If 0.99...=1 What about 0.89...=0.9?

371 Views Asked by At

I notice the general pattern is that ?.??999... equals to 0.??1 more than the repeating 9 part. Is it true?

1

There are 1 best solutions below

0
On

Consider $$x=a_0.a_1\cdots a_{n-1}a_n\bar 9$$ then $$ x=a_0.a_1\cdots a_{n-1}a_n+0.0\cdots 00\bar 9 $$ and $$ 0.0\cdots 00\bar9=0.\bar9\times0.0\cdots 01=0.0\cdots 01$$ hence $$ x=a_0.a_1\cdots a_{n-1}a_n+0.0\cdots 01.$$