37: Mama, I have a question.
37: Why don't people like the "irrational numbers"?
The woman turns around, gazing her daughter with warm affection.
77: Mmm, do you feel the same about them, 37?
37: Mmm, maybe a little?
37 looks down and kicks away a pebble.
37: You see, Mama. I can count to 3, and to 17, but I can never count to something like 0.01001000100001, let alone use it in calculations.
37: The endless, non-repeating decimals would make a mess of the results, and I can't keep the equations as simple and elegant as I want.
77: But your favorite, the circle—the ratio of its circumference to diameter is also an irrational number, is it not?
37: π is special. I know how it was found and what it represents, so I trust it and use it in my calculations. The same with e, log2, and √2.
37: But not many irrational numbers are this convenient to work with.
37: Some have no patterns, no simplest forms, and no end. I can't work out their digits, write them out, or calculate them.
37: Not only are they impossible to pinpoint on the number axis, but there's an infinite amount of these irrational numbers!
77: Haha.
37: What's so funny, Mama?
77: You're a clever, silly goose, my dear.
77: You don't dislike them. You just don't understand them enough.
Leaning down, she picks up a soggy twig.
Her hand moves, sliding the tip of the twig across the sand.
Four equal perpendicular lines are drawn, with two more diagonal lines intersecting in the middle.
It is the shape of the familiar square.
77: Take our old friend √2 for instance.
77: The irrationality of this number can be proven through basic arithmetic, as it cannot be expressed as an irreducible fraction of integers a/b ...
77: A simple proof by contradiction is enough, with no knowledge of irrational numbers required.
77: The presence of √2 is prevalent in nature, and it is particularly noticeable along the diagonals of a square. This means—
37: Oh, I know!
37: This means the system built solely on the ratio of integers was flawed!
77: Yes. √2 is simple, elegant, and one of the greatest discoveries in mathematics. It showed the existence of infinite incommensurable numbers, with √2 being the most obvious one to find.
77: This was how the tower of old ideas crumbled, paving the way for revolutionary breakthroughs that catapulted mathematical analysis into uncharted frontiers.
77: We discovered a kingdom beyond our traditional methods—one that is immeasurable, incommensurable, and inexhaustible.
77: And the key to its gates is hidden in plain sight—in the diagonal of a square.
37: A kingdom of irrationality?
The girl listens intently. As always, she understands quickly.
37: I see now, Mama. We are not all-knowing beings. There will always be numbers beyond our awareness. Such is the nature of irrational numbers.
37: So, if we get to know the irrational numbers better, we can be friends with them, too!
37: But, you haven't told me why people on the island hate them.
77: ...
The image of her mother's face fractures.
But her smile is clear—gentle, quiet, and almost ... pitying.
37: Mama?
Sophia: ...!
Sophia: 37, you're awake!
The girl sits upright in bed, her eyes staring into the air in dismay.
37: ...
37: Sophia, how long was I out?
Sophia: You've been in a coma for a week. We ...
Her words are cut off as 37 jumps off the bed, rushing her way out.
Sophia: No, stop! You're not well yet!
37: There's no time!
37: I have to find 6. I have to tell them now—
37: Our "circle" has been broken.
