問い:次の式を計算し,結果を循環小数で表せ。
0.3636・・・×0.3222・・・
(左側は36を,右側は2を永遠に繰り返す。)
解:
0.3636・・・
=0.36+0.0036+0.000036・・・
=(36/100)/{1-(1/100)}
=(36/100)・(100/99)
=4/11
0.3222・・・
=0.3+0.02+0.002+0.0002+・・・
=(3/10)+[(2/100)/{1-(1/10)}]
=(3/10)+(2/100)・(10/9)
=(3/10)+(1/45)
=29/90
∴ 0.3636・・・×0.3222・・・
=(4/11)・(29/90)
=116/990
=(1/10)・(116/99)
=(1/10){1+(17/99)}
=(1/10)[1+{17/(10^2-1)}]
=(1/10)(1+[(17/10^2)/{1-(1/10^2)}])
=(1/10)(1+0.17+0.0017+0.00017・・・)
=0.1+0.017+0.00017+0.000017・・・
=0.1171717・・・
(17を永遠に繰り返す。)