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  "body": "\n## Definition\n\nLet ${}^*\\R$ be a field of [[hyperreal numbers|421QMTFG9T]] containing the [[real numbers|08J2M1JYAG]].\nA hyperreal $x$ is *finite* if $\\abs{x} \u003c r$ for some real $r$.\nFor each finite $x$ there is exactly one real number $r$ such that $x - r$ is infinitesimal.\nThis $r$ is the *standard part* of $x$, written $\\operatorname{st}(x)$.\nThe *standard part function* is the map $\\operatorname{st}$ from the set of finite hyperreals to $\\R$.\n\n## Remarks\n\nExistence: $\\operatorname{st}(x)$ is the [[supremum|0R3H5PANRQ]] of the set $\\set{q \\in \\R : q \u003c x}$, which is nonempty and bounded above when $x$ is finite.\nUniqueness: if $x - r$ and $x - r'$ are infinitesimal for reals $r$ and $r'$, then $r - r'$ is a real infinitesimal, so $r = r'$.\n\nExample: for a positive infinitesimal $\\varepsilon$, $\\operatorname{st}(3 + \\varepsilon) = 3$, since $(3 + \\varepsilon) - 3 = \\varepsilon$ is infinitesimal.\n\nNon-example: $H = 1/\\varepsilon$ is not finite, and for every real $r$ the difference $H - r$ is larger than $1$, so it is not infinitesimal and $H$ has no standard part.\n\nThe finite hyperreals form a subring of ${}^*\\R$, and $\\operatorname{st}$ is a [[ring homomorphism|0VTH8KPCG2]] onto $\\R$ whose kernel is the set of infinitesimals.\nIt is also monotone: $x \\le y$ implies $\\operatorname{st}(x) \\le \\operatorname{st}(y)$.\n",
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