Friday, February 29, 2008

Procrastination and internet

Sometimes the internet consumes an illogical amount of time.. It is very important for everyone to know how to use it in such a way that it will not bring you problems. This applies especially to those who do research or have homework to do. While the internet has a positive impact on our world, it is a tool that turns out to be to problematic for us sometimes, even if we use it correctly. About incorrect uses I don't even want to talk. The spam I got this week is more than enough for me..

I hope that you will do the right choice in such situation..

Please do remember that if you will sit down in front of your computer instead of doing your work you may soon lose both your computer and your work...

As of now I have two blogs and a site to manage... It is a not damaging my life, but it does take a lot of time.. I used to read productivity sites, until I understood that on this mater there are no good advices, except, maybe the advices you give to yourself. So I am not going to write a "How to stop the internet from ruining your life" guide... Just Google it if you want advice, there is a lot of sites dedicated to this topic.

I love strange situation (when they don't happen to me) and surprising results. In my opinion the penetration of the internet into our society is a good example of a very interesting phenomena. We usually don't think much about it. We just use the internet without thinking how life would look without it, and what changes it brings in our daily life.

Thursday, February 28, 2008

Unwanted comments

I was very surprised two days ago when I was notified that someone left comment on one of the post on math pages.. This at itself is a rare things to happen, but this is not what surprised me. The comment contained only three words and two links to some site. Classical spam. Since visiting on this site will almost surely result in getting a virus or a spyware program I am not putting a link to it. I deleted the comment of course, and turned on comment moderation - just in case. I thought that the story will end, but it didn't.

Yesterday I received another such comment. Slightly different words, but still two links to the same site. I just deleted it. I was sure that this will be the end to the story... Today I got another one. Deleted it and reported the one who left it - he was using a blogger account. Hope this will end the story...

I am surprised why I am getting this spam at all.. My blog is relatively new and has a low page rank. I see no reason for spammers to waste their time putting links here..

Update 2.3.08
I am still getting these comments... I really don't understand why. The person who leaves them uses blogger accounts, so the only result is that I report him as spammer. Well it is his choice..
The last comment was slightly longer than the previous - a whole 10 words - with a very obvious link to the same site.

Tuesday, February 26, 2008

Optical Illusions

One of the most complicated thing in existence is our mind, It is a tool designed to and capable to solve difficult problems. But it has its own limitations. One of them is in the way we see things. Very often what we see is only our mind interpretation of what actually is.
Also the construction of our eyes allows to make surprising images. Lets look on the following examples:

Perhaps it seems to look like an ordinary house, but it is clearly impossible to build such a structure.

This is a real photo. It is not possible to construct this structure, but it is possible to construct a box that when viewed in the right way looks like this. .

This one is another example of games with perspective.

And so is this one.

In the picture above all the lines are parallel. Surprised? This is an example of how our mind sees things... It is very easy to see something wrongly.

These illusions are all excellent examples of a failure of our senses. We are all guided at least partially by intuition and by what we see. and yet sometimes this is not enough... The very existence of such images is a proof that we must not go always by intuition but by knowledge. This is best seen in mathematics. In math you cannot make the smallest step if it is not based on logic and a logic that other people consider correct.

Friday, February 22, 2008

Where the movement occurs?

Continuing the discussion on Zeno paradoxes, lets examine the moving arrow paradox. In the case you missed the two previous posts about Zeno, you can read about another two of his paradox here and here.

In this paradox the main problem is to explain when things move. It seems a strange thing to ask, but it is hard to give any answer to this one. We don't usually see any problem with our perception of time and space. But what will happen if we will start to analyze the world around us from a logical position? The results are often surprising.

Photo by Odalaigh

The paradox arises from the following line of thought: Lets take an arrow that is flying, and look on it in an infinitesimal time. More precisely, lets look on it in a point in time. In such a short time "period" (not even a period in fact, but a mere point in time), the arrow clearly doesn't move at all. The time is made from such points. Now, since the arrow doesn't move in anyone of these points, how can it be that it moves at all? It must remain at rest - and thus time and motion are illusions.

The explanation to this is very simple. In order for this paradox to work time must be infinitely divisible. But this is not true. There exists a basic unit, called Planck time. I already wrote about this in the first post about Zeno paradoxes, so I am not going to repeat myself here. What is interesting about this answer however is that it shows that Zeno paradoxes touch the very structure of our world. As such they are valuable, because they show us were we live.

The most interesting property of these paradoxes is that they show how easily we can deduct amazing facts about our world from pure logic. Zeno lived a long time ago and yet managed to create paradoxes that were only fully resolved scientifically by showing that our world is discrete. This was done only very recently. I believe that this is a great lesson both about logic and about science.

Friday, February 15, 2008

How to square a circle

A few days ago, I wrote a post in which I introduced a simple and elegant method to "square" a circle using only compass and straightedge. The solution in that post gave a value for that was accurate to 7 decimal places.
One of the readers left a comment asking for a way to get a better approximation for , using only compass and straightedge.

While it is impossible to square a circle, because is transcendental, there exists a simple algorithm to get a construction for any desired approximation. In this post I will not describe a specific construction, but a general algorithm for finding a construction, given an error you want to get.

Lets look on the following circle:

This is a unit circle, with the point marked in red. Our goal is to get a construction that will give us a point B on the radius so that the distance between B and the red dot is as small as we want.
Point C is totally random - it can be any point on the circle or even outside of the circle. I put it only for convenience.

Step one : Connect C with O, the center of the circle.
Step two: The red dot is to the right of O, so mark a dot at the center of OD. Connect this new dot with C.
Step three: The red dot is still to the right of , so mark a dot at the center of D. Connect it with C.
Step four: Now the red dot is between and . Again mark the dot at the center of this segment. Connect it with C.
At this point the length of the line A is close to . If this is not good enough continue in the some way. With each step you will be getting closer and closer to the red dot.
You should get something like this:
It is very easy to see on this picture that each step brings you closer, and it happens relatively fast.
After you will find the sequence of steps that bring you close enough to the red dot, simply repeat it without marking the dot first.

Lastly, an exercise to the reader: prove that is impossible to square the circle this way.

Wednesday, February 13, 2008

Counting the points

In the previous post, I wrote about an ancient paradox. The author of this paradox, Zeno of Elea, presented 7 paradoxes. In this post I will focus on another of his paradoxes.

This paradox arises from the following situation. Suppose you need to go to a shop to buy some milk. You live very near to the shop so you decide to take a walk. Lets suppose the shop is 100 meters away.

Photo by lime*monkey

If you read my previous post, you probably already guessed that Zeno claimed that it is impossible for you to get to the shop...

Where is the problem? Lets look on it in the following way - in order to get to the shop you need firstly to pass half of the distance to the shop. But even before this you have to pass a quarter of the distance. And before this an eighth of the distance... And so on.
It turns out that in order to get to the shop you need to pass an infinite number of such mid points. Zeno reasoned that this is impossible, because you would need an infinite amount of time for this. Therefore you cannot move at all. No matter how small the distance you move we can divide it in the some way and get an infinite number of mid points again. Conclusion - movement is only an illusion.

There is a simple solution to this paradox. Firstly, if we will sum all the time intervals the sum will be a finite number. This is the accepted solution to the paradox.
But this is not a full solution. In order to sum the time intervals we need to know two things - distance and speed. We know the distance, it is given in the problem. But we don't know the speed. We can decide that the man in the problem has a speed of ten meters per minute. If we would do so, it is trivial to find how much time it would take him to get to the shop:

And for all the mid points we would get the following infinite series:

5+2.5+1.25+ .... =

As you can see the sum of all the intervals is indeed a finite number. But can we assume that speed exists at all? The whole point of the paradox is to show that motion is only an illusion. By assuming that something called speed exists we assume that the paradox is wrong from the beginning.

What is interesting in this paradox is that it doesn't describe our reality as we know it. There is a hidden assumption in the paradox that the distance can always be divided. But this is not true from a physical prospective. Our world is discrete - there is a basic unit of length called Planck length. Below this length distance is simply meaningless. So in our world we have only a finite number of points between us and the milk shop, so there is no problem to get there in a finite time.

Tuesday, February 12, 2008

Squaring the circle

This is a very ancient question. To put it in a more concrete way, is there a way using only straightedge and compasses (Euclid's toolkit), to construct a square with exactly the some area as a given circle in a finite number of steps?
There is a simple answer, but lets first look at the following method to square the circle:

1. Draw a horizontal line in the middle of the paper and mark a point R at the right end.
2. Open your compasses about an inch (roughly) and mark a point T on the line.
3. With unchanged compasses, mark a point O on the line so that distance OT=2*TR.
4. Draw a circle centered at O, and label point P at the left of the diameter line. This will be the circle POR whose area we are trying to equal with a square.
5. Construct point H halfway along the line PO, (you should know how to do that).
6. Raise a perpendicular at T, intersecting the circle at Q.
7. Draw a chord RS of the same length as QT.
8. Join PS and draw MO and NT both parallel to RS.
9. Below the horizontal diameter, draw a chord PK of length PK=PM.
10. Construct a vertical tangent PL of the same length as line segment MN.
11. Join RL, RK and KL.
12. Label point C on RK so that RC has length RH.
13. Construct CD parallel to LK, meeting RL at D.
14. Construct a square on base RD. It will have the same area as the circle! You can verify this by measuring the length RD as the radius RO times the square root of (~1,7724539). So the area of the square is *r2.

This is very convincing. It is wrong, but it is not obvious were the problem is. It turns out that using this method we created a square of area (355/113)*r2. This fraction is a representation of that is good to 8 digits. However it is only an approximation. If we would try to do this construction on a large enough circle, it would be possible to see that there is a mistake. But for this we would need a circle with a one mile radius.

It is however impossible to square the circle with the restrictions listed above. It is so because is transcendental. If such a construction was possible it would mean that is an algebraic number and this is not true.

The construction shown in this post is due to Srinivasa Ramanujan. He is considered to be the greatest mathematician ever by some people.

Monday, February 4, 2008

Zeno's paradoxes

Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic, and Bertrand Russell credited him with having laid the foundations of modern logic. He is best known for his paradoxes.

In this post I want to focus on one of his paradoxes. The paradox arises from the following situation:
A Greek athlete named Achilles is approached by a turtle. The turtle wants to compete against him in a 100 meter run.

Photo by lander2006
Since Achiles is the fastest runner, he decides to let the turtle start 20 meters ahead. The turtle speed is 1 m/s while Achiles can run at 10 m/s.
Now who do you think is going to win in the race? It seems that the logical answer is that Achiles will of course win. But is it indeed so?
Two seconds after the beginning of the race Achiles run 20 meters, bur he still needs to run 2 meters to reach the turtle. 1/5 of a second later, he runs 22 meters but the turtle is still ahead of him - by 1/5 of a meter. We can continue this way to infinity - Achiles will always need to pass some extra distance until he reaches the turtle.

It is clear that Achiles will never reach the turtle. No matter how much time will pass the turtle will always be slightly ahead of him. Moreover since there is no final step in this series, Achiles need to run for infinite time. This is not what happens in the real world, but what is the explanations to this?
One proposed solution is that as the distance between Achiles and the turtle approaches zero so it the time needed to pass this distance, and the sum of these time intervals is finite. Lets write it mathematically:
The intervals are:
2, 0.2, 0.02, 0.002 ......
This is a simple geometric series, and the sum is:

This is approximately 2.3 seconds. And this is of course a finite time.

However this is only a partial solution of the paradox. The reason why Zeno published this paradoxes was to defend the claims of his teacher who claimed that space, time and motion don't exist. In this paradox he tried to show that if we assume that space and time exist, what we get is that Achiles cannot outrun the turtle. And since this is not what happens in the word around us time and space don't exist - they are merely illusions.
This line of thought is probably exactly the opposite of the goal of this blog - as I see it, the goal of Math Pages is to show that math and the world around us are logical. Zeno however tries to claim that what we see is an illusion. Lets therefore take a look on another version of this paradox, a version that clearly shows the connection between this paradox and the claim that there is no time.

In this version, Achiles runs after the turtle and counts the instances in which he reached the previous position of the turtle. Thus when he reaches 20 m this is one, when he reaches 22 meters this is 2 and so on. Lets suppose he outruns the turtle. To what number he counted?
As in the previous version, there are infinitely many such points so it turns out that he had to count to infinity. But this is impossible.
As far as I know there is no mathematical answer to this problem. It doesn't help that the time intervals become smaller, because he still needs to accomplish an infinite task in a finite time period.

There are two possible solution to this paradox. Firstly it depends on the fact that time is divisible to infinitely small parts. This is wrong however. It turns out that there are exists a basic unit of time - it is called Planck time. It is approximately 5.4*10^-44 seconds.
The problem with this answer is that this time is the smallest unit of time because physics begin to "break" on shorter time intervals.
The second solution is the Holographic universe. It is a new and weird theory that claims that our universe is in fact one particle that is simply viewed from different points of view. If so, then space is a sort of illusion and the paradox proves it's point. I don't think that I believe in this theory, but since it offers a logical explanation and not merely claims that space is an illusion, it is acceptable for me. By acceptable I mean of course acceptable as a theory and not as a fact.

There are two other Zeno paradoxes about which I want to write, but they will have to wait for the next post :)

Saturday, February 2, 2008

Stopping learning fractions?!

I just read an excellent article on the Good Math, Bad Math blog. The author, Mark Chu-Carrol, is a PhD Computer Scientist, who works for Google as a Software Engineer. His blogging goal is, as he puts it: "Find the fun in good math, squash bad math and the folks who promote it".

It appear that there is a "innovative" idea floating around- to stop teaching fractions at school. This idea was proposed by an award-winning professor Dennis DeTurck. This idea is of course about as innovative as stopping to use electricity.

DeTurck does not want to abolish the teaching of fractions and long division altogether. He believes fractions are important for high-level mathematics and scientific research. But it could be that the study of fractions should be delayed until it can be understood, perhaps after a student learns calculus, he said. Long division has its uses, too, but maybe it doesn't need to be taught as intensely.

In his post Mark "squashed" both this idea, and also the position of those who oppose DeTurck, and claim that: "Math is hard. The idea that somehow we're going to make math just fun is just a dream."

In my opinion fractions, long division, calculation of square roots and by-hand multiplication of long numbers are the building blocks of our understanding. It is not possible to really understand the world of numbers without it. There are many examples of people who do totally unbelievable mistakes because of this. Not so long ago I read about a woman who when asked to pay 100 dollars for her purchase gave the seller four 20$. She was completely sure that 4*20=100 and even went to complain to the manager, when told that she needs to add 20$. At this part you are probably thinking: well that can happen to her but not to me. If it didn't happen to you yet it probably won't, but with such changes in education as DeTurck proposes it is more than likely to happen to your children.

There is also a very interesting claim, that was put both by DeTurck and by one of the readers of Mark blog - that fractions should be taught in college after the student already knows calculus. Perhaps we should also teach them how to read and write after they will graduate? It is also not very exciting to learn. Even better, I think only the professors should be able to know how to read and write. And for the rest of the population we can make computerized readers that will be able to read them what ever they need.

How much time is needed to go back to the stone age? It took a long time to get up, but it is always easier to go down than to go up.

Eventually we get what we are paying for. If we don't want to know math, we will not know it. The problem is, we are not gaining anything by not knowing math. If fractions will not be taught in school the children will not gain anything - except perhaps a headache when they will try to figure how to divide a pizza into three halves.

It is good that we have calculators but we shouldn't depend on them. In one of the comment to the article high school teacher says that he has students who come to him and don't know how to calculate 3+5 without a calculator. This sound weird, but there is no reason to think he is lying.
Even if we do know how to use a calculator, there is no guarantee that it will help us. About a month ago I heard a story about a second year physics student at the Hebrew university. He needed to calculate the wavelength of a wave, in an exam. The calculator told him that it was 10^40. This is impossible - the diameter of our galaxy is only about 10^20 meters. However he wrote it, confident in the calculator.

It may sound surprising to you but my calculator often "lies" me. I noticed that sometimes when I put in simple calculations with fractions, the result is not correct. It happens because the calculator doesn't know how to treat infinite decimal fractions, so when I do:

It might show that the result is not zero, but 10^-6 for example. I know why it happens, but obviously it would be insane to trust the calculator after this.

Friday, February 1, 2008

Explanation of a Geometry Paradox

Last month I wrote about an interesting geometry trick, that resulted in a rather strange picture. In the post I explained why it happened, but it wasn't completely clear to some of the readers. So I decided to explain it again in a more complete way.

The problem is the following picture:

In the previous post I just wrote that the slope was different. For me such explanation seemed complete so I just continued ot talk about importance of proofs. In this post I will try to explain what it means that the slope is different.
First of all, it should be obvious that the area of both triangles should be equal, because they are built from the some blocks. This means that the missing segment area is still in the picture, but in a different place. Lets look on the following illustration:

In this picture the D dot is moved slightly to create a change in the slope. It is easy to see that the second triangle (on the right) is identical to the first (on the left) but it has another triangle added to it. This is exactly was was done in the original picture. The tiles were rearranged in such a way that the line connecting the points A and C is no longer straight. If the distance that the dot D has moved is small enough we don't see this, and the triangles seem identical.
Lets calculate how much the dot needs to move in order to account for the missing segment area.
In our picture the size of the triangle is: AB=5, BC=13. Thus according to Pythagoras:

The missing area is a 1x1 square. If the area of the ADC triangle is also equal to 1, we get the following equation:

And therefore h (the height of the ADC triangle) is: h=1/7 of a square. I rounded the figures a bit but the result is close enough. The result seems large, but it is not. It is of course possible to see, but it is not immediately visible.

The only thing that remains, is to show that indeed in the original picture the slope is different, and that the rearrangement of the pieces creates an extra triangle on top of the original one. This is very simple to do. All you need is to download the image to your computer and open it in an image editing program that supports layers and transparency. Now, cut the bottom triangle and put it in a different layer. Place it above the top triangle and play with transparency. Or you can watch the video below:

The program used to edit the picture is Gimp. It is a free program, capable of replacing Photoshop. The only problem with it is that it is not very intuitive.

I hope this post answered the question. If not, let me know in the comments.