Math and Firefox wallpapers

I added new wallpapers to my collections of Firefox and math wallpapers, follow the links to download.

From Firefox Wallpapers

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.

Unique captchas

Apparently some sites use captchas to assure the intelligence level of their users:

Project Euler

I recently found an interesting site called Project Euler. This site attempts to provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context. This is being done by publishing different mathematical problems of varying difficulty.
I liked the concept so I thought about joining the site, but after taking a closer look on the problems I lost my motivation - most of the problems are meant to be solved using a computer. I just can't feel motivated to write a program in order to sum all the primes less than 2 million (this is problem number 10). However, if you like this type of problems this is clearly an excellent site. They have lots of problems of varying difficulty and new problems are constantly added.

For those who like my don't like using computer to solve problems there are problems that don't require a computer to solve - for example the first problem: "Add all the natural numbers below one thousand that are multiples of 3 or 5." This one is a very simple problem, so I guess it will be fine to post a hint to a solution. All you need to do is to sum the arithmetic progressions 3,6,9,..... and 5,10,15..... If you add the sums you will get the result, but the numbers that are multiples of both 3 and 5 will be counted twice.
I also really liked problem number 205. It is pretty easy to solve, but it requires some thinking and there is no need for a computer.

Infinitesimals

In the previous post, I wrote about division by zero. In this post I want to talk about one particular case when such division, and its definition are important. As you probably guessed from the title, this post is about calculus. (In this post I am talking only about one variable calculus).

One of the most basic questions in calculus is finding slopes of functions. The simplest example of such a problem is to find the slope of a linear function, f(x)=mx+b.
In this case we get a straight line, so the slope is uniform. To find is we need to calculate the difference in y divided by the difference in x:

(f(x+h)-f(x))/h=(m(x+h)+b-mx-b)/h=m(h)/h=m

For a less friendlier functions, we cannot talk about globe slope, but only about slope of a certain part of the function, or even only in one single point. And this is were the problem appeared. We want to know the slope in every point, but how can we calculate it? Firstly we need to define what such a slope is. For example lets look on the function f(x)=x^2 and on the point x=3. Lets look on the points f(3.5) and f(3). If we connect those two points by a straight line, we can calculate the slope of that line. If we draw this on a paper, you will see that the function and the line are really close to each other on a small area around 3. Therefore we can thing about the slope of this line as an approximation to the slope of the function. But, obviously if instead of 3.5 we will take 3.1, we will get a better approximation. In the end we can think about the slope as:

slope=(f(3+h)-f(3))/h , h=0

And here we have it - devision by zero. It is obvious that there is no way around it. If h is not zero we get only an approximation and one which can be improved easily enough. The solution to this problem was found by Newton and Leibniz. Their idea was to define a non negative "number" that is smaller that any positive number. Such a number is called infinitesimal. The only real infinitesimal is zero, but if we agree to imagine that there is another such number, they we get many such numbers. This is because if dx is an infinitisimal that 0.5dx is also an infinitisimal. Since dx is not zero, we can divide by it. And becasue it is smaller than any positive number we can disregard it as if it was zero. It is simple to find the slope of x^2 usinig dx:

slope=(f(3+dx)-f(3))/dx=(9+6dx+dx^2-9)/dx=(6dx+dx^2)/dx=6+dx=6

Although the result is correct, we no longer use infinitesimals, but use limits instead. The reason for this is that infinitesimals are problematic. The problem lies in the very definition - it is not clear what do we mean by a number that is smaller than any positive number, but not negative or zero. We also treat it as both zero and as not zero. However, they are still in use in physics. The reasons for this is that while they are not rigorous enough for mathematicians to use, they give good intuition and they appear rather naturally in physical problems.

Division by Zero

We all know that division by zero is undefined. But what does it mean? Firstly, it is not "completely" undefined. For example, it is possible to say that:

1/0=lim1/x, x->0

This particular definition is not very good, because the limit doesn't exist. But we can always suppose that we look only on positive x and the the limit is defined to be infinity. Alternatively, it is possible to use geometric series. The formula for the sum of the geometric series says that:

1+q+q^2=q^3+...=1/1-q,

If we will take q to be 1 we will get: 1/0=1+1+1+1+....=infinity

As you can see I just wrote two different definitions to division be zero. However, this doesn't solve anything. The problem is that infinity is not a number. We could just as well say that 1/0=watermelon. Mathematically it is basically the same. Therefore when we divide by zero we no longer deal with numbers, and this what makes such division undefined.
Now, what about other definitions? A very long time ago, when "zero" appeared in mathematics there was an attempt to define division be zero by simply stating that division by zero gives a fraction whose denominator is zero. But this is again not a number.
We also cannot define 1/0=a where a is a number. The reason for this is that in this case we get:

1=0*1/0=0*a=0

And if this happens we get that the only number that we have is 0, because all the "other" numbers are equal to it. Obviously this is not an interesting situation. Because of this it is necessary that the devision by zero is not defined.

Lets look on some other example of undefined identities. The first one is 0^0. The reason this one is undefined is very simple:

0^0=0^(1-1)=0^1*0^-1=0/0=0*1/0

And we are back to division by zero. It is important to note that it is sometimes assumed that 0^0=1, but this is mostly done as an alternative to saying that in a polynomial x^n in which n can be zero x is not zero.
A more complex example is (-3)^x where is a real number (that is, x is not rational). Obviously, there is nothing special in number (-3), this expression is undefined for all negative numbers. For positive numbers the definition uses limits. For example:

3^x=lim(3^q), limq=x, q- rational

This definition works well for positive numbers, but for negarive numbers we have the problem that the square root is not real. Because of this if we don't allow complex numbers we get many undefined points in the series, and if we allow such numbers the series doesn't converge.

As a bonus, here is a little proof that 2=1. Can you see where is the error?

a=b.
a^2=ab ,
a^2+a^2=a^2+ab,
2a^2=a^2+ab,
2a^2-2ab=a^2-ab,
2(a^2-ab)=1(a^2-ab)
2(a^2-ab)/(a^2-ab)=1(a^2-ab)/(a^2-ab)
2=1

The reliability of wikipedia

As you all know there are many articles on wikipedia, and their quality varies greatly. However the main problem is that in many cases it is impossible to tell if a particular article is accurate or not. For example, I just found the following article on wikipedia:

Beard-second
The beard-second is a unit of length inspired by the light year, but used for extremely short distances such as those in nuclear physics. The beard-second is defined as the length an average physicist's beard grows in a second, or about 5 nanometers[1].

One beard-second equals 50 Ångströms (10^-10 m). 20000 Beard seconds equal 1 RCH. 2000 Beard seconds = 1 RBC.

Google search supports the beard-second for unit conversions.[2]

Unsurprisingly, this article is being considered for deletion in accordance with Wikipedia's deletion policy. Suprisingly, the part about Google supporting beard second in unit conversion is actually true. After a quick search on the internet, it seems that the beard second was invented as a teaching exercise a few years ago.

In this case, the article seems to be a joke, but in reality it is about a funny teaching exercise....