George Peacock: Treatise on Algebra (1830)

 

(1791–1858)

On a surface level, Peacock’s Treatise on Algebra seems a rather humdrum textbook on algebra, with no particularly original innovations to be found in it. Yet, what is most important in the treatise are not any purely mathematical results, but the more philosophical considerations on the nature of algebra and its relation to other parts of mathematics and especially arithmetics.

What Peacock is particularly interested in is the question whether algebra should be somehow based on arithmetics, which is based on concrete groups of objects, like seven apples. Peacock’s answer is a resounding no: arithmetics is far more constrained, allowing e.g. no negative numbers, since you cannot produce a group of, say, minus seven apples, although negative numbers are a common occurrence in algebra.

Not restricted by characteristics of arithmetical numbers, what is to determine what rules are to be assumed in algebra? Peacock makes the bold suggestion that these rules are just assumed: algebra concerns only symbols, and we could in principle choose any rules to govern our calculations in it.

Still, Peacock does not yet reach the modern notion of algebra, where we can have many different systems of calculation with different rules. Instead, he thinks that algebra should be especially the most general system of calculation, where all sorts of calculation are possible. Thus, he accepts not just calculations leading to negative numbers, but also roots of negative numbers, which we nowadays call imaginary numbers. Peacock goes even so far as to suggest that such seemingly nonsensical symbols like 0/0 could be used in a meaningful manner (although he mentions this possibility only in passing, he is referring to certain readings of the infinitesimal calculus).

Peacock still wants that algebra would be of practical use. Here, he suggests that the rules of algebra should at least correspond to the rules of arithmetics at least in those cases where the calculations and their results make arithmetical sense. Then again, he continues, results of algebra could also be used in other mathematical sciences, where some of them might be of relevance. Thus, negative numbers do make sense, for instance, when speaking of debt or of movement to an opposite direction. Even imaginary numbers can have concrete meaning, as describing movement perpendicular to a line.

Algebra becomes then, for Peacock, like a general tool for mathematical problem solving. When a concrete problem is given, it is turned into algebraic symbols: what quantities are known and what are unknown that are supposed to be determined in terms of the known quantities? After doing the algebraic manipulation of the symbols, one must still interpret the results. Often the context of the problem restricts this interpretation, and although algebra would lead to a number of possible results, only some of them might make sense in the context. Sometimes no result suggested by algebra would make sense, and then the problem itself will be impossible.

Évariste Galois: Dissertation on the conditions of solvability of equations by radicals (unpublished)

I’ve already discussed Galois as an example of a mathematical wunderkind. Like with many wunderkinds, his life was not long, as he was killed in a duel, when he was just twenty years old. Galois had not published much before his untimely death, and indeed, the work he is mostly remembered fpr nowadays, Mémoire sur les conditions de résolubilité des équations par radicaux, was rejected by the Academy of Sciences, due to being incomprehensible in its present form. Indeed, the evaluation of the Academy was not completely without grounds, since Galois often passes over required steps in his proofs, if he considers them to be sufficiently obvious. Still, it seems clear that at least part of the rejection was due to Galois’ method being so original and unseen.

The topic of Galois’ paper concerns the possibility of solving polynomial equations, that is, of equations of the form a1xn + a2xn-1 + …. an = 0. Methods for finding the possible solutions of the equation, using only the operations of addition, subtraction, multiplication, division, raising to power and taking a root of the coefficients, had been known for all cases where n is at most 4 (or as the mathematicians would say, where the degree of the equation was at most 4). In mathematical terms, such methods were known as solving the equation by radicals (radical being the symbol for taking the root). It had also been shown that already when the degree of the equation was 5, there are cases where the equation could not be solved by radicals. What Galois added was a criterion by which all equations solvable by radicals could be recognised.

Galois’ method of finding this criterion was preceded by the idea of Lagrange to study the solutions, or to put it in mathematical parlance, the roots of the equation, before they were actually known. Lagrange had especially considered what happens, when in combinations of these roots, using only addition, subtraction, multiplication and division, the places of these roots were changed - permuted, as they say in mathematics. Now, Lagrange had shown that the structure of the equations had something to do with these permutations, although he had not yet managed to find the complete tale.

Galois notes that if a polynomial equation with degree n has n different roots (the maximal number it could have), one could always form a combination of these roots, where every permutation of the roots would change the value of the combination. With a characteristic leap of thought, Galois just states that some combination of the form Aa + Bb + Cc + … would suffice (a, b, c… being roots of the polynomial and A, B, C … being appropriate integers), without proving this statement or even giving any method how to find such combination (the statement is true, but let’s just take it on faith and not go in full detail how to prove it). Furthermore, he points out that Aa + Bb + Cc + … can be assumed to be irreducible, that is, not expressible as a product of other polynomials.

Another point that Galois makes rather quickly is that all of the individual roots can then be expressed as roots of a polynomial in terms of V, which is the numerical result of the combination of the form above, where the root expressed is, say, the first in the combination. In other words, if V = Aa + Bb + Cc + …, then we can find another polynomial F, such that F(V, a) = 0. Even more, Galois points out, just by permuting the roots a, b, c … in a suitable fashion in the combination, one can find for each root (say, b) a numerical result of a suitable permutation of the combination (say, V’), which can then be used to express the root with the same polynomial F (e.g. F(V’,b) = 0). Note that it is quite arbitrary, what is the order of the roots in the first V: it is the permutation of them to make V’ out of V that is of great importance. Indeed, even the exact values V, V’, V’’ … corresponding to the roots a, b, c … are not as important, Galois notes, as the way in which the roots have to be permuted in order to construct the values V, V’, V’’ … . This group of permutations, or group of the equation, as Galois calls it, has the interesting characteristic that whenever some arbitrary function F(a, b, c …) has a value expressible in the by now familiar terms of addition, subtraction, multiplication and division of the roots, then this function will have the exact same value, if the places of the roots are changed with a permutation in this group.

Galois seems to have thought of his group of the equation as a sort of matrix formed of roots, where one line describes one permutation of the roots. Next, Galois makes the interesting suggestion that while the Vs determining the group of the equation are defined by integer coefficients A, B, C …, we could, as he says, adjoin a new quantity, beside integers, that could take the place of the coefficients (this quantity could be, for instance, an irrational root of some integer). With this addition or adjoinment in place, it might become possible to express the polynomial Aa + Bb + Cc + … as a product of further polynomials. If so happens, the group of the equation, expressed in terms of possible values of Aa + Bb + Cc + …, is, as Galois says, partitioned or decomposed into smaller groups. All of these smaller groups happen to be of the same size and also of the same form in the sense that one subgroup can be turned into another with a rulelike permutation.

Now, if just suitable quantities to be adjoined can be found, the procedure can in principle be continued further. In this case, we can move to polynomials of smaller and smaller degrees and eventually hit the rock bottom, when the corresponding group contains nothing but one row. If this can be done, the original equation will be solvable by radicals.

While the aim of the paper is to find the method for recognising polynomial equations solvable by radicals, Galois himself seems to become more and more interested in the supposed mere means for this goal, namely, the group of the equation and the permutations involved. Indeed, this is the road that the development of mathematics and especially algebra was to take: it was not anymore just a method for solving equations, but a more intricate study of such abstract structures like groups of permutations.

Évariste Galois: Demonstration of a theorem about periodic continued fractions (1828)

(1811 - 1832)

Mathematics is a field where wunderkinds are a real possibility. It is a field with abstract enough objects that can be grasped even with very little experience, even if dealing with them and their interrelations might require some innate or acquired skills.

A good example of such mathematical wunderkind is Galois, who had managed to make significant strides in algebra before his death, when he was just a few months over twenty year old. Indeed, his first paper, ”Démonstration d'un théorème sur les fractions continues périodiques”, was published just a few years before his death, making his life as a mathematician remarkably short.

The paper itself contains just one beautiful theorem, with a proof simple enough to understand, once you just see how Galois does it. First, some terminological explanations are in order. The topic of the article is continued fractions. To understand what this means, one can start from a sum of a whole number and a fraction of the form a + 1/b. Then let us suppose that the denominator of the fraction is not just another whole number, but another sum of a whole number and a fraction: the whole is then something like a + 1/(b+1/c). We can clearly continue doing the same again, adding fractions within fractions. Like in many other cases we have witnessed, we can think of this development continuing indefinitely into ever smaller and smaller fractions and progressing closer and closer toward some definite number.

Now, the simplest kind of a continued fraction to deal with is such where the numbers a, b, c… follow some clear rule. A specifically simple case is such, where after a while, the numbers a, b, c…continue again from some previous number of the series, repeating after that the same series of numbers over and over again. In such a case, the continued fraction is called periodic.

Before Galois, mathematician Lagrange had showed that periodic continued fractions had a special relationship with quadratic equation, Ax2 + Bx + C = 0. The root of such an equation - that is, a number, which makes the equation correct when put instead of x - can be expressed as a periodic continued fraction, and conversely, periodic continued fraction is a root of such an equation.

Galois continued Lagrange’s work and demonstrated another relation, involving especially immediately periodic continued fractions. If the number where the repetition begins is same as the first number a - that is, if there is no additional sequence of numbers at the beginning of the series - the continued fraction is in the terminology used by Galois called immediately periodic.

Let’s say we have an immediately periodic continued fraction, where the repetition of the numbers begins after the fourth, d. Then the fraction is of the form a + 1/(b + 1/(c + 1/(d + 1/(a + … If you look very carefully, you can see that the denominator following the first d repeats the whole series representing the fraction. In other words, if we designate the fraction with x, then we can say that x = a + 1/(b + 1/(c + 1/(d + 1/x))).

Now, it is a simple task requiring simple operations to dig out the x on the right side of the equation. I’ll just give the first few steps to explain what I mean: a - x = - 1/(b + 1/(c + 1/(d + 1/x))); 1/(a - x) = - (b + 1/(c + 1/(d + 1/x))); b + 1/(a - x) = - 1/(c + 1/(d + 1/x)) and so on. The end result of this manipulation of the equation is this: -1/(d + 1/(c + 1/(b + 1/(a - 1/x))) = x. In the place of the inner x, we can again start writing the series from the beginning: -1/(d + 1/(c + 1/(b + 1/(a + 1/d…

In other words, Galois has found by this simple method a new periodic continued fraction. This new fraction also works as a root of the same quadratic equation as the original periodic continued fraction. In addition, the numbers of the new fraction move in opposite direction from the original one. Furthermore, if we assume that all the numbers a, b, c and d are positive, the original fraction expresses a positive number larger than 1, while the new fraction is a negative number smaller than - 1.