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And the box “What Are Imaginary Numbers?” should have said that rational numbers include the integers, not that “rational numbers are the integers.”

Even such bonkers-looking numbers, however, together with all the rational numbers, make up only a tiny fraction of the real numbers, or numbers that can appear along a number line.

If the two new arrangements contained repeating patterns, the length of their translation vectors should have been related to each other—specifically, their ratio should have been a rational number.

The conjecture predicts that the corresponding field will be “the smallest field that you get by taking sums, products and rational number multiples of these roots,” Arthur says.

Even the rational numbers—the set of numbers that can be written as a/b for integers a and b—were countable.