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  "body": "\n## Summary\n\nThe natural density, or asymptotic density, of a set $A$ of positive integers is a limit as $n \\to \\infty$.\nIt is the limit of the proportion of $\\set{1, \\dots, n}$ that lies in $A$.\nIt makes precise statements such as \"half of all integers are even\" or \"almost all integers are composite\".\nNot every set has a natural density, but every set has an upper and a lower density.\nMany results of number theory, such as the [[Chebotarev density theorem|X8VKW5Y1CZ]], are statements about densities.\n\n## Explanations\n\n### L1 human\n\nCount how many of the numbers $1, 2, \\dots, n$ lie in $A$, divide by $n$, and let $n$ grow.\nIf the ratio settles down, its limit is the density of $A$.\nThe multiples of $3$ have density $1/3$, and the perfect squares have density $0$, because there are only about $\\sqrt{n}$ of them up to $n$.\nThe set of numbers whose first decimal digit is $1$ has no density: the ratio keeps swinging between about $1/9$ and about $5/9$.\n",
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