Is $9\ast (1/9)=\mathrm{0.99999999...}$ statement correct?

We can tell that

$\mathrm{0.11111111...}\cdot 9=\mathrm{0.999999999...}$

And that

$\frac{1}{9}=\mathrm{0.11111111111...}$

Therefore

$\frac{1}{9}\cdot 9=\mathrm{0.999999999...}$

However, we know that

$\frac{1}{9}\cdot 9=\frac{9}{9}=1$

Note: I'm taking in account that ... are the other rational digits left.

What am I making wrong? What is misunderstood?

Thanks for the help in clearing this problem.

We can tell that

$\mathrm{0.11111111...}\cdot 9=\mathrm{0.999999999...}$

And that

$\frac{1}{9}=\mathrm{0.11111111111...}$

Therefore

$\frac{1}{9}\cdot 9=\mathrm{0.999999999...}$

However, we know that

$\frac{1}{9}\cdot 9=\frac{9}{9}=1$

Note: I'm taking in account that ... are the other rational digits left.

What am I making wrong? What is misunderstood?

Thanks for the help in clearing this problem.