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也是发现了一个数学问题
0.9˙ □ 1\Large0.\dot9~\square ~10.9˙ □ 1
通过以下方法可以快速理解
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F1.分数法 ~~~~~~~~~~~~~~~~~~~~~~~~~ (感谢 NINJACQ 的投稿)
因为 13=0.3˙\cfrac{1}{3}=0.\dot331 =0.3˙,而 13×3=1\cfrac{1}{3}\times3=131 ×3=1,所以得出 0.3˙×3=0.9˙=1 (13×3)0.\dot3\times3=0.\dot9=1~\left(\cfrac{1}{3}\times 3\right)0.3˙×3=0.9˙=1 (31 ×3)
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F2.未知数法 ~~~~~~~~~~~~~~~~~~~~~~~~~ (感谢 皮皮虾_天之神_衣云滔(不加团) 的投稿)
设 x=0.9˙x=0.\dot9x=0.9˙,那么 10x=9.9˙10x=9.\dot910x=9.9˙。10x−x=9.9˙−0.9˙=910x-x=9.\dot9-0.\dot9=910x−x=9.9˙−0.9˙=9,9x=9,x=19x=9,x=19x=9,x=1,则 0.9˙=10.\dot9=10.9˙=1。
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F3.除法 ~~~~~~~~~~~~~~~~~~~~~~~~~ (感谢 WEMMBU 的投稿)
因为 19=0.1˙\cfrac{1}{9}=0.\dot191 =0.1˙,29=0.2˙\cfrac{2}{9}=0.\dot292 =0.2˙……,那么99=0.9˙=1\cfrac{9}{9}=0.\dot9=199 =0.9˙=1。
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F4.等比数列 ~~~~~~~~~~~~~~~~~~~~~~~~~ (感谢 原豌 的投稿)
0.9999…=0.9+0.09+0.009+0.0009+…0.9999…= 0.9+0.09+0.009+0.0009+…0.9999…=0.9+0.09+0.009+0.0009+…
这是等比数列,首项:0.90.90.9,公比:0.10.10.1
无穷等比数列求和公式:首项÷(1−÷(1-÷(1−公比)))
0.9÷(1−0.9)=10.9÷(1-0.9)=10.9÷(1−0.9)=1,所以0.999…=10.999…=10.999…=1。