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  "body": "\n## Summary\n\nQuantum calculus is calculus without limits.\nIt replaces the derivative by a difference quotient taken at a fixed scale.\nThe [[q-derivative|PMJT3FZE0Y]] compares $f(qx)$ with $f(x)$, and the $h$-derivative compares $f(x + h)$ with $f(x)$.\nThe matching antiderivatives are the [[Jackson integral|7EMPA6M8S7]] and finite sums.\nIt is the natural language for [[q-analogs|GFSDMJHHVA]] in combinatorics, for basic hypergeometric series and for [[q-difference equations|FQPYGXJJN0]].\nIt recovers ordinary calculus in the limit $q \\to 1$ or $h \\to 0$.\n\n## Explanations\n\n### L1 human\n\nIn ordinary calculus the slope of $f$ at $x$ is the limit of $(f(x + h) - f(x))/h$ as $h$ shrinks to zero.\nIn quantum calculus one stops before the limit and keeps a fixed step.\nWith the multiplicative step $x \\mapsto qx$, the function $x^n$ has $q$-derivative $[n]_q x^{n-1}$, where $[n]_q = 1 + q + \\cdots + q^{n-1}$.\nThis number $[n]_q$ is the $q$-analog of $n$, and it becomes $n$ when $q = 1$.\nMany formulas of calculus keep a $q$-version, which often counts objects with an extra weight.\n",
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