Why do I get 0.098765432098765432... when I divide 8 by 81? I got this remarkable thing when

Charity Daniels 2022-05-23 Answered
Why do I get 0.098765432098765432... when I divide 8 by 81?
I got this remarkable thing when I divided 16 by 162, or, in a simplified version, 8 by 81. It's 0.098765432098765432 , or more commonly known as 0. 098765432 ¯ with all the one-digit numbers going backwards...except for 1. Yeah, it's missing the 1. One, how do I get this remarkable outcome and two, why is it missing the 1?
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Answers (2)

Leah Conley
Answered 2022-05-24 Author has 12 answers
Maybe you can also see it by the following calculations
x = 0. 098765432 ¯
10 9 x = 98765432. 098765432 ¯
( 10 9 1 ) x = 98765432
x = 8 / 81
We could actually find the number y = 0. 0987654321 ¯ by similar logic.
10 10 y = 987654321. 0987654321 ¯
( 10 10 1 ) y = 987654321
y = 109739369 1111111111
So maybe the reason why you're number is cool is that it has a easy fractional representation and a decimal representation with interesting properties. Any interesting repeating decimal representation is of course a fraction and can be calculated, but it may not have an easy fractional representation as shown with y.

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hughy46u
Answered 2022-05-25 Author has 2 answers
More generally, in base b (any integer > 1)
b 1 b 2 + b 2 b 3 + + 2 b b 1 = ( b 2 ) ( b b 1 1 ) b b 1 ( b 1 ) 2
so that
b 1 b 2 + b 2 b 3 + + 2 b b 1 + 0 b b + b 1 b b + 1 + b 2 b b + 3 + = j = 0 b j ( b 1 ) ( b 2 ) ( b b 1 1 ) b b 1 ( b 1 ) 2 = b b 1 b b 1 1 ( b 2 ) ( b b 1 1 ) b b 1 ( b 1 ) 2 = ( b 2 ) ( b 1 ) 2
What you have is the case b = 10

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