Set of all real numbers with a '$2$' in their decimal expansion

Question

It is a common question in an elementary probability or combinatorics class to ask "How many integers between $1$ and $10^k$ have at least one '$2$' in their decimal expansion?" This result can then be generalized further, and analyzed as $k$ approaches infinity

However, I'm curious about something slightly different; rather than the set of all integers with a '$2$' in their decimal expansion, I want to consider the set of all real numbers with a '$2$' in their decimal expansion. More specifically, I have the following questions:

1) Which has greater cardinality: the set of all real numbers with a '$2$' in their decimal expansion (which we will denote by $S$), or the set of all real numbers without a '$2$' in their decimal expansion (ie $\mathbb{R} \setminus S$)?

2) In comparing the above sets, $S$ and $\mathbb{R} \setminus S$, by 'how much' is one set's size greater than the other? I realize this is a poorly phrased question, and am not really looking for a formal answer. Anything that gives some general intuition would be much appreciated.

For question 1, I am leaning towards thinking that the latter set, of all real numbers without a '$2$' in their decimal expansion, would be larger. To show this, I've tried using alternative bases and formulating some sort of mapping between the sets, to no avail. I also thought about simplifying the problem to just include numbers in the set $[0, 1]$, and then generalizing for $\mathbb{R}$. However, I haven't found anything by doing this either. I haven't yet attempted the second question, although I think the way I approach the first will largely dictate the answer here.

I'd appreciate any and all help with this problem. Thanks!

Answer 1

The cardinalities are the same, namely the same as the cardinality of $\mathbb R$ itself. We can form injections from $(0,1)$ into either $(0,1)\cap S$ or $(0,1)\setminus S$ by the rules

• $x\mapsto 0.2 + \frac1{10}x$, or
• Replace all digits 2 in the decimal expansion by 37 and all (original) digits 3 by 35.

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By Lebesgue measure, however, the difference is dramatic: $(0,1)\cap S$ has measure $1$ whereas $\mathbb R\setminus S$ has measure $0$. In other words, almost all real numbers (and this is a technical term) have a 2 somewhere in their decimal expansion.

We can see this by writing

$$ \mathbb (0,1)\setminus S = \bigcap_{n\ge 1} \{ x\in(0,1)\mid \text{the $n$th decimal of $x$ is not $2$}\} $$ This is the limit for $k\to\infty$ of $$ \bigcap_{n=1}^k \{ x\in(0,1)\mid \text{the $n$th decimal of $x$ is not $2$}\} $$ whose measure is $0.9^k$ -- each time we exclude 2 from another position, we lose $\frac1{10}$ of the numbers we have left. And $0.9^k\to 0$ for $k\to\infty$. So the measure of $(0,1)\setminus S$ is $0$, and similarly for every other unit interval.

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In fact, it is known that almost all real numbers are normal in base 10 (or any other base), and therefore have infinitely many 2s in their decimal expansion.