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Post details: Rational points on the unit circle

Rational points on the unit circle

Unit circle is a very simple object. It is just a set of all points in the plane which happen to be exactly one unit away from the center which you can think of as the usual origin in the coordinate plane. Like everything living in the coordinate plane, every single point of it corresponds to a pair of numbers, expressing its location with respect to the x and y axes. If you are of curious nature, as some mathematicians are, you may start wondering if there are any rational points on this circle? That means points whose both coordinates are common fractions.

Knowing how ubiquitous common fractions are, your gut instinct would advise that there must be tons of them. But if you start looking for examples you realize the answer is not as obvious as it seemed. Let us pick a common fraction for one coordinate - say x=1/2. Since any point on the circle satisfies the equation x^2+y^2=1 you can calculate the other one easily: y=Sqrt(3)/2. And there is the rub - square root of three is not a common fraction. So let's take another well known fractions for the first coordinate: x=1/3 or and x=2/3 and see if we get luckier. We won't. The corresponding y-coordinate turns out to contain those pesky square roots again. At this point you may think that situation is becoming hopeless and if there are any rational points on the circle at all, they are extremely rare. Indeed, unless you have a perfect square to begin with - and perfect squares are few and far between - taking a square root of a whole number happens to be profoundly irrational and thus not eligible to pose as a common fraction.

But overly pessimistic viewpoint is misleading as well. Despite the slim odds of hitting a perfect square, there are in fact infinitely many rational points on the unit circle. If you want to see that without breaking too much algebraic sweat, you can connect the point (-1,0) with any rational point on the y-axis (and of those there are plenty because there you do NOT have to worry about that other coordinate) and then see where the resulting line intersects the circle. It is a fun calculation and it leads to points like (3/5,4/5) or (5/13,12/13) or (20/29,21/29) which are clearly rational and (slightly less clearly) lying on the unit circle. In fact, if you paid attention in high school algebra classes, you may recognize them as sides of Pythagorean triangles. They are among the most amazing objects of all mathematics.

For lay people this is one the most puzzling paradoxes in elementary mathematics. On the one hand these points are rare and difficult to calculate, but on the other there are infinitely many of them and they fill the circle densely to boot. That means no matter how tiny arc of the circle you consider, there will always be infinitely many points on it whose coordinates are common fractions. In other words, no part of the circle is fraction free. I sometimes imagine myself as a tiny ant standing in the center of that circle and watching all those rational points on it as if they were stars on a night sky.

Now if you moved that little ant to one such point on the circle, it would see the coordinate cross from a point which is algebraically simple (a pair of fractions) and unlike any other. In a certain sense these points are like people. They see the Cartesian geometry of the circle from their very unique and rational view point. Just like every person sees the circumstances of their lives from a very special, and hopefully rational, vantage point.

2012 will be a tough year and many important decisions will have to be made. Let's hope they will stay rational - like those Pythagorean points on the circle. They may be hard to find, but hey - there are infinitely many to choose from.


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