2014-11-02

自然対数の底 e を暗算で計算してみた．
$1+1+\frac{1}2=2.5$
$\frac{1}6=0.1666\cdots$
$1+1+\frac{1}2+\frac{1}6=2.6666\cdots$
$\frac{1}{24}=\frac{0.1666\cdots}4=\frac{0.08333\cdots}2=0.041666\cdots$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}=2.708333\cdots$
$\frac{1}{120} =0.008333\cdots$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120}=2.716666\cdots$
$\frac{1}{720} =0.0125\times 0.111\cdots=0.013888\cdots$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}=2.7180555\cdots$
ここまでは厳密．

$\frac{1}{720\times 7} \sim \frac{1}{5000} =0.0002$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5000}=2.7182\ 555\cdots$
ここまでなら暗算でいける．

もうちょっとがんばると，
$\frac{1}{720\times 7} =\frac{1}{5040}=0.0002\times \frac{1}{1008} \sim 0.0002\times 0.99=0.0001\ 98$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}\sim 2.7182\ 53$
$\frac{1}{720\times 7\times 8} =\frac{1}{40000} =0.000025$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}+\frac{1}{5040\times 8}\sim 2.7182\ 78$
$\frac{1}{720\times 7\times 8\times 9} \sim \frac{1}{360000} \sim 0.000003$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}+\frac{1}{5040\times 8}}+\frac{1}{5040\times 8\times 9} \sim 2.7182\ 81$

もう1桁がんばると，
$\frac{1}{720\times 7} =\frac{1}{5040}=0.0002\times \frac{1}{1008} \sim 0.0002\times 0.992=0.000198\ 4$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}\sim 2.718253\ 9$
$\frac{1}{720\times 7\times 8} =\frac{1}{40000\times 1.008} \sim 0.000025\times 0.992=0.000024\ 8$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}+\frac{1}{5040\times 8}\sim 2.718278\ 7$
$\frac{1}{720\times 7\times 8\times 9} \sim \frac{0.0000248}9\sim 0.000002\ 7$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}+\frac{1}{5040\times 8}+ \frac{1}{720\times 7\times 8\times 9}\sim 2.718281\ 4$
$\frac{1}{720\times 7\times 8\times 9\times 10} \sim 0.000000\ 3$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}+\frac{1}{5040\times 8}+ \frac{1}{720\times 7\times 8\times 9}+\frac{1}{720\times 7\times 8\times 9\times 10}\sim 2.718281\ 7$

さらにがんばると，
$\frac{1}{720\times 7} =\frac{1}{5040}=0.0002\times \frac{1}{1008} \sim 0.0002\times 0.992\ 064=0.0001984\ 128$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}\sim 2.7182539\ 683$
$\frac{1}{720\times 7\times 8} =\frac{1}{40000\times 1.008} \sim 0.000025\times 0.992064=0.0000248\ 016$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}+\frac{1}{720\times 7\times 8}\sim 2.7182787\ 699$
$\frac{1}{720\times 7\times 8\times 9} \sim \frac{0.0000248}9\sim 0.0000027\ 555$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}+\frac{1}{720\times 7\times 8}+\frac{1}{720\times 7\times 8\times 9}\sim 2.7182815\ 254$
$\frac{1}{720\times 7\times 8\times 9\times 10} \sim 0.0000002\ 755$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}+\frac{1}{5040\times 8}+ \frac{1}{720\times 7\times 8\times 9}+\frac{1}{720\times 7\times 8\times 9\times 10}\sim 2.7182818\ 009$
$\frac{1}{720\times 7\times 8\times 9\times 10\times 11} \sim 0.0000000\ 255$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}+\frac{1}{5040\times 8}+ \frac{1}{720\times 7\times 8\times 9}+\frac{1}{720\times 7\times 8\times 9\times 10}+\frac{1}{720\times 7\times 8\times 9\times 10\times 11} \sim 2.7182818\ 264$
$\frac{1}{720\times 7\times 8\times 9\times 10\times 11\times 12} \sim 0.0000000\ 021$
$1+1+\frac{1}2+\frac{1}6+\frac{1}{24}+ \frac{1}{120} +\frac{1}{720}+\frac{1}{5040}+\frac{1}{5040\times 8}+ \frac{1}{720\times 7\times 8\times 9}+\frac{1}{720\times 7\times 8\times 9\times 10}+\frac{1}{720\times 7\times 8\times 9\times 10\times 11}+\frac{1}{720\times 7\times 8\times 9\times 10\times 11\times 12} \sim 2.7182818\ 285$

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e の暗算

2000乗して40になる数