Why do divisions like 1/98 and 1/998 give us numbers continuously being multiplied by two each time in decimal form?

 Why do divisions like 1/98 and 1/998 give us numbers continuously being multiplied by two each time in decimal form?

Nice observation. We can write

198=11002=1100112100

Now for |x|<111x=1+x+x2+x3+

Apply that result to the expression above and we have

198=1100(1+(2100)+(2100)2+(2100)3+)

That gives the pattern you observe.

You can make a similar analysis for 1998.


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