Ordinals

I once read about a theory that said that numbers can be described as a common property of two groups that have nothing in common excluding their size. For example the number three is a common property of the following groups - three deers, three stones and three trees.
In modern mathematics we have a sort of an extension to this idea - ordinals. An ordinal is a well ordered set such that if A is an ordinal and x is in A and y is in x then y is in A. The first ordinals are phi (=empty set), {phi}, {phi,{phi}}, {phi,{phi}, {phi,{phi}}}. Those ordinals correspond to 0,1,2,3.
As you probably noticed there is a very simple rule that produces the next ordinal - if A is an ordinal than A(union){A} is the next ordinal. From this we can conclude that: The set of all ordinals is a well ordered set and the union of any number of ordinals is an ordinal.

What makes the ordinals truly interesting for me is the fact that in for them "infinity plus one" is not equal to infinity. This is very simple to see, infinity is the so called least infinite ordinal - w. It can be defined as the union of all finite ordinals. The next ordinal is w+1=w(union){w}. It is rather obvious that the two sets are not equal and therefore w+1 is not equal to w.
Ordinals are not the only example of infinity not being equal to infinity and one, but in my opinion they are extremely intuitive in this regard. After all, all we basically do with ordinals is to constantly "add one". This is the same thing we did with natural numbers long ago, but it appears that the natural numbers don't follow our basic intuition that says that "it is always possible to add one"

In the beginning of the post I told that numbers can be described as a common natural property. This however brings an interesting philosophical question - if our intuition is a product of our world than why do natural numbers that come from it don't follow our intuition after a specific point? A possible answer is that "infinity is not natural" and therefore there is no reason for it to follow our intuition in any way. However, infinity appeared as a concept a lot of time ago. At first it appeared as "many" which basically told that there was no known number large enough.When a new number (or even a number system) where invented the "many" was replaced by an appropriate number. And this brings us to the following thought: Is it possible that we are in the same condition again? That is, should we use ordinals instead of natural numbers? After all, they are pretty much an extension of the natural numbers.

Geometrical representation of numbers

This post s a response to a comment I got today. I was asked if there are numbers that cannot be represented at all using geometry.
The answer to this question depends greatly on what you consider to be a number, and what you consider to be a geometrical representation.

What is a number?
One possible approach is to define that x is a number if and only if it belongs to R (the set of all real numbers). There are different ways to define this set, but it can be proven that all those definitions give the same set, so this definition of a number doesn't have any problems.
However, there are objects that don't belong to R but are considered numbers (at least by some people). The most obvious example is the complex numbers. It can be easily shown that the "number" sqrt(-1)=i doesn't belong to R, so if we want to consider it as a number we need to extend our definition to include all the complex numbers - in other words x is a number if and only if it belongs to C (the set of all complex numbers).
It turns out however that even this definition can be extended. In addition to i we have other imaginary "numbers" - the infinitesimals (a non negative number smaller than any positive number) and infinity. Neither of them belong to R or to C.
Surprisingly, even this is not the end. There are also cardinal numbers - a cardinal number is basically a size of a set, so for finite sets this is just a natural number. For infinite sets a cardinal is a "number" (and there are infinitely many different sizes of infinite sets, so there is an infinite os such cardinals) that doesn't belong to any of the sets mentioned above (except for the cardinal of the set of the natural numbers).
It is possible to bring more examples of objects that can be called numbers but, in my opinion the best definition is the first one - x is a number if and only if it belongs to R. But you are free to chose the one you like.
Geometrical representation
It is important to notice that there are two different definition of what a geometrical representation is. The definitions are:
1. A number has a geometrical representation if there is a point on the real line that corresponds to this number.
2. A number has a geometrical representation if a line segment of a corresponding length can be constructed geometrically (using compass and straightedge alone).

Since R can be viewed as the real line, it follows immediately that all the numbers in R has a geometrical representation according to the first definition. It turn out that if you extend this definition of the numbers to include all C, there is also a geometrical representation, because every complex number can represented by a point on a plane. Infinitesimals, infinity and cardinals don't have such a representation.

The second definition is much more strict. The ancient Creeks never asked if there are numbers that cannot be represented in such a way, but it turned out (at about the 16 century I think) that there are such numbers. To better understand this, lets first see some examples. Lets look on the following numbers - 2, 9, sqrt(2) , pi.
It is obvious that we can construct the first two. All we need to do is to decide what we call a line segment of length one, and we are done. We can now draw 2 such segment to get 2 and 9 to get 9. For sqrt(2) it is a bit more complex, we need to make a right triangle with sides 1 and then we will get that the third side is sqrt(2). Generally it has been proven that a number that is a root of a polynomial with rational coefficients can be somehow constructed under the restrictions we put on ourselves. It was also proven that a number that is not a root of any such polynomial cannot be constructed in such a way. Not so long ago (in the last century) it was shown by Cantor that most numbers (numbers according to the first definition) are not constructible. Such numbers are called transcendental.
It is important to note that numbers that belong to C also can be constructed (not all of them, but some of them) this happens because C and R are sets of the same cardinality so you can assign a number in R for any number in C.

Formula for Primes

I stumbled on two interesting formulas for primes today. The first formulas allows to check if a certain number is prime or not, and the second formulas gives you the Nth prime. They both use factorials, so neither of them is efficient for large numbers. And since n! becomes large rather rapidly, it means that without a computer it would be a problem to use this formula even for n=20 for example.

Both of the formulas were invented by C.P Willans. The | | stands for the floor function.

If x is prime, the result will be 1 else it will be zero.



As you can see both of them are higly ineficcient - the numbers becomes laege extremely fast.

Copyright infringement

First of all a disclaimer: This post contains information that may be used to obtain illegal copies of music, videos, books etc. All of this information is presented here for informational purposes only. By reading it you agree that I am in no way responsible for any damage resulting from using any methods described in this post.

Now lets get to the main idea.
In this post I am going to discuss a method that can be used to obtain any piece of copyright work, that can be uploaded to the internet, for free.
It is a very simple method, although it may require a lot of time. It all starts with numbers, real numbers. As you probably know the numbers can be divided in two groups - rational and irrational numbers. Surprisingly irrational numbers can be used to get staff for free..

The difference between rational and irrational numbers, simply put, is that irrational numbers have infinite decimal expansion - and it doesn't repeat itself as the decimal expansion of 1/3 for example. An example for such a numbers is . It decimal expansion is approximately 3.14159.... Now what will happen if we will write it down in binary? We will get an infinite string of numbers 1 and 0.

After reading this you probably already understand what I mean, but lets see an example. As I said you can use this to get free copies of music. Suppose what you want is to get the latest song of your favorite artist. This song can be stored as a mp3 file. In this form it is also s string of 1's and 0's. Just like . But there is a very important difference - it is finite. It is very long but it is still finite. Therefore it is contained in . From some point the 1's and 0's in string will be exactly the some as the 1's and 0's of the mp3 file. All you need to do is to find this point. After finding this point you can make a new mp3 from this part of the string. Moreover since you didn't copy the first mp3 but created it yourself it belongs to you and not to the artist.

It may seem impossible and a joke. But in fact it is not. While today this is no more then a fun thought, all we need is more powerful computers to achieve this. And then all we will need to do is to break in chunks and see what they turn to be if converted by appropriate program.
Finding a specific song in this way is of course impossible.

Visualizing large numbers

This is a very hard thing to do. While it is easy to think about small numbers it is very difficult to imagine large numbers. It is largely because they seem to be totally unrealistic and not related to reality. For me it is sometimes difficult to visualize even not very large numbers - 100 for example. And visualizing say 10^50 is nearly impossible for everyone.

A bit of trivia: what is the largest named number?

Surprisingly it is a very well known one - googolplex. It is .
Why I said it is well known? Well the Google main office complex is named after it. So it is a very well known number (for such a huge and totally not connected to reality number) .
At this point you are probably wondering why I said that it is totally not connected to reality, that is, there is nothing in the physical world that it describes. After all the universe is so huge. So why wouldn't it contain a googolplex of something? However it turns out that there is only about atoms in the whole universe. This is almost zero compared to the googolplex!!

When I first learned that there is only atoms I actually said: "and this is all? Such a small number... " I said so because I couldn't visualize it. It had no meaning for me, and I new that there were numbers much bigger.

Lets see another example. Suppose we take a sheet of paper 0.1 mm thick and we will cut it in half, stack it and then cut in half again. We will repeat this 100 times.
In the end we will have a stack of paper sheets. Since the stack height is representing the number, it should help us visualize it, right? Well lets see how high will the stack be.

=1,267,650,600,228,229,401,496,703,205,376

This is in 0.1 mm. In light years it is about 13.4 billion light years. This is close to the radius of the whole universe...

The problem with visualization of such numbers should be rather obvious now. Nothing we have in our world can help us visualize them. But what about smaller numbers - 10^6 for example?
Such numbers are possible to visualize using small objects. But it is still difficult.
For example you can visit this site to see how a 10^18 pennies looks like.

For most of us this is not an issue. We don't use such numbers were often, and we are not asked to visualize them. However, there is an interesting underlaying fact in all this. Those numbers can be written, we can use them in equations (I see no reason why but it is possible) but we have no use for them. They don't describe anything in our world. Then why do they exist? And do they exist at all, if they are useless? As I already said, I believe that math is discovered and that it has a "life of its own". While the existence of such numbers is not a proof to this view, it is still something to think about - if most of the numbers that we have are of no use to us then why we invented them, if they were invented by us?

Readability level

I recently did a check of this blog readability level on website grader. The site checks different specs including readability, google rank and so on. Strangely I got a Phd level. Since I tried to write it as simple as I could I was very surprised and amused by this result.

Anyway, I recently stumbled upon a site about octomatic number system. It is a system of base 8.
As of itself it is not very interesting - as I already wrote in a previous post, Evolution of numbers, all number systems that can be used for geometry are isomorphic to each other. However on this site they also offer to use different signs for the numbers:

What is interesting is the second row from below. The signs here are directly connected to binary. You can see it for your self if we will line to be 1 and empty space zero we will get:
000 001 010 011 100 101 110 111
And from the image you can tell immediately that 111=7. The same rule works for large numbers: 100111001 is for example 4*8*8+7*8+1=313.

The bottom row is a handwritten form. Personally I doubt that it is a good form, but it is their choice...

Evolution of numbers

We all know how numbers look. And sometimes it is even hard to imagine that they can have a different form that what is familiar to us. But while math is a rather ancient science our number system and our numbers are not very old. This photo shows their development in time:

As you can see even numbers in the middle ages were different from modern numbers. And I am sure that you would not even think that there is any connection between our numbers and Hindu.

It is a surprising fact however that there is just one number system. You will probably say it is not correct. But it is. Yes, there were a lot of different number system in history. Even in the present we have more then one. Take the binary system for example.
But if we want a number system that is good enough for both algebra and geometry, that is a system that lets us measure distance we discovery that there is only one such system - they are all isomorphic to each other. It means that you can give different names to numbers, you can have it in base 17 (for example) and not in base 10 but it will still be the same thing. In mathematical language - we have only one mathematical structure that is a complete ordered field.