3

Today, I was trying to prove Cantor set is uncountable and I completed it just a while ago.

So, I know that the end-points of each $A_n$ are elements of $C$ and those end-points are rational numbers. But since $C$ is uncountable, $C$ must contain uncountable numbers of irrational numbers. Then, is their a way to prove that, a specific irrational number (say $1/\sqrt2$ or $1/4\pi$) belongs to the set $C$ or not?

(Description of notation can be found in the link given above or here)

Lets say,

Prove or disprove that $1/\sqrt2\in$ Cantor set on $[0,1]$.

Can we do that? Or is their a way to solve such problem?

topology_001
  • 321

3 Answers3

1

No, it does not belong to the Cantor set. In base $3$, the expression of $\frac1{\sqrt2}$ is $0.21\ldots$ Because of that $1$ there, the answer is negative.

José Carlos Santos
  • 427,504
  • 3
    You need more digits, since it could happen that the number is $0.2111111\dots=0.22$. – lisyarus Nov 05 '17 at 13:13
  • 2
    @lisyarus Then $\frac1{\sqrt2}$ would be rational, which is not. – José Carlos Santos Nov 05 '17 at 13:15
  • 2
    Of course it isn't, but I feel that it should be treated somehow in the answer. Also, $0.21\dots_3$ is wrong, since $0.21\dots_3 \times 0.21\dots_3 = 0.1211\dots_3 > 0.(1)_3 = {1 \over 2}$. – lisyarus Nov 05 '17 at 13:20
1

The Cantor set can be describes in terms of ternary notation: a number $x \in [0; 1]$ belongs to the Cantor set if and only if it has a ternary representation consisting of $0$s and $2$s only. Since $\frac{1}{\sqrt 2} = 0.2010021102221121022211100021011\dots_3$, it does not belong to the Cantor set.

lisyarus
  • 15,517
1

Any number in $[0, 1]$ which has (at least one) base $3$ expansion without a $1$ will be in the Cantor set. More precisely,

$$\mathcal{C} = \left\{ \sum_{n=1}^{\infty} \frac{c_n}{3^n} : (\forall n)( c_n \in \{ 0, 2 \} ) \right\}.$$

It is known that a number as above is irrational if and only if the expansion is non-recurring. So for example $0{.}202202220222202222202\ldots$ will be an irrational number in the Cantor set.

But you seem to be looking for an irrational number expressed by radicals, for example $\frac{\sqrt{7-\sqrt{3}}}{4}$. This appears to be difficult, since there is no obvious connection between an expression of an irrational number by radicals and it's base $3$ expansion.

As others have indicated, $\frac{1}{\sqrt{2}} \notin \mathcal{C}$. This is quite understandable, because it's like a random guess, and since the Cantor set has measure $0$, it's difficult to hit the Cantor set with such.

Adayah
  • 10,468