Is this an injection from $\mathbb{R}_+$ to $(0,1)$? [closed]

Question

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 $(0,1)$?

Answer 1

Your proposal is rather unclear, but per your comments it does not give a function with codomain $\mathbb{R}$ at all; this is because things like

[a sequence] of the form 0.0000000000…33333333… where [the] amount of zeroes preceding the first 3 is some infinite number

is not actually (a valid representation of) a real number in the first place.

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However, you may be interested to know that something similar to your idea does actually work! Specifically, letting $d(\alpha)$ be the number of digits in the integer part of $\alpha$ (so e.g. $d(\pi)=1$ and $d(12.3)=2$), the function $$g:\alpha\mapsto \alpha\cdot 10^{-2d(\alpha)}$$ gives an injection from $\mathbb{R}_{>1}$ to $(0,1)$. Some example values include:

• $g(31.234)=0.0031234$.

• $g(3.3333...)=g({10\over 3})=0.0333333...={1\over 30}$.

• $g(103)=0.000103$.

The injectivity of $g$ comes from the fact that we can "read off" of $g(\alpha)$ the size of the integer part of $\alpha$, by looking at the number of zeroes in the decimal expansion (a bit of care is needed here around numbers with two decimal expansions, so I'm leaving the detailed proof of injectivity as an exercise).

Note that there's a crucial asymmetry here: the "left-of-decimal-point part" of a real number is required to be finite. This is why the above actually results in something well-defined, while your proposal runs into trouble with things like $1\over 3$.