Wednesday, June 25, 2008

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:


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:


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:

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:


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:


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:


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


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


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:


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:


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.


Abraham Akinin said...

typo: Adel should be Abel.
cool blog dude

Anatoly said...

Hello Abraham,
Thanks for noticing the typo. I corrected it and a few others.