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  "body": "\n## Summary\n\nThe hyperreal numbers form an [[ordered field|0P3VCBDEE2]] ${}^*\\R$ that contains the [[real numbers|08J2M1JYAG]] together with infinitesimal and infinite numbers.\nThe usual construction is an [[ultrapower|AX9Q86GB8M]] of $\\R$: sequences of reals modulo agreement on a set in a nonprincipal [[ultrafilter|J64Y3JC0RE]] on $\\N$.\nBy [[Łoś's theorem|AGYBSTD7BG]], ${}^*\\R$ satisfies the same first-order statements as $\\R$, which makes it the basic number system of [[nonstandard analysis|WM62GKCBWN]].\nIt is not [[Archimedean|0XH79S06J3]]: an infinite hyperreal is larger than every natural number.\n\n## Explanations\n\n### L1 human\n\nThink of a hyperreal number as a sequence of real numbers, where two sequences count as equal if they agree \"almost everywhere\" in a precise sense.\nThe constant sequence $(3, 3, 3, \\dots)$ is the real number $3$.\nThe sequence $(1, 1/2, 1/3, \\dots)$ is positive but eventually below any positive real, so it is an infinitesimal.\nThe sequence $(1, 2, 3, \\dots)$ is larger than every real number, so it is infinite.\n",
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