 # How to Find Rational Points Like Your Job Depends on It

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2021-07-22 15:30:16

You’re sitting at the end of a long conference table, interviewing for your dream job. You’ve made it this far, but there’s just one more question you have to answer.

You might think this only happens in story problems, but it happened to me. “Rational points” are points in the plane whose coordinates are all rational numbers. For example, $latex \left(\frac{12}{5},-\frac{2}{3}\right)$, $latex \left(3,\frac{1}{2}\right)$ and  (11, 4) are rational points, but (4, $latex\sqrt{2}$) and (π, -1) aren’t, since $latex\sqrt{2}$ and π are irrational. Rational points are important to number theorists and cryptographers, and they even lie at the heart of one of the most famous mathematical theorems of all time. But the question before me had to do with a line that passes through the origin, which means it has at least one rational point, namely (0, 0). Could it avoid passing through another? I didn’t know the answer right away, so I had to think about it.

At first it seems like the answer must be no. Around any point in the coordinate plane there are infinitely many rational points close by. With rational points so densely packed, it seems impossible that a line could avoid them all.