Rational points on the unit circle and the Pythagorean triples.

Trigonometry translated from it Greek roots means "the measure of triangles", most notably right triangles. The most important aspect of right triangles is the Pythagorean Theorem, usually stated as a² + b² = c².

The unit circle on the xy plane is given by the formula  x² + y² = 1. There are infinitely many points on the circle and if you pick one truly at random, it will very likely be irrational, which means neither the x or the y be written as p/q, where p and q are whole numbers. The only way it is possible for a point on the unit circle to be rational is to have two fractions of the form a/c and b/c where a² + b² = c². For example,

(3/5)² + (4/5)² = (5/5)² = 1.

In Excel, I generated 26 rational points on the unit circle, the largest denominator being 1,105. As you can see, there are a lot of gaps, but you can make out the general shape of a quarter circle.

Here's what it looks like with 662 rational points plotted. You can still see some gaps, notably near the horizontal lines at 0.8 and 0.6. If all the rational points were plotted, and there are infinitely many, you would not be able to see any gaps with the naked eye or at any magnification. In math, we say that the set of rational points is dense on the unit circle.

But here's one of the goofy things about infinity. There are also infinitely many points on the unit circle where both values are irrational AND infinitely many points where x is rational and y irrational AND infinitely many points where x is irrational and y rational. Let me write those sets out.


Set #1: { the set of points on the unit circle where both x and y are irrational}
Set #2: { the set of points on the unit circle where both x and y are rational}
Set #3: { the set of points on the unit circle where x is rational and y irrational}
Set #4: { the set of points on the unit circle where x is irrational and y rational}

All four of those sets are dense on the circle, meaning that in any segment of the circle no matter how small, there will be elements from all four of the sets.

Here's something even weirder about infinity. Sets #2, #3 and #4 are all the same size of infinity, but Set #1 is a bigger size of infinity, bigger than the other three combined.

Tomorrow, we move from goofy facts about infinity to more practical math aimed at answering the question "Is climate change or global warming real?"