14/06/2018

In mathematics, problems can be good.

Reproduction from the IMPA Science & Mathematics blog, from O Globo, coordinated by Claudio Landim.

Tatiana Roque, professor at the Institute of Mathematics at UFRJ

Having a problem is often seen as a bad thing. It means facing a deadlock, going through a difficult situation. In mathematics, it's the opposite. Having a problem is the first step towards successful research. Especially if it's a good problem, because there are bad problems too – those that lead nowhere. A good problem is one that bears fruit, that leads to the creation of propositions, concepts, and tools that will become useful for mathematical research.

One example is the invention of irrational numbers. Back in the 4th century BC, mathematicians had a problem: how can the diagonal of a square (and other analogous line segments) be measured?

To understand the issue, we need to remember that measuring is comparing. We take a segment of fixed size that we call a "unit of measurement." Then, we see how many times this segment fits inside another. The result is the measure of this other segment. But does this process always work? Let's take the case of the diagonal and one side of the square in the figure. Is it possible to subdivide the side in order to obtain a segment that fits a countable number of times inside the diagonal? Defying intuition, the answer is no. Geometric procedures already showed this even before the time of Euclid, the best-known Greek geometer, who wrote his "Elements" in the 3rd century BC.

This concerns the problem of incommensurability between the diagonal and the side of a square. The side and the diagonal are not commensurable, that is, they cannot be measured by each other: even subdividing the side of the square into equal parts at our pleasure, that is, into parts as small as we want, we will not find a segment that fits a countable number of times inside the diagonal. The consequence is the non-existence of a number to measure the diagonal of the square within the universe of numbers existing at the time, those that are obtained by counting and that we call today "natural numbers".

Contrary to numerous legends that claim the discovery of incommensurables was a scandal, historical evidence suggests that, at the time, geometers embraced the problem productively. Euclid's "Elements" testifies to this, as it contains sophisticated geometric procedures for working with geometric magnitudes as such, without associating them with numbers, that is, without measuring. For example, constructing, using a (non-graduated) ruler and compass, a square whose area is equal to that of a given rectangle. Or showing that, given a right triangle, the area of the square constructed on the hypotenuse is equal to the sum of the areas of the squares constructed on the legs. This last theorem is a purely geometric version of what we know as the Pythagorean theorem (which was hardly stated by Pythagoras).

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