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  "body": "\n## Definition\n\nConsider a [[floating-point|RTKRSC3KPH]] system with base $\\beta$, precision $p$ and minimal exponent $e_{\\min}$.\nThe smallest positive normalized number is $\\beta^{e_{\\min}}$.\nA *subnormal number* of the system is a real number\n$$x = \\pm\\, (0 . d_1 d_2 \\ldots d_{p-1})_\\beta \\times \\beta^{e_{\\min}},$$\nwhere the digits $d_i$ are integers in $\\set{0, \\ldots, \\beta - 1}$, not all zero.\nEquivalently, $x = \\pm k \\, \\beta^{e_{\\min} - p + 1}$ with $k$ an integer and $1 \\le k \\le \\beta^{p-1} - 1$.\n\n## Remarks\n\nExample: in IEEE binary64, $\\beta = 2$, $p = 53$ and $e_{\\min} = -1022$, so the subnormal numbers are $\\pm k \\, 2^{-1074}$ with $1 \\le k \\le 2^{52} - 1$; the smallest is $2^{-1074} \\approx 4.9 \\times 10^{-324}$.\nNon-example: $2^{-1022}$ is not subnormal in binary64; it is the smallest positive normalized number, with leading digit $1$.\nAll subnormal numbers are multiples of the same spacing $\\beta^{e_{\\min} - p + 1}$, which is also the spacing of the normalized numbers in $[\\beta^{e_{\\min}}, \\beta^{e_{\\min}+1})$.\nA subnormal number has fewer than $p$ significant digits, so the relative rounding error in the subnormal range can exceed the unit roundoff.\nOther names are *denormal* and *denormalized* numbers.\n",
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