数式ミステリー㉛『数学者のメモ』Page4

「0,99999…=1」

Hey, you.
You think 0.99999… is a little smaller than 1.
But that is not true.
Let me show you that.
If 0.99999… and 1 are equal, then 1−0.99999… should become 0.
Let’s check it.

1－0.9＝0.1
1－0.99＝0.01
1－0.999＝0.001
1－0.9999＝0.0001

It will become 0 someday.
Let’s go on.

1－0.99999＝0.00001
1－0.999999＝0.000001
1－0.9999999＝0.0000001
1－0.99999999＝0.00000001

Wait… when will it become 0?
Let’s keep going.

1－0.999999999＝0.000000001
1－0.9999999999＝0.0000000001
1－0.99999999999＝0.00000000001
1－0.999999999999＝0.000000000001

It still doesn’t become 0.
Let’s keep going more.

1－0.9999999999999＝0.0000000000001
1－0.99999999999999＝0.00000000000001
1－0.999999999999999＝0.000000000000001
1－0.9999999999999999＝0.0000000000000001
1－0.99999999999999999＝0.00000000000000001
1－0.999999999999999999＝0.000000000000000001
$\hspace{9pt}\vdots$
1－0.9999999999999999999…9＝0.0000000000000000000…1

Huh? It’s still not 0?
That is strange, very strange.
Are 0.99999… and 1 really equal?

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