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量について

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$\newcommand{\X}{\mathrm{X}}\newcommand{\kg}{\mathrm{kg}}\newcommand{\m}{\mathrm{m}}\newcommand{\s}{\mathrm{s}}\newcommand{\K}{\mathrm{K}}\newcommand{\J}{\mathrm{J}}\newcommand{\N}{\mathrm{N}}\newcommand{\A}{\mathrm{A}}\newcommand{\b}{\mathrm{b}}\newcommand{\B}{\mathrm{B}}\newcommand{\Ki}{\mathrm{Ki}}\newcommand{\Mi}{\mathrm{Mi}}\newcommand{\Gi}{\mathrm{Gi}}$

次元の除去

$4 \pi G = c = \varepsilon_0 = \mu_0 = k = h/2 \pi i = 1$とする．

$pq - qp = h/2\pi i$だけ見ても，$h = 2\pi i$とみなしたくなる．

これらの量を慣用の単位系で表して
$c = 299792458\ \m\ \s^{-1} = 1$
$h = 6.626070040 \times 10^{-34}\ \kg\ \m^2 \s^{-1} = 2 \pi i$
$G = 6.67408 \times 10^{-11}\ \kg^{-1} \m^3 \s^{-2} = 1/4\pi$
とすれば，$\mathbb{C}$で解いて
$\kg = 18.908488.125\ \X = \exp(18.908488 + 2\pi i \times 0.125)$
$\m = 78.844871.125\ \X$
$\s = 98.363472.125\ \X$
となる．ついでに
$k = 1.3806488 \times 10^{-23} \J\ \K^{-1} = 1$
$\mu_0 = 4\pi \times 10^{-7}\ \N\ \A^{-2} = 1$
とすれば，
$\K = -72.765617.125\ \X$
$\A = -56.280327 \ \X$
となる．また，
$\b = \log 2 = 0.693147 = -0.366513\ \X$
$\B = \log 2^8 = 8 \log 2 = 1.712929\ \X$
$\Ki\B = 2^{10}\ \B = 8.644400\ \X$
$\Mi\B = 2^{20}\ \B = 15.575872\ \X$
$\Gi\B = 2^{30}\ \B = 22.507344\ \X$
などとなる．

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