers increase by 2. So their sequence is:

1, 3, 5, 7, 9 …

Even Natural Numbers: They are the numbers which can be divided by 2 with remainder 0 (zero). First even number is 0. Their sequence is:

0, 2, 4, 6, 8 …

Integers: They are the sum of natural numbers and negative of counting numbers. They can be shown like an infinite line where left goes to negative infinity as right goes to positive infinity. Their sequence is:

… -3, -2, -1, 0, 1, 2, 3 …

Prime Numbers: They are greater than 1 and they can only be divided by themselves and 1 to give and integer answer. They are:

2, 3, 5, 7, 11, 13, 17 …

“What is 7 times 8? Bravo! You are so fast!”

We need to define these, so there is no problem up to this point. What is done wrong with traditional approach is that it rapidly focuses on problems. There is nothing wrong with asking questions to kids, although with the traditional approach important thing isn’t the question: It is the answer! After a while kids focus on the answer rather than the question. That is one of the reasons why when asked (I am not talking about mathematics per se) children try to answer even without thinking. Sometimes even families contribute that with awarding how fast the answer is given.

Even before they start high school most of the kids lose the ability to think in mathematics classes.

What needs to be done?

If you are good at observation, you probably realized that those definitions above include notions such as negative, zero and infinity. These three notions are relatively new to humans and they all possess deep meanings underneath them. For instance only 1350 years ago humans first explained the number zero mathematically. Actually first use of the number zero in Europe was around 12th century which was almost 500 years after its first discovery in India! (I will get back to this subject in another article.)

Nowadays we define negative numbers and we expect kids to do calculations with it immediately. Actually negative numbers were known to civilizations way before the number zero. Although in the 16th century important European mathematicians were referring to negative numbers as “wrong numbers”. It has been only 200 years or so since humans made peace with the negative number notion. Maybe it is clearer now to you how wrong it is to expect kids understand negative numbers instantly.

Infinity might seem the hardest of these three notions. There is a popular definition for it: Something that never ends, keeps going until the end of time, everlasting. However there are much more profound meanings for infinity in mathematics. For example, some infinities are countable and some are bigger than others.

Hilbert’s Infinity Hotel

This paradox was thrown out for consideration by David Hilbert, one of the leading mathematicians of 20th century. In Hilbert’s hotel there is infinite number of rooms. Imagine that you are the manager of this hotel and you work on commission. One day business was ticking and every room was filled with customers.

  1. Would you turn down if a new customer arrives? If you decide to give this customer a room, which room will it be? How can you decide that room number?
  2. A couple of hours later you see infinite number of customers in front of the hotel. How can you arrange rooms for this many people?
  3. Just when you thought you could relax for a bit, you hear honking coming outside of the hotel. You go outside and see infinite number of buses in the parking lot. Not only that, each bus has infinite number of tourists. WHAT WILL YOU DO?!

hilbert

Answers to these three questions will help students comprehend different number notions. You should stay away from internet in case you have never heard of this paradox. Give yourself time, even if it is limited, at least 1-2 hours. Just think on these questions. Try to remember this: It is not about “when”, it is about “how”.

M. Serkan Kalaycıoğlu

Real Mathematics – Special Numbers #2

Sieve of Eratosthenes

Eratosthenes (whenever I write his name, I can’t stop thinking about a nice and crusty toast) was not only a one-job man: He did not only measure the circumference of Earth, he also contributed to various fields, mathematics in particular. Ancient Greek philosophers dealt with geometry a lot which led to a misunderstanding as if they didn’t work for other branches of mathematics. Reality shows us that lots of Greek philosophers worked for mathematics in a broader aspect. Eratosthenes was one of them and he discovered an ingenious method for finding prime numbers.

Today that method is known as the Sieve of Eratosthenes. This method is really easy to learn and very affective for finding prime numbers in a list. But method is rather slow and takes too much time in larger number lists.

Let me show you how Eratosthenes’ method works between numbers 2 and 100. (I am skipping 1. I’ll be talking about 1 and why it is not considered as prime in another article.)

IMG_6051

Thus we start with the number 2, which is the smallest prime number. Here is how the method goes:

  • Move in the list until you find a prime number.
  • When you find your prime, stop there and square it.
  • Go to the square of the prime in the list, and eliminate all the numbers that are multiples of the prime.
  • When you finish eliminating, go to your prime in the list and move until you find your next prime number.
  • Repeat the same process until there are only primes left in the list.

Example: Let’s do it for 2-100.

2 is prime. Square it: 4. Start eliminating all numbers that are multiples of 2.

Go back to 2. Move to the next number, which is 3. It is prime. Square it: 9. Go to 9 and start eliminating all the numbers that are multiples of 3.

Go back to 3. Move to the next number; 4. It had been eliminated already, thus move to 5 which is a prime number. Square it: 25. Start from 25 and eliminate all the numbers that are multiples of 5.

Go back to 5. Move to the next available number which is 7 and it is a prime number. Square it: 49. Start from 49 and eliminate all the numbers that are multiples of 7.

Go back to 7 and move to the next available number which is 11. It is prime, thus square it: 121. 121 is out of our list. This means every available in our list is a prime number.

IMG_6061

New In the Olympics: Walking the Square

Game 1:

  • Draw a circle and place eight dots on it.
  • You will walk around the circle from the dot that is on top. You’ll take 1 step each time you move, that means you will be moving to the neighbor dot.
  • You’d visit every dot with this method until you finish your walk.

Game 2:

  • Again start with the same circle and eight dots on it.
  • This time you will take 2 steps in each time. That means always move to the second dot.
  • When you finish your walk, you’ll see that you visited only half of the dots.

Game 3-4-5-6-7:

Try walking with 3, 4, 5, 6 and 7 steps.

One wonders…

In which steps you would visit all the dots? Do you see anything special about those numbers of steps?

Can you find a general conclusion from this game?

What would you see if you play the same game for 9, 10 and 11 dots?

M. Serkan Kalaycıoğlu

Real Mathematics – Special Numbers #1

Understanding Time

How did humans keep track of time before the invention of clock?

First thing comes to mind is that humans observed rise and fall of the Sun that is 1 day. People needed to know time in a broader aspect as civilizations emerged so that they could understand crucial time intervals such as rain season etcetera. For that, they used a cycle that was up there in the sky, changing its shape in a pattern: Phases of the moon. They observed and calculated that Moon phases are in cycles of 29 days.

Ayın-Evreleri-NelerdirNasıl-Oluşur

A lunar month is roughly 29 days. There is about 1 day difference between lengths of a month and a lunar month.

29 is a whole integer.

If you try to divide it into two equal parts, it will give you 14,5 which is not a whole integer.

If you continue dividing 29 into equal whole integers, you would fail for every number between 2 and 28. Only there is one 29 and twenty nine 1s. Nothing else can divide 29 into equal whole integers.

IMG_6013

In other words, there are only two whole numbers which gives 29 when they are multiplied: 1 and 29. These kinds of numbers are called “prime numbers”.

Today “one month” refers to 30 days. One of the main reasons behind turning 29 days into 30 is that 30 is not a prime number. It can be divided equally into whole numbers such as 2, 3, 5, 6 (2*3), 10 (2*5) and 15 (5*3).

Prime Numbers and Encrypting

Another beautiful property about 30 is that it can be shown as the multiplication of 2, 3 and 5. All these three numbers are prime. Therefore we can write any whole integer as the multiplication of prime numbers.

Let’s take this one step further: Every whole integer has one and only one representation what is a multiplication of prime numbers. Hence different integers have different prime number demonstrations.

This property has a key importance in our modern day. Assume that I have to create a password for my online banking account. Bank’s application requires me to create this password with using numbers only. Naturally I would like to create a password that would be very hard (preferably impossible) to crack. Also I would like to use an easy method when I create the password so that I can remember the method in case I forget the password itself.

Trapdoor Function: Roughly, it is a method that is easy to compute in one direction, yet difficult to compute in the opposite direction. You could imagine it as a machine that takes a number (input number) and easily processes it into another number (output number). Although it would be very hard or even impossible to use the output number and find the input number with it.

If I take one small and one huge (let’s say 6-digits) prime numbers, it would take only a few seconds to multiply them using a calculator. Resulting number, my new password, can only be found with the multiplication of those specific prime numbers. Then, if someone tries to log in to my bank account, he/she must know the primes I’ve chosen.

IMG_6039
Creating 1115035 is very easy: Just multiply prime number 223007 with another prime 7.

Finding the prime multiplication of a random number is a really hard and long lasting thing to do.

Why?

Because, even today we don’t have a method that finds the prime multiplication of any given number. Thus, using prime numbers is an effective way to secure information.

Crack the Code

I’ve built an encryption system that uses prime numbers and alphabet. According to my system, every letter in the alphabet is assigned to a number:

IMG_6010

Method:

  • I choose two prime numbers between 1 and 20.
  • I multiplied them.
  • Then I added the result to each letter in the alphabet.
  • For instance if I choose 3 and 5, letter A would go to (3*5=15, 15+1=16) the 16th letter that is the letter M. Hence letter M in my encrypt means that it is the letter A in reality.

Now, try to crack the encrypted sentence below:

RUTM ROBK XUIQ T XURR

M. Serkan Kalaycıoğlu