Buying a refrigerator

Unfortunately, I have to buy a new refrigerator. The old one has been working for a long time, but it appears that it is time for him to go to the great electronics beyond. Because of this and some other things, I spent most of the afternoon in shops today. Not a good way to spend half a day...

Now to the reason I am writing this post. I was asking people for recommendations, what company has good quality, and where to look for good deals for some time already. So I figured it would be a good idea to post this question to my blog. Currently I am looking at LG, Sharp, Amcor and Siemens. The Siemens are all too tall, and Sharp is expensive. LG looks like it will break any minute. Amcor is nice, but I heard that they all have a little defect, because of which they require constant maintenance. If you have any experience with these please share it with me..

Since this is a math blog - a bonus question. If the following integral (the borders are 1 and plus infinity) is finite, and f(x) is continues everywhere:

Does it means that the limit of f(x) when x approaches infinity is zero? I will publish the answer after a few days, meanwhile feel free to say what you think in the comments.

Fractal Oranges

I am too tired today to post about polynomials (because it requires to use latex for typing math formulas and a bit of drawing), so this is a post about fractals in every day life. I was sent an interesting example of such fractal by a friend, and I thought to post about it here. Look on the following photos:

This is the "fractal stage"

The final product

And this is how it all started.

You probably wonder how it is connected to fractals. The answer is simple - the process in which this orange was made is iquivalent to creating a fractal. We took a symmetrical complex shape (the third photo) and then we squished it without loosing the symmetry. The result (the first photo) is in fact the original one, in many copies. We will need to slice it, but if we will do it, we will get a lot of smaller copies of the original shape. Which means we can repeat the process again and again. Now if we will look on this process in four dimensional space (with time as the fourth dimension), it is obvious that the result is a fractal. The more we "zoom in" (zooming is going forward in time) the more our figure becomes complex and with smaller details. However, it always resemble the original shape. Thus, by definition this is a fractal. You can find more photos here.

Quantum cat

A quantum cat. I am sure Schroedinger didn't think that his little experiments will become the reason for so many jokes..

Science is fun sometimes, isn't it?

Since in this case we see the cat, we are sure that he is in the box (and we know he is alive). And therefore this is not a paradox. However, it is always good to think about puzzles of quantum mechanics...

Solving the Quintic

In my previous post about finding the roots of polynomials, I wrote that no general solution exist for polynomials whose degree is higher than 4. However, we still need to be able to find the roots of these polynomials. In this post I want to talk about some of the methods for finding roots, and about their limitations.

Finding the roots of polynomial might sound an abstract goal, but it is actually a very practical one. Look on the following problem for example - Given a line segment of length one AC, divide it in two parts AB and BC in such a way that AC/AB=AB/BC. This is a purely geometrical problem, but its solution is equivalent to solving the following equation:

If we will simplify it a bit, we will get:]

1-x=x^2
x^2+x-1=0

This is a very simple and geometrical example, but the need to solve an equation similar form may arise in different problems in engineering. In this example, it is easy to see that the solution is irrational. This brings us to the first problem with solving polynomials. The answer might be irrational - and if so, even a computer will not be able to solve an equation by trial and error.

Lets look on the following quintic equation (fifth degree polynomial):

This one cannot be solved using a formula. But it is obvious that 1 a root of this polynomial. However, such a polynomial must have five roots (according to the basic theorem of algebra). What can we do to find them? A simple way would be to try to divide this polynomial by x-1. After such division, the degree of the polynomial we need to solve becomes less by one. We will get then that our original polynomial can be written as:

So, we got now a problem which is very easy to solve. All we need to do is to use the formula we know for the forth degree polynomial, and we are done. If we don't remember the formula, we can try to guess another root, lest call it y. and then divide by x-y. For this polynomial, it is again obvious that 1 is a root. So we can divide by x-1 again. We get:

If this is still too complicated, we can divide again.

I wanted the above example to be as simple as possible, so I selected the polynomial (x-1)^5. However, what about more complex examples? It is practical to try to solve some random polynomial in this way? The answer to this is yes. Firstly, we can always divide two polynomials. This result we get from a theorem I don't want to talk about now. Secondly, while it is often impossible to guess the root, this is a rather practical approach. Unless all of the roots are irrational, the chances to guess them are pretty good (especially if you get this question in an exam). The reason for this is that the last monom (the free number) must be multiplication of all the roots, and the second must be minus the sum of all the roots. In the polynomial I solved above, for example the roots are (1,1,1,1,1). There sum is 5 and there multiplication is 1. And in the polynomial this is exactly what we get - in the second place we have (-5x^4) and in the last we have 1.

Lets look on the following polynomial:

This particular polynomial cannot be solved by the method specified. The reason for this is very simple - are the roots are complex. For complex roots there is a very interesting property - if, lets say, (2-i) is a root than (2+i) is also a root. This makes the process slightly easier. If you guess one root you get another one as a bonus. Now, I don't know about you, but I have no idea what are the roots of this particular polynomial, and I don't know how to find then.

So why I am saying that the method I presented in which you need to guess a root is practical? It is only practical when you need to solve a polynomial that is "nice and easy". If the polynomial you need to solve cannot be solved by simply guessing, it is better to use the computer to find the solution or at least an approximation to the solution. Unfortunately, this is the only thing we can do without a closed formula.

The problem we have with the quintic, and other polynomials is not unique. We have similar problem with integrals. It is often perfectly clear that an integral have a solution, but we have no way to find it. For example:

.

It is proven that this integral cannot be written using elementary functions, but it exists. And there are many others like this one. In modern mathematics there are more than enough things that are proved to be impossible, but we would like them to be possible. And this is (in my opinion) a strong argument for the claim that math is discovered and not invented - we get result which are hardly expected or desired.

In the next post I will write about how simple polynomials were solved in the ancient world, and I will also try to find time to write about the formulas for the third and forth degree.

A bit of Ads

You probably noticed that I have changed the positions of ads on MathPages. The reason for this is that I am trying to test if moving the ads to this position will give better results. From what I read on the web, this position should make them more visible.
I tried to make them not very obtrusive, so I doubt they will be noticed too much. But they will be noticed better (I hope). I don't expect to make money form blogging, but I feel it would be wrong not to attempt by not putting ads. If you feel annoyed by them just install adblock to block them.

Problems with my ISP

I am heaving some weird problems with my ISP today. Yesterday everything was fine, but when I tried to connect to the internet today at the morning, it just didn't connect. I tried to ping google.com and got the message "unknown host".

Since I am writ8ing this post, I finally manged to connect. Unfortunately, this involved changing the DNS name servers. It appears that my ISP either changed his DNS servers, or they are broken for today (I cannot even ping them). I had to use the name servers of another Israeli ISP to connect to the internet. For now it works. I will try to use OpenDNS latter today or tomorrow. Hopefully everything will work fine.

I doubt that I will be without access to the internet, during the weekend, but if there will be no new posts in the next couple of days, this is the reason why. Also, I managed to injure my hands by typing too much, and writing a lot this week, so it now hurts to type :(.

Update: I am now using OpenDNS instead of my ISP DNS servers. I changed the script I use to connect to the internet so it will now automatically use these new servers instead of any Israeli ISP servers. For some reason I started to have problems with gnome-settings-daemon. If I connect to the internet and then logout, it fails to start when I log back in. It casues the logon to be extra slow, and the themes are all wrong. I found a workround, but it the problem is still there. Probably I should do some serious fixing to my computer when I will have time.

Quantum physics in a grafiti form

For some reason, I feel that this simple pictures talks well about such properties of quantum physics as probability and our inability to observe the quantum processes.




What do you think about it?

The hunt for the roots

One of the most basic questions in mathematics is finding solutions to equations. In this post I want to make a short overview of the ways to solve some of the common forms of equations, and I also want to discus the history of how this solutions were found. This is only the first post about this subject, so it is mostly introductory.

For the simplest equations, the solutions are known for a long time, so I will only mention since what period the solutions were known, and will not discuss the ways they were developed. The most simple equation is an equation of the form:

x=4-3x

This is a simple equation in one unknown, and what is important for this post, the only power of x that appears in it is 1. This is a first degree polynomial. Such equations are very easy to solve. This particular one is solved by simply moving 3x to the other side and dividing by 4. The solution is x=4. If we will add more unknowns we will get a linear system. For example:

x+3y+5z=0
3x+6y+z=0
7x+9y+2z=0

Solving such system is slightly more complex, but it is not difficult. The more unknowns we have the more time we will have to spend on this system, but any such system can be solved. In fact, simple linear equations were solved even in ancient times - in Egypt and Babylon. Since then the methods used have evolved greatly. In the ancient world, such equations were frequently solved geometrically. Now we use matrices and determinants to solve such equations. Also, the algebra notations allows to solve simply linear system by simply "moving" the numbers from side to side. In ancient time this was not the case - mathematics was often done with no symbols at all.

The linear equations we now how to solve. So, what other types of equations we have? The next in line is the equation of the form:

x^2-b=0

This is a simple second degree polynomial. The solution seems trivial. If for example, b=4 than x=2 is a solution. Solving such equations is also something people knew how to do for a long time. There are examples from ancient Babylon of such equations being solved at schools. However, it must be noted that in the ancient world such equation often didn't have a solution. If for example b=2, it is impossible to write the solution, because the root of two is irrational. Also if b=4, a student at ancient Babylon would write that the solution is 2. But we know that (-2) is also a solution. The reason for this is that the concept of negative numbers was not familiar to the Babylonians - it was first introduced by the Muslims.
If we will take b=-1 we will have an even harder problem. The equation becomes then x^2=-1. Until Gauss at the 18 century, such equations were considered unsolvable.
There are other, slightly more difficult to solve, second degree polynomials. The most general form is:

ax^2+bx+c=0

Obviously, a is not equal to zero (we get a first degree polynomial otherwise). From now on I will call the solution of an equation the root of the polynomial, or just a root. To find the root for such general polynomial, all we need to do is a few simple steps. Firstly we will divide by a, and then multiply by -1 if a was negative. After this we will get a polynomial of the form:

x^2+bx+c=0

The b in this polynomial is not the same it is the original be divided by a. The next step is to add to both sides (b^2)/4. We would be able then to make one of the sides a square:

x^2+bx+(b^2)/4+c=(b^2)/4

(x+b/2)^2=(b^2)/4-c

We have a square on one side, so it is logical to take the root, and simplify a bit:

x+b/2=

We can now move b to the second site, and get the equation:

x=

It looks very similar to the equation we all learned in school, but an a is missing. However, our b is in fact the original b divided by a and so is c. If we will write b/a and c/a and move them a bit we will get:

x=

Any second degree polynomial can be solved using this formula. In ancient Babylon, the problems were solved using this formula, but the solution process went without any symbols. It was done completely verbally. The Greeks also knew how to solve such problems, but they used geometry to solve the problem. I will write about how they did it, and the reasons for using geometry for such tasks in another post.

Now we know how to find the roots for second degree polynomial. But what about the third degree? Solutions for simple polynomials were known to the Babylonians. But they didn't know a formula for a general third degree polynomial. The same is true about the Greeks, and the Muslims. I once heard that when Archimedes was killed, his last words were a curse on those who will try to find the general solution for the third degree polynomial: "Cubics you shall not solve". I don't know if this is a true story. Probably it is just a legend. However it took a lot of time to find the solution for this problem. The solution was finally published by Cardano, a French mathematician, in the 16 century. He was the first one to publish it, but he himself got the solution from another mathematician - Tartaglia. In the 16 century the competition between mathematicians was very strong, so Tartaglia who was the first to find the solution, didn't want to publish it, but preferred to keep it to himself. When Cardano discovered that Tartaglia knows the solution, who put a lot of pressure on him to make him tell the solution. Finally Tartaglia agreed, but asked Cardano to make an oath that he will not publish the solution before he does. A few years passed and Cardano found out that the solution Tartaglia told him was found before by another mathematician and went unnoticed (the communication wasn't very good then). Upon discovering this, Cardano published the solution. The solution is anything but simple. Since it is significantly longer than for the second degree I will not write it in this post.

The next step is obviously a forth degree polynomial. This one was also solved in the 16 century, by a student of Cardano. Again, the solution is too long to be written in this post. If I will have time, I will write another post in which I will fully solve both of these questions. There two important facts about both of these solutions - they both are solutions by radicals, and they both effectively turn the problem into finding the roots of a polynomial of second (for the cubic) and third degree (for the forth degree polynomial).

And finally we get to the fifth degree - the quintic. The quintic is a polynomial of the form:

ax^5+bx^4+cx^3+dx^2+fx+g=0

After seeing the solutions of the previous problems, everyone was sure that this problem would be solved as well. But no solution was found for a long period, until Abel came to the scene. He had an interesting idea - he started to question the generally accepted thought that there was a solution. He wrote a rather large (about 950 pages) proof that showed that it wasn't possible to find a general solution for any degree larger than 4. This proof wasn't accepted well. He sent it to Lagrange, but didn't got an answer. When he tried to get an official response from the Academy, the response was that "They don't find it usefully to look into his proof". Just before he died he received a letter from Cauchy, in which Cauchy wrote that he believes his proof is right. It was found out letter than while there is indeed no general formula for a degree larger than 4, there was a mistake in his proof. He skipped on one step, because he thought it was obvious - but it wasn't so. Anyway, it is now a generally accepted fact that there is no general formula.

There is also another interesting result that this discovery brought. You probably recognized the formula for the second degree polynomial, but I doubt you know the formula for the third degree or for the forth. They are no longer studied and are of no importance. It is possible to get a B.S. in math and don't know these formulas. It turned out that it is more practical to be able to solve specific examples, than to solve the general case. And for a specific problem we can use a computer who knows the formula.

However, we still need to be able to solve the quintic, as well as other polynomials. In the next post I will describe some tricks that are used for this purpose, and the general methods for finding solutions.

Relativity and quantum mechanics

In my previous post on this topic I have shown that while faster that light travel is impossible, it is possible for the electrons to move from one energy level to another in zero time. The distance it travels is like the distance between planets on our scale, however the time is zero.

The reason for such a result is very simple - relativity and quantum mechanics cannot be used together. It is not possible to apply relativity where one should use quantum mechanics. When we discussed individual electrons the relativity theory simply stopped working. The results that were correct for large scale become wrong on this scale.

But why is it so? There is after all a general agreement that a theory that works only under specific conditions should transform gradually to a different theory when the conditions it requires are changed. This part probably sounds a bit confusing, so here is a simple example:

Photo by wili_hybrid

A long time before Einstein, people noticed that for two system that are moving with a constant speed compared to each other, the system of coordinates has to be transformed when you move from one system to another. If, for example, you are on a train that moves with 50 kph east relative to the earth and you see someone who is sitting on the field outside, than from his system of coordinates you are moving with speed 50 kph east, but from your point of view he is moving with the same speed to the west. The transformation used to move from one system to another is the Galileo transformation. If your coordinates in one system are (x,y,z,t) than your coordinates in a system moving away from you with a constant speed v are (x',y',z',t'). If at the moment t=o the both observers where in the same place and the movement is only on the x axis we get that:

x'=x-vt
y'=y
z'=z
t'=t

However, according to relativity this is not correct when v is big enough. In relativity we use Laplace transformation instead of Galileo's. Under the same condition we will get:

y'=y
z'=z

The formulas look very different. Partially this is because the of units used. However, if we will go to the limit were c is significantly large than v (that is c is regarded as infinite), they will turn into the Galileo transformation. It is very easy to see - the only impostarnt part is to notice that the units need to be balanced, after this it is trivial. We can say therefore that relativity turns into classical mechanics when the speeds are low in comparison to the speed of light.

However, this doesn't happen with quantum mechanics. It is divided from relativity by a scale barrier, and when this barrier is approached the two theories start to contradict each other. A lot of work have been done to solve this problem. The main approach is to try to unify all the basic forces. Those forces are - Electricity, Magnetism, Gravitation, Strong and Weak. The first two are already unified for a lot of time. The weak force also can be unified with them. I also heard that the strong force was unified with the weak force, but I don't know the details. Gravitation is a problem however. For the other forces particle carriers where found - but not for gravitation. In fact, the question what gravitation really is, is still without answer. It is a mystery waiting to be solved...

A bit of trivia - It is a surprising fact that Einstein contributed a lot to both of these theories, but while he helped quantum mechanics to take roots he wasn't happy with the result. He was the one who proposed the duality of the photon, and he was the one who helped to promote the understanding that all particles have this duality.

I ended my previous post with a question - Was Einstein wrong? The answer should be clear, but I will say it anyway. He wasn't wrong. It is simply that by going to this very small scale I left the domain of relativity and there the rules are different.

You probably noticed that this post raises a very interesting question. Since faster than light travel is equivalent to time travel as I have shown, does the fact that the electron can move such great distance (on his scale) in zero time means time travel is possible? Nope. There is no time travel in this case. The reason for this is simple, but it requires getting used to. The electron belongs to the "quantum world". We can think about this world as being separated from our world by a "shield". This shield is called The Heisenberg Uncertainty principal. What it says is very simple - the uncertainty in the location and energy are always bigger than some constant number. It means that we simply cannot see too well what is going on in this "quantum world".
In fact the way I used to show that faster than light travel is equivalent to time travel depends on accurate measuring of distance and time. Since we cannot do this with the electrons, even this general result just doesn't apply.

By the way, if you want to be remembered forever in the history of science, finding a way to unify electricity and gravitation will surely achieve this goal...

Scraping status update

Unfortunately, instead of continuing to write about faster than light travel, I have to write about some "problems" I am having with this blog (it doesn't really amount to a problem but still requires to be taken care of) . Hopefully I will continue posting more interesting staff tomorrow.

I recently wrote that some of my posts were scraped. You can read more about this here. To my surprise I found out that most people don't know what scraping is. I wrote about it in the post I linked, so if you don't know what it is read it first.

As of now, I know about 4 blogs that scrap my posts - that is, they post short paragraphs from my posts on their blogs (all of the content of such blogs is just paragraphs collected from other blogs) without proper attribution, and spam me with backlinks. I found the email addresses of all the owners of these blogs (I suspect that there are only two owners - one of them runs three of these blogs). Two of the addresses were fake, but the other two worked.

The email I sent:
------------------------------------------
Hello
I have found out that you took content from my blog post (link) and
placed it in yours (link) without proper attribution. This is a
copyright infringement, please fix it. And don't put my content in
your posts in the future.
---------------------------------------

I didn't got a response (that would be too much) but I got some reaction. One of these blogs deleted itself from the Technoroti database. However, the paragraph he took from my post still appears there and it is still not attributed to me.

The other scrapper just ignored me. Also, I found out yesterday that another of my posts was scrapped. I think this happened because I mentioned Einstein in the post - I suspect that the program that selects the content to be scraped uses his name as a keyword, so if I mention him my post is likely to be scraped.

I am thinking about publishing the two email addresses I have. It probably wouldn't help much, but it is something to consider.

I don't think I will write more about this subject - I have more interesting topics to write about. Probably I will start to delete the backlinks I am getting from these blogs, and perhaps I will publish their emails.

Disclaimer

I was recently contacted by Kevin Brown, the owner of MathPages.com.You probably noticed that my blog is named Math Pages Blog. However, I have no connection to MathPages.com.

When I chose the name "Math Pages" I didn't know that it was already taken - otherwise I would have chosen another name. Unfortunately, the name of the blog (or site) is the first thing you need to chose when you start your blog, and it is very problematic to change later because it is usually connected with the URL (and it is connected in my case). Also, I suspects that even if I will change the name, it will have little influence unless the URL is also changed. And changing the URL basically means to start a new blog. I am blogging only for a few months, but starting a fresh is not a happy thought.

Because of this, I am now adding a disclaimer to the "About me" section in the sidebar.

There is no real reason to think that my blog will in any way influence the traffic MathPages.com is getting, and by putting this disclaimer I will hopefully prevent any confusion about this blog connection to MathPages.com.

Faster than light

I finished my post about time travel paradoxes with a promise to write about one very specific example in which faster than light travel (which is equivalent to time travel) is possible. I am going to write it in a slightly weird way - firstly I intend to explain why this particular example is not possible, and then to show that it happens nevertheless.


We all know how an atom looks like - it is just a little ball (nucleus) surrounded by a cloud of electrons. Most of the atom is empty space. The atom has a very interesting property: the electrons that orbit the nucleus can be only on a finite number of orbits = energy levels. I am not going to talk about why it is so, but what is important to note is that the distance between any two energy levels is very large. In fact, the atom can be viewed as a solar system in miniature - the distances between the orbits are comparable to the distance between planets. The electrons decide on which energy level to be according to the energy they have. The more energy an electron has the higher (further from the nucleus) it will be..

Thee is a simple way to cause an electron to jump from orbit to orbit - we just need to hit it with a photon. If the energy the photon has is the energy needed for the electron to move to a higher orbit, it will absorb the photon and move to this higher orbit. After discussing this we can go to the main question I will deal with in this post:

How much time does it take for the electron to move from one energy level to another?

Answer: Zero.

To explain this, lets firstly look on why this time must be more than zero. In the following paragraph I will develop a mathematical formula - if you don't understand what I am writing just skip it and go to the next paragraph.

Lets look on a regular piece of wire. If we will connect it to a battery (if you want this battery to be of any use lately don't do it) a current will flow in it. How much current? The usual way to define current is to look on the area of a cut of the wire = A. The current than is the amount of charge that passes in infinitesimal time=dt. To define it precisely we need first to define the density of charge flow. The density of charge flow (J) is the number of free electrons in the wire multiplied by their speed and by their charge: J=nvq. Note - J is a vector. The current is than defined as:

I=

dA is an area vector of an infinitesimal part of the cut area A. If, as it is with regular wire J=const we get that I is simply: I=JA. The formula with dA is good for any object in which a current can flow. Because of this we can use it in a slightly different way. Lets think about a general bounded 3d object. This formula is good for it - but instead of saying what the current is, it tells what is the rate of change in the charge of this object.
What the formula says exactly? It says that the rate of change of the charge is an integral on the charge flow density. Thus, the charge (electrons) that left this object is exactly the charge that passed the surface of this object - the integral is basically a flow throw the surface area of our object. This means that electrons cannot just jump outside - they must cross the surface somewhere. This conclusion seems obvious, but it isn't. Without this formula, having only the law that the total charge is conserved, we can say that if the electron jumped outside the total charge in the space is the same - so it is possible. But this simple formula says no. However, it doesn't say that it is impossible for the electrons to jump inside our object. To show that this is also impossible we need another formula, so lets develop it.

What is Q? It is the total charge of our 3d object. Therefore it can be expressed as:

In this integral rho is the charge density (it is not J). If we will put this expression into the previous formula, and also use the Gauss theorem to change the right integral in our formula from JdA to divJdV we will get:

The both integrals are dV, so we get that:

It is usually written in this form:

The plus sign is a convention - it doesn't really matter, because the positive and negative flow direction is defined according to what is easier to work with.
What does this new formula says? It says one important thing - the change in the density of charge in an infinitesimal is equal to minus the divergence of J. This is exactly what we were looking for - now even the electrons inside our 3d object cannot jump. They all must flow passing all the midpoints.
Therefore it is impossible for the electron to jump from one energy level to another in zero time.

But is this correct? No. The math I used is correct, but there is one single thing that is left out - quantum mechanics. The formulas I developed don't take it into account. They are still correct, but they cannot be used on very small scales.

It is now time to explain what happens with out electron. When the electron is being hit by a photon, it consumes the photon and gains its energy. This energy is exactly the energy needed for it to jump to the next energy level (otherwise the photon will not be consumed). The collusion between the photon and the electrons happens in one instance - it is not a process. What this means is that the electron gets energy that corresponds to the next energy level. Therefore it must be in a higher energy level. It cannot just stay. If there would be no photons, than the electron would consume the light energy gradually and would "rise" to the next level. But the energy is transfered in a collusion.
Mathematically, the potential energy is space must be continues. Because of this it is not possible for the electron to have a higher energy while being in a place that corresponds to a lower energy. The reverse is also true - when the electron descends to a lower energy level it releases a photon and jumps down - in zero time.

So, was Einstein wrong in saying that faster than light travel is impossible? Were I wrong when I said in the previous posts that faster than light communication (or travel, it doesn't matter) is equivalent to time travel? This I will answer in the next post.

What degree should I get?

I just did a test that is suppose to tell you what advanced degree is the best for you. The result is not very surprising:

You Should Get a PhD in Science (like chemistry, math, or engineering)

You're both smart and innovative when it comes to ideas. Maybe you'll find a cure for cancer - or develop the latest underground drug.

What Advanced Degree Should You Get?

A lot of bloggers whose blogs I read did this test, and we all got the same result. This is suspicious, so the test quality is doubtful (besides, it is a very short test). However, this is a fun thing to talk about...

Learning History of Mathematics

The title of this post is a bit misleading - I am not going to write about History of Mathematics in this post. This I plan to do in another post.

I am now doing a short introductory course about History of Mathematics. When I took it I though it is going to be a simple and easy course. However, it turned out to be problematic. The course itself is only two hours a week, but we are also supposed to read books about History of Mathematics. The professor recommended History of Mathematics by Boyer (very good book), and also some other books. I was reading this book in the library for some time, but somebody took it so I can no longer use this copy. When I found out the Boyer was taken, I took another book - Development of Mathematics by Bell. I cannot say I like it, he is very critical and he is also jumping a lot. One of the purposes of this book was to show to students what areas are now alive in math and what areas are of no interest now, however this goal makes it more difficult to read. Unfortunately, the part of the library where this book is, was closed today and will be closed for the following month.
I requested Boyer and will get it for 3 weeks on 9/7, so this is not a problem. I also have two books on my computer - A History of Mathematics From Mesopotamia to Modernity by Luke Hodgkin and History of Mathematics An Introduction 6th Edition by David M. Burton. I don't like to read from the computer screen, but the books are very good. Another resource that the professor recommended is MacTutor. It is a very good site with a lot of articles about math history and it also has a lot of mathematicians biographies.

However, after doing this course for six weeks it is obvious that reading one book is not enough. I feel that to really understand and learn this subject I need to search for extra information to feel the gaps which are not filled by the books I have. This means that I will have to "research" on the net the topics that are not clear to me.
Since I will be doing this research anyway, I will write about these topics on Math Pages. I have a post about faster than light travel and another one about definite integrals to write, but after finishing them, I will write about math history. Probably I will start with solving the cubic equation.

Time travel paradoxes

I recently wrote a post about why faster than light communication is impossible. I finished this post showing that such communication would be equivalent to time travel, and it is logical to assume that time travel is impossible.
However, is this indeed true? Time travel is often talked about in science fiction, but is there a reason to think that it is possible?

Photo by jonrawlinson

Firstly we need to distinguish between two types of time travel - to the past and to the future. We all know that it is possible to go to the future. We all do it, it is called living. It is also possible to travel to the future with a "faster speed" than normal life. All you need for this is to accelerate yourself to a high enough speed, and according to relativity you will travel to the future - when you will return to the place you started your travel you will see that your watch is late. How late depends on your speed, it might be a second or 1000 years. This effect is very real, we even have to take it into consideration when we communicate with satellites.

Traveling back to time is a different story. Unlike traveling to the future it was never done. So the only thing we can do is to discuss different theories. There are three main theories about traveling to the past:
1. Time travel is not possible - there are two versions: either there is no way or it will destroy the universe in the process.
2. Time travel is possible but the past cannot be changed.
3. Time travel is possible, but it will destroy part of the universe.

Lets look on all of the three theories:

1. Time travel is not possible:
There are two versions of this theory. The first one comes from a literal understanding of time travel. According to it in order to travel back one second in time, you need somehow to return the whole universe to the exact some state it was one second ago. It means that you need to place every single particle exactly at the same place. However this is clearly an impossible task.
The second version of this theory is based on time travel paradoxes. The paradox I will talk about here is called the grandfather paradox: Lets suppose that time travel is possible. Lets also suppose that someone (Mr X) traveled back in time. While he was in the past he 9accidently) killed his grandfather. In doing so he preventing himself from being born in the first place. But if he wasn't born how could he travel back in time?
There is also a more general version of this paradox - by traveling back in time you change the world past, so in the very moment you will get to the past the world you come from (the future) will no longer exist. And therefore, you never traveled back in time.
The conclusion from this is that if time travel is possible, you will destroy the whole universe by traveling back.

2. The past cannot be changed:
This theory is an answer to the question arising from the previous one. In the grandfather paradox, we assumed that it was possible for Mr X to kill his grandfather and to prevent his own birth. But what if this is not true? What if there is a law that not allows people to influence the past? In this particular example, we can suppose that Mr X will be stopped by police just in the right moment, or it would turn out that the men he killed wasn't his grandfather at all. For the more general version of this paradox, we can assume that either the results of the activities of time travelers just slowly vanish so they don't affect the future in any way, or that there is a fixed time line in which time travel appears together will all other things and nothing can be changed.
This theory allows for time travel and solves the paradoxes I presented, but there is a problem with it. Lets do the following thought experiment: Suppose you have a time machine, and a laser that shoots a bit of light into the time machine. The machine send the light back in time, so it goes out of it on the opposite side and two minutes before the laser was fired. On the wall after the time machine there is a detector that when hit by the laser been will send a signal to put a barrier between the laser and the time machine. It looks like this:Now if the time machine works, the laser will prevent itself from firing, but this would mean that the detector didn't close the barrier so the laser worked - and this is a contradiction. Note that it doesn't matter how the time machine works, and to how long ago the light pulse is sent.
If, as the theory says, we will assume that somehow "it all worked" it follows that something is broken - if for example the detector is broken, no paradox will be created. However, this is a very simple system. The only thing that is likely to always malfunction is the time machine itself. This means that the time machine doesn't work, and therefore time travel is impossible.

3. Local destruction:
This one is the attempt to unite the previous two theories. Basically it says that time travel is possible but because of the paradoxes described above, it will destroy the universe. However, the universe is a very large thing. So only a small part of it will be effected. Time travel will create a "wave of destruction" which will move over some finite distance, destroying everything. As it moves it will slow down and become less distractive, so after some finite distance it will just stop. Beyond this distance (this is, beyond a sphere with the time machine in the center), the universe will remain as it was. Inside this sphere however nothing will exist - a singularity will be formed.
From this is should be obvious that this theory also doesn't allow for time travel - because it can be used only as a weapon, and it is not possible to return or to do anything.

Conclusion:
Time travel is not possible, and therefore faster than light communication is also impossible.
But - not always. In the next post about this topic, I will write about some very specific situations in which faster than light travel and time travel are possible.

Storing books online

I just discovered that Google Docs recently included support for pdf files. I don't know when it happened exactly - I discovered it only an hour ago. I am now uploading some of the books I have in pdf format to it. Unfortunately the maximum file size is only 10 mb. I have books that are larger, and I would like to be able to store them there.
I want to collect a collection of mathematical books and to store them on the net - I tried to do this on my computer, but the collection was lost after the computer failed.
I am also going to use it to store lecture notes and exercises - I can get those in pdf format from the Hebrew University site.

I wanted to do this for a long time, but I failed to find a site with a nice gui and a good set of options. For now Google Docs seem much better than any other service I tried. When I will have enough books in there I will put a link to the collection on this blog (provided I will be able to find enough books with licenses that would allow to publish them in such way).

By the way, this is my 100 post.... And it only took 10 month.

Blog readability level

I found a site that checks blog readability level. As you can see on the image below, Math Pages readability level is High School:

Since I want this blog to be readable for as much people as possible, this is a good result. I did a similar test when I only started blogging and the result was "Advanced degree" - which was rather surprising.
However, some of the topics I mention on this blog need better knowledge of math that what you get in school (at least in Israel) so the result is not very accurate.

Why faster than light communication is impossible

Lets consider the following situation - You are 4 light years away from the Earth, on a distant planet X. While you there aliens land on that planet and capture you. They make you tell them from where you came, and tell you that they will go and destroy Earth. You manage to escape, and get to your spaceship. You cannot stop them, and they don't want to look for you - they just go to Earth.
Lets suppose that:
1. Their spaceship can fly at a speed lower than light, but faster than yours spaceship.
2. If the people on Earth are informed in advance about these aliens coming, they will likely be able to defend themselves.
3. You have a device that allows you to send message to Earth that will travel there faster than light.
4. You use the device and send the message in the exact same moment that the aliens leave that planet.

Now, after we have all the information, lets see what will happen.
From the view point of out story hero, he send the message and now goes home.
From the view point of Earth the message is received, and they prepare to fight the aliens.
But what the aliens see?

Lets denote the starting point (the planet X) and the starting time (sending of the signal) by (0,0) respectively, and the end point (Earth) and end time (signal received) by (x,t).
The speed of the alien spaceship is v<1 c="1)" style="text-align: center;">x'=

t'=

For point (0,0) we get (0,0), which is not surprising. For point (x,t) we get ( ,).
Firstly lets suppose that t=0. This means that the message went to Earth in zero time. We will get: (). The minus sign means that this event happened before t=o. Thus the signal was received before it was sent. This is clearly impossible - unless a time travel is involved.

Now, what happens if the message is not instant but is still faster than light? In this case t=x/(1+h) - time is distance divided by speed - where h is a positive number (remember I am working with c=1).
Lets look what is the condition for t'>0, for any v (we must find condition for any v because we don't know what is the speed of the alien spaceship):

>0

We can divide:

>0

And finally we get:

1>v+vh

Rearrange:

1-v>vh

If we will choose v=1-<1>

>(1-)h

If we will now take the limit when epsilon approaches zero, we will get that h=o. This means that it is not possible to send messages faster than light, because otherwise there are always be an observer for whom the order of the events changes - which means time travel.
Note that the way the signal is send is unimportant. The only thing I used is a spaceship which is totally unrelated to the way the device works.

Strictly speaking, this doesn't prove that it is impossible to comminicate faster than light - but the only way to do this involves time travel. Therefore it is logical to assume that it is simply impossible for anything to travel faster than light, including information.

Update: Read part two of this post - Time travel paradoxes.

Poland math Olympiads

I use Google Analytics to gather stats for Math Pages. Besides other information it provides, it also shows from which countries my visitors come. I was somewhat surprised today to see that Poland is on the fifth place on the list. India is on the six, but I know people who live there, so it is not surprising. Visits from Poland are a surprise however.

Because of this positive surprise, this post is about math Olympiads in Poland. I am not going to talk much about them, just to solve some of the problems I found on their (official?) site. I never participated in a math Olympiad (and likely will never participate in one in the future), so writing this post is an attempt to fix this partially.

First problem:

Let n ≥ 3 be a positive integer. Prove that the sum of the cubes of all natural numbers, coprime and less than n, is divisible by n.

Firstly lets test this for n=3, and for n=4:

1^3+2^3=9, 9/3=3

1^3+2^3+3^3=36 36/4=9

So far the theorem works. Unfortunately, it is not correct. It fails on n=10:

1^3+2^3+3^3+5^3+7^3=504 504/10=50.4

It is easy to see that the numbers 1,2,3,5,7 are coprimes - the only common divisor for any two of them is 1. The numbers not included 4,6,8,9 are not coprimes with the rest because:
4=2*2 not coprime with 2, 6=2*3 not coprime with 2, 8=2*4 not coprime with 2, 9=3*3 not coprime with 3.

Second problem:

Prove that the number:
is divisible by

Speaking about usefulness of large numbers.. It is not difficult to solve this one, but it requires writing a lot in Latex - so I am leaving this one for the reader. Clue - you need to prove a general statement and then the problem is solved as a special case.

Third Problem:

In a group of n ≥ 3 peoples each member of the group has an even number (perhaps zero) of acquaintances in the group. Prove that there exist three members of the group which have the same number of acquaintances in this group.
Remark: Assume that nobody includes himself into the set of his acquaintances and that A knows B if and only if B knows A.

This one is actually very easy. Lets see what is going on for n=3: Because the number of acquaintances must be even and less then 3, it can be only zero or two. Lets call each of the three members A(x).B(x),C(x), where x is the number of the member acquaintances. Firstly, if A(0) than it must follow that neither B or C know A. Therefore, because x can have only even values we get B(0), C(0). In the same way it is possible to show that if A(2) then B(2) and C(2).

Now lets prove the general case by induction. Lets suppose that we proved for n=k, k≥3. Thus in this group we have three members with equal number of acquaintances. If we will add one more member to the group there are two options - either he knows one of these three members or not. If he doesn't know anyone of them than there are still those three members with the same number of acquaintances that were before we added him. Now lets suppose that he knows one of these three members, lets call this member A. But then the acquaintances number of A will grow by exactly one and become odd. This is not allowed. Q.E.D.

One of my posts was scraped

A few days ago I wrote a post about Einstein connection to math. When I went to Technorati yesterday I saw that someone linked to this post. When I followed the link, I came to a blog that is clearly used for scraping. From Wikipedia: "A scraper site is a website that copies all of its content from other websites using web scraping. No part of a scraper site is original."

I am not putting a link to it in this post, if you want to visit it go to Technorati and look for blogs that link to Math Pages, or go to the post about Einstein- the link is on the bottom of the page (for now).

The blog in question didn't copy all of the post - just a small paragraph, a quote of Einstein (it was clearly copied by some program). The post there links back to Math Pages, but my name is not mentioned. Such blogs usually use auto generated names - I appear there as "snakeman11689".
I have nothing against getting links, but such blogs have bad reputation. From what I heard getting a lot of links from scrapper sites may hurt the my blog page rank.

For now only one of my posts appeared there, but it is probably not the end of the story. For some reason Blogger does not notify me about new backlinks so it is not simple to discover them.
I am thinking about contacting the owner of this blog and ask him not to scrap my blog. It would also be nice if he used my name on the post he scraped and not an auto generated one. What do you think I should do?

Update: Well, only an hour past after I published this post and I already need to update it.. I just found out that my post Learning Math was also scrapped. It is a different blog this time, but I think it the same person. I tried to use whois to find information about him, but not successfully. If I will manage to find anything interesting I will put it here.

Update 2: Correction - three of my posts were scraped. I noticed the third only now, but it was scraped about a week ago. By the way all of the three scraper blogs domains are of the form "something.net" - without even www at the beginning of the URL.

Update 3: I was told that from reading my post it is not clear if scraping is a good or bad thing. To clarify this point - it is a bad thing. Such blogs are set up to generate money, and they also function as link farms. But this is not why they are a problem. Search engines (especially Google) has a reputation for not liking spam, and this is how such blogs are viewed but search engines. It takes some time for them to get classified as spam, but when it happens it means problems to those they took content from. This is because when they take even a short paragraph from your post, they also leave a backtrack on your blog. For Google it looks like you are linking to them, and linking to sites that are classified as spam can have negative results (page rank reduction for example). If only a post or two was picked up by such site, I doubt there will be any effect. However, three of my posts were scraped during one week - and I doubt it will stop.
To clarify it a bit further - I am reacting to this in such a way mainly because I am currently really annoyed by spam. I get 1-2 spam emails every day to an email address I have to keep (got it from my ISP), one person regularly sends me chain letters and just yesterday I was contacted by some spammer who tried to sell me his "best method for getting money online".

Update 4: I found emails of the owners of the blogs in question - and I sent them an email asking to credit me as the author of the content they scraped and to not post anything taken from my blog again. One of the the email bounced back, and I got a message that another is still not delivered.

learning Math

This post is an answer to a question one of my readers asked using the Skribit suggestion widget.

We all have to study math at some point at our lives. Even if we fail to find a use for it in the future, we still have to learn it at school. At this post I want to present a few general tips for learning different aspects of math.

Photo by foundphotoslj

The math taught at school can be divided into two main subjects - algebra and geometry. In university the difference between the two becomes less obvious, yet most of math can be put under one of these subjects. I am using the terms algebra and geometry in a rather general way in this post - all involving drawing is geometry and the rest is algebra.

There is no "royal road in math" - all who study math face some difficulties. However there are methods that can be used to improve your result.
Math requires two qualities - good memory and the ability to use the formulas and ideas learned effectively. Not all of us have these qualities, but there are ways to acquire them.

Memory:
1. The first thing you need to memorize is the multiplication table. Remembering it well helps a lot (at least thats what my experience taught me). The easiest way to learn it is the following: Firstly memorize all the squares - preferably up to 13 or 14. Then memorize all the multiplications by 5 - it should be very simple to do. Finally memorize all the multiplications by two and three. Now you can multiply any two numbers. All you need to do for this is to use the closets pair whose multiplication you remember and then add to it. For example, to multiply 6*7 you would do the following:

5*7=35
35+7=42
42=6*7

I also would recommend to memorize 3*15, 3*7, 4*7 - for some reason these three are often encountered.

2. A simple example to a formula we need to memorize in school is the sum of geometric series. The geometric series is a series in which every number is the previous number multiplied by some number q. For example:

1,2,4,8,16,32,64

These are the first 7 numbers of a geometric progression that starts with one, for q=2. The formula for the sum of such a progression is:

A1 is the first number in the series, and n is the number of numbers in the progression. Obviously the formula doesn't work for q=1. To prove that this formula is correct you just need to multiply - I am not going to prove it here. It is very easy to remember the formula itself - but I personally found it difficult to remember that it is n and not n+1. To remember it you just need to do a simple calculation:

1+2+4+8=15

3. The example shows how to remember the derivatives of inverse functions. One way to do it is to remember a short formula, but it is not a nice one to remember, although it is possible to use the method in the second example for this. However there is a more user-friendly way. Lets for example find the derivative of arcsin. It goes the following way:

arcsinx=y

x=siny

1=dx/dx=cosy*y'

y'=1/cosy

y'=

y'=

And this is exactly the derivative.

4. It happens sometimes that you need to memorize a lot of theorems that are relatively simple (even obvious) but are hard to remember, because there are so many of them. The approach here is to remember the general idea of a theorem. Lets look on this theorem:

Theorem: If two polynomials p(x) and q(x) of degree n-1 and n difference numbers .... are such that p()=q() for all i between 1 and n, than p(x)=q(x).

The theorem is relatively simple - perhaps even too simple to be called an theorem. But the exact form of the theorem might prove hard to remember, especially when it is just one out of 40 you need to remember. However it is simple to remember the general idea of the theorem - it basically states a condition for equality of two polynomials. If you remember this idea, it is easy to derive the rest. The only thing you need to remember is that polynomials of degree n-1 form an n dimensional space - so you need n numbers and the rest follows easily (for those who didn't study Linear Algebra this should be meaningless) . Another way is to remember the Wan-Der-Monde matrix.

5. Sometimes you just fail to remember if there is a theorem that says what you need to solve the problem. In this case it is sometimes useful to see if the claim you need seems obvious and general enough. If so it is safe (to some degree) to assume that there is a theorem and then formulate it. This might work, but it is better not to use this during a test - unless you can prove the theorem during the test itself.

6. For geometry, an analytical approach is often useful. This is, it is a good idea to use tools of analytical geometry and vectors to check if you remember a theorem correctly. The example I thought of is rather trivial, however it shows the point. We all know the Pythagoras theorem - for a right triangle with sides x,y,z the following equality holds:

x^2+y^2=z^2

Now lets write the same equality using vectors. For any triangle:

v+u=w

Lets suppose that v=(x,0) u=(0,y) and w stand for the third side. It follows than that w=(x,y). Lets calculate w^2:

w^2=z^2==x^2+y^2

And this is exactly what we wanted to show.

In these examples you saw the following methods:
1. Memorize some parts, and then use simple operations to calculate the rest.
2. Use simple example, for which you can easily calculate the result, in order to remember the more general case.
3. Learn how to get to the result by hand. The last method is especially useful for physics - you often have a lot of formulas that can be easily derived by hand.
4. Go from general idea to concrete theorems using logic.
5. Make an educated guess.
6. Use different methods and approaches to derive your conclusion, and to check it.

Also. don't forget the old tried method of writing the formulas/theorems on paper until you start seeing them in your dreams. It is a lot of hard work, but it works.


Using what you learned:
This is difficult to do. The only true method that works is to sit down and solve problems. It doesn't help much if you know all the formulas/theorems perfectly - you still need to practice to be able to solve a problem. One of my lectures in the University told the class once that while he knows the course material very well, he doubt that he would be able to solve out homework. He didn't practice solving problems for a long time, all he did was proving theorems. I doubt that he was serious, but this is something to consider.
What helps is positive thinking, and being really stubborn. Perhaps it will take an hour (or more) to solve a problem - you will solve it in the end, if you will try hard enough. And never feel that you wasted your time. Just today I fought for about an hour with a really simple question - for some reason I decided to try to solve it in a way in which it was impossible to solve. Finally I tried a different approach and I solved the problem in half a minute. I don't care that it took so much time - it was spent trying different approaches, which will hopefully contribute to better understanding of this particular subject.


Final thought:
Math is portrayed as a hard subject. There are also people who want to change this mindset and who claim that "math is your friend". In my opinion, both approaches are wrong.
I believe that math is interesting and fun. Sometimes it presents us with difficulties that are too great for us, but even by doing so it lets us try and attempt to solve this problem. However, it is also certainly not friendly to all people - it is a "friend" to only those who already like it. If you will decide that you hate math, don't expect it to be fun for you. Look at this little story from Clientcopia, for example.