I am wondering if $f: \mathbb{R}_+ \rightarrow (0,1)$ is an injection if $f$ just moves the decimal point to the left of each number an equal amount of times as how far the decimal point is from the right of the begining of the real number which has the decimal point featured to the furthest point right, adding a $0$ every time we move the decimal point left and there's nothing there: e.g. if we had $3.14, 314.1, 34567.2$ which $f$ was mapping from, $f$ would output $0.0000314, 0.00314,$ and $0.34567$ respectively.

I am unsure because I don't know whether there is a well defined number which constitutes the greatest number of spots to the right which the decimal point is featured in a real number. If this isn't an injection, is there a way to use a similar idea to define an injection from $\mathbb{R}_+$ to $(0,1)$?

I am wondering if $f: \mathbb{R}_+ \rightarrow (0,1)$ is an injection if $f$ just moves the decimal point to the left of each number an equal amount of times as how far the decimal point is from the right of the first digit of the real number which has the decimal point featured to the furthest point right from its first digit, adding a $0$ every time we move the decimal point left and there's nothing there: e.g. if we had $3.14, 314.1, 34567.2$ which $f$ was mapping from, $f$ would output $0.0000314, 0.00314,$ and $0.34567$ respectively.

I am unsure because I don't know whether there is a well defined number which constitutes the greatest number of spots to the right which the decimal point is featured in a real number. If this isn't an injection, is there a way to use a similar idea to define an injection from $\mathbb{R}_+$ to $(0,1)$?