Question

Can you prove `0.bar9 = 1?`

Answer 1

Yes, `0.999999…=1`

But note that the above holds true only if the `9`'s are recurring infinitely (which cannot be well-defined with only basic algebra concepts)

Explanation:

Here is one non-rigorous proof.

Consider `x` such that
`x=0.999999…`

Multiply both sides by `10`:
`10x=9.999999…`

Subtract the first expression from the second expression:
`10x-x=9.999999…-0.999999…`
`9x=9`
`x=1`

Thus, `x=0.999999…=1`.

To make it more rigorous, instead of saying `1=0.999999…` where `9` is recurring, we should say `0.999999…`, where the number of `9`'s tends to `oo`, converges to `1`. However, this would involve the concepts of limits and convergence, something beyond algebra.

Answer 2

Yes, `0.bar9=1`

Explanation:

Here is one of the proofs:

`1/3=0.bar3`

`1/3*color(blue)3=0.bar3*color(blue)3`

`3/3=0.bar9`

`1=0.bar9`

Since all of the algebraic manipulations was done correctly, the statement holds true.