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  "body": "\n## Statement\n\nLet $x$ and $y$ be real numbers with $x \u003c y$.\nThen there is an [[irrational number|080T2TXVQV]] $z$ with $x \u003c z \u003c y$.\n\n## Remarks\n\nProof: by the [[density of the rationals|0STX0WRDNG]], there is a rational $q$ with $x / \\sqrt{2} \u003c q \u003c y / \\sqrt{2}$ and $q \\neq 0$. Then $z = q \\sqrt{2}$ is irrational and $x \u003c z \u003c y$.\nThus both $\\mathbb{Q}$ and $\\mathbb{R} \\setminus \\mathbb{Q}$ are dense in $\\mathbb{R}$.\n\n## Explanations\n\n### L1 human\n\nBetween any two different real numbers, there is also a number that is not a fraction.\nSo fractions and non-fractions are mixed together everywhere on the number line.\n",
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