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  "body": "\n## Definition\n\nNonstandard analysis is the part of mathematics that develops analysis with the following tools.\n\n1. A proper extension ${}^*\\R$ of the [[real numbers|08J2M1JYAG]], such as the field of [[hyperreal numbers|421QMTFG9T]], that contains nonzero infinitesimals.\n2. For each set, function and relation on $\\R$, an extension of it to ${}^*\\R$.\n3. The [[transfer principle|3RT74FQYG1]], under which a first-order statement about $\\R$ holds if and only if the corresponding statement about ${}^*\\R$ holds.\n\nNotions of analysis are then expressed with infinitesimals and the [[standard part function|1ZBY4W4E81]].\n\n## Remarks\n\nExample: in nonstandard analysis, a function $f \\colon \\R \\to \\R$ has [[derivative|0V1EJCT8G6]] $L$ at $x$ if $\\operatorname{st}\\bigl(({}^*f(x + \\varepsilon) - f(x))/\\varepsilon\\bigr) = L$ for every nonzero infinitesimal $\\varepsilon$, where ${}^*f$ is the extension of $f$ to ${}^*\\R$.\nFor $f(x) = x^2$ the quotient is $2x + \\varepsilon$, whose standard part is $2x$.\n\nNon-example: smooth infinitesimal analysis also uses infinitesimals, but they are nilpotent and its logic is intuitionistic.\nIt does not use an extension of $\\R$ with transfer, so it is not nonstandard analysis.\n\nThe theorems of nonstandard analysis about standard objects can also be proved by ordinary methods, but the nonstandard proofs are often shorter.\n",
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