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  "body": "\n## Summary\n\nA rational point of a variety defined over a field $K$ is a point whose coordinates all lie in $K$.\nOver $K = \\Q$ these are the rational solutions of the defining equations, so rational points are the geometric form of [[Diophantine equations|08B1RTW7VZ]].\nDeciding whether rational points exist, and describing them when they do, is the central problem of [[arithmetic geometry|824JAXQ5M6]].\n\n## Explanations\n\n### L1 human\n\nThe circle $x^2 + y^2 = 1$ has many points, but only some have both coordinates rational, such as $(3/5, 4/5)$.\nThese are its rational points over $\\Q$.\nEach one corresponds to a Pythagorean triple, here $3^2 + 4^2 = 5^2$.\nThe circle $x^2 + y^2 = 3$ has real points but no rational points at all.\nWhich curves and surfaces have rational points, and how many, is a deep question.\n",
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