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| (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.
