MacTutor
F W Levi teaches algebra
The following is taken from F W Levi's Algebra textbook. It was also published in The Mathematics Student 10 (3) (1942), 129-134. We give a version below.
A student/tutor dialogue
Student. When I started reading Algebra, you advised me to study carefully the systems of linear equations. So I did. I further read general Algebra and continued fractions. At first it was hard work, but later on, I was quite successful.
Tutor. Alright, but I do not think you have met me for this. You are looking rather despondent.
Student. Indeed, I am again in the wilderness.
Tutor. Why, is there some difficulty in the book which you want me to explain?
Student. It is not for that, but the whole subject became problematical to me again.
Tutor. I wonder how.
Student. Yesterday an engineer asked me to solve a certain algebraic equation. I replied that, in consequence of the fundamental theorem of General Algebra, there exist roots in a suitable extension of the field of the coefficients, and that from the fundamental theorem of classical Algebra, this extension can be chosen as the field of the complex numbers. Thus there exist complex roots of the equation and some of them might be real.
Tutor. Was the engineer satisfied by your reply?
Student. Not at all; he said that I seemed to be a great philosopher, and that I had missed the point completely. He was interested in real roots only. And he had no doubt about their existence. He has found out that the force (expressed in kilogrammes) acting on a certain part of an engine was bound to satisfy that equation. He was asking me to compute that force and nothing else.
Tutor. And you could not. The polynomial was too complicated.
Student. It looked very simple. It was something like this . From Eisenstein's theorem it is irreducible and therefore its real roots must be irrational. This I told the engineer.
Tutor. Perhaps the good man did not know anything about irrationality.
Student. He did! But he was not at all interested in my statement. He said, "I do not want to have an infinity of decimals, even if you can provide me with them. Compute the kilogrammes; I leave the grams etc. to you." Now, for any positive the polynomial takes positive values only. So, I told him that the real roots of the polynomial must be negative.
Tutor. And, was this statement of any use to the engineer?
Student. No, he knew already that the force was directed to the negative side; and then he said. 'The direction of the force is not very interesting to me, as there is little difference whether the material is exposed to stress or to pressure. If you give me a solution with 30% of error and a wrong sign, I could make some use of it. But your philosophical talk is worth nothing." He was quite rude eventually.
Tutor. And you?
Student. I am bewildered. After having read about 200 pages of the book, I am still unable to solve a very simple algebraic problem, not even if 30% of error and a wrong sign are admitted. Though I got very interested in Algebra, the Engineer's argument has impressed me. I am afraid that all my hard work has seen spent uselessly.
Tutor. I rather think, you have stopped reading at the wrong place. If you continue, you will be able to provide your friend with a solution, with considerably less than 30% of error.
Student. I had already a glance on the next chapter but I do not see any connection between its content and the preceding parts of the book (e.g.) general Algebra; and then, there is another thing that strikes me, every solution is given only approximately. I should like to know the solutions correctly. If for a particular application, a few decimal places only are requested, then I may neglect the higher terms of the correct result. But as a student of Pure Mathematics, I must know at first the proper solution, before admitting some error for the sake of abbreviation.
Tutor. How do you want to represent the solution if it happens to be an irrational number?
Student. There are many ways of expressing irrational numbers. For instance √2 is irrational. It cannot be expressed as a ratio of 2 integers, but nevertheless, √2 is a number. Everybody knows what is √2.
Tutor. Suppose that I do not know and you try to explain.
Student. √2 is the positive number whose square is equal to 2.
Tutor. Well, I take it for granted that one and only one such positive number exists. Let be positive and . Then = √2, or √2 is the positive root of . I think this statement is completely equivalent to yours.
Student. It is.
Tutor. You seem to be satisfied with this manner of expressing irrational numbers.
Student. Of course, I am. If I could express the roots of every algebraic equation in this way, there would be nothing to complain of.
Tutor. My point is that in this case, the roots are expressed by a tautology.
Student. I cannot follow you.
Tutor. Listen, which are the roots of the polynomial ?
Student. √2 and -√2.
Tutor. Whereby √2 is nothing else than a symbol for the positive root of . Besides the statement, that has two real roots and that their sum is equal to zero, your solution of the problem to find the roots of is a mere tautology. Your conclusion goes like this: "Who is Amal?" "The Brother of Bimal "-"And who is Bimal?" "Amals' brother." That means only that there exist two brothers Amal and Bimal, but it does not explain who is Amal.
Student. But √2 is a well-known number; Mathematicians have got used to it and they calculate with √2 as they do with 23 or 1: 7. For me, there is no problem about √2.
Tutor. Is it for the symbol √, that you hold this opinion?
Student. √ as a symbol is a mere convention, the mathematicians could use any other notation instead of it, but I do not see any reason why symbols familiar to everybody should be replaced by new ones.
Tutor. I fully agree with you, but for the sake of our conversation let us denote the real roots of a polynomial , as far as they exist, by in their order of magnitude starting with the greatest root.
Then your explanation of √2 simply means that .
Student. Now you want me to admit that for every which has a real root, the symbol must be considered as a solution of the equation . You propose that there is no higher justification in considering √2 as the given number than e.g. . But there is a huge difference between these two cases.
Tutor. How is that?
Student. We know more about √2 than that it is positive and that its square is equal to two.
Tutor. What do you know about √2?
Student. √2 = 1.4142...
Tutor. 1.4142 is a rational number, whereas √2 is irrational.
Student. Certainly, it is an infinite decimal fraction, but they have computed 200 decimals or even more of them. You cannot deny that √2 is very well known.
Tutor. There still remains a certain error.
Student. But a negligible one.
Tutor. That depends on the purpose of the calculation. I was told that certain students of Pure Mathematics must know the proper solution before admitting some error for the sake of abbreviation, was it not so?
Stu, But √2 is uniquely determined as the only positive root of .
Tutor. Yes there exists one and only one such root. This is a statement on existence and on uniqueness but nothing more than that. I think we have agreed already about this item. On the other hand, I admit that we know more than that about √2. For instance approximately equal to 1.4142. Or to put it more clearly, √2 lies between 1.4142 and 1.4143. One can find out easily small intervals where √2 is situated. There is no limit to the improvement of the approximation, and the diminution of the error. This error is not a kind of mistake which is the result of a negligent treatment. On the contrary it is an essential part of the solution of the problem. One cannot determine irrational numbers otherwise than approximately. This fact is concealed by symbols we are using. Numbers represented by them are uniquely determined in the sense that there exists one and only such number but one cannot determine the place of an irrational number on the real axis otherwise than approximately.
Student. Thus a formula like is only a recipe how to determine an irrational number approximately.
Tutor. The formula denotes the greatest root of and it shows of course a recipe how to compute that number approximately. A mental calculation furnishes 3 as a first approximation which would satisfy your friend completely.
Student. Suppose one could represent every root of a polynomial by the help of similar symbols. This would furnish a recipe to determine every root approximately.
Tutor. As a matter of fact, not every root is representable in that manner, and even if it is, one prefers a different method sometimes.
Student. My impression is that those methods have no connection with General Algebra. Theory of Approximation and General Algebra apparently belong to different branches of Mathematics if not to two different sciences.
Tutor. They are complementary to each other. You already mentioned the two fundamental theorems which state the existence of roots in certain fields. They must be supplemented by investigations about where the roots are situated in the field. The methods of investigation must tally with the structure of the particular field under consideration; they cannot be of a general nature. The real numbers are ordered linearly, whereas the complex numbers correspond to the points of a plane. Hence one subdivides the real axis into intervals to determine the real numbers and similarly the plane is subdivided into certain domains (e.g. rectangular or circular ones] to locate complex numbers, Both ways lead to an approximate determination of number (e.g.) roots of a polynomial.
Student. As the roots of the polynomial are uniquely determined by the numbers , there must exist functions which show the distribution of roots in the complex plane. One should investigate these functions; this would be a worthy continuation of General Algebra.
Tutor. If you take the word function in the most general sense, there exist indeed such functions e.g. the set of the roots itself is one. The problem is how to represent these functions; you should not expect that all of them are polynomials in . Every polynomial can be represented by a point of an -dimensional space. There are theorems stating that if certain inequalities in hold i.e., if is situated in a particular domain of the -dimensional space, then roots are distributed in the complex plane in a particular manner. These investigations are very interesting, but at the present time, the approximation of the roots is based more on the methods of calculation than on these theorems. In many cases, the theorems seem to be the result of the practice of calculation. For this reason, the author has started from Horner's scheme which gives the clue to the whole theory. I advise you to work out many numerical examples, it will help you to understand the theoretical portion!
Last Updated July 2026