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  "body": "\n## Statement\n\nLet $n$ be a natural number.\nThen $\\sqrt{n}$ is irrational if and only if $n$ is not a perfect square.\n\n## Remarks\n\nIf $n = m^2$, then $\\sqrt{n} = m$ is rational. The content is the other direction.\nA proof: if $\\sqrt{n} = a/b$, then $a^2 = n b^2$. By [[unique factorization|084HH1NDCB]], each prime has an even exponent in $a^2$ and in $b^2$, so also in $n$. Then $n$ is a square.\nFor example, $\\sqrt{2}$, $\\sqrt{3}$ and $\\sqrt{12}$ are irrational.\nThe same argument shows that the $k$-th root of $n$ is irrational unless $n$ is a $k$-th power.\n\n## Explanations\n\n### L1 human\n\nThe square root of a whole number is either a whole number or an irrational number.\nIt is never a fraction such as $\\frac{3}{2}$.\n",
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