Rational And Irrational Points

In the Euclidean plane, the points which have rational x and y coordinates are rational. For example P(0,1) or P(-4,1/3) are rational points, by definition. P(sqrt(2),5) isn’t, since the x coordinate is an irrational number.

You probably know, that every circle on this plane covers an infinite number of rational points.

What you probably don’t know, is that most lines in this plane, don’t intersect with a single rational point. Most lines on the plane go exclusively through irrational points!


2 thoughts on “Rational And Irrational Points

    • Every line goes mostly through irrational points and sometimes through countably many rational points, too. For example, y=x goes through many (aleph zero) rational points. Still aleph one irrational points.

      When you think about it, that rational points are dense everywhere in the plane, what means that there are infinite number of them inside every circle, no matter how small it is … those lines are pretty damn good at avoiding all of rational points, all the time. Almost all of them.

      And nearly every plane in the space also. It has no rational points at all.

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