Proof that the irrational numbers are uncountable

Question

Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the irrationals are uncountable?

Answer 1

Given that the reals are uncountable (which can be shown via Cantor diagonalization) and the rationals are countable, the irrationals are the reals with the rationals removed, which is uncountable. (Or, since the reals are the union of the rationals and the irrationals, if the irrationals were countable, the reals would be the union of two countable sets and would have to be countable, so the irrationals must be uncountable.)

Answer 2

Just for fun, we can prove this using way more machinery than necessary.

Assume for contradiction that the irrational numbers are countable. Now let $q_1,q_2,\ldots$ be an enumeration of the rationals, and let $r_1,r_2,\ldots$ be an enumeration of the irrationals. Now set $F_i=\mathbb R\setminus \{q_i,r_i\}$. Then the sets $F_i$ are open and dense in the usual topology on $\mathbb R$, and so by the Baire Category Theorem, $\bigcap_{i=1}^\infty F_i$ is dense in $\mathbb R$. However, $\bigcap_{i=1}^\infty F_i=\emptyset$ which is not dense, and hence the irrationals mustn't have been countable.

Answer 3

One can use a variant of the familiar Cantor diagonalization argument. Let us suppose that $x_1,x_2,x_3,\dots$ is an enumeration of the irrationals. Let $d_n$ be the $n$-th decimal digit of $x_n$ after the decimal point.

Let $w_n=5$ if $d_n$ is not equal to $3$ or $4$, and let $w_n=6$ if $d_n$ is equal to $3$ or $4$.

Now we give the decimal representation of an irrational $y$ not in the list $x_1,x_2,x_3,\dots$. Very roughly speaking, it is the number whose $n$-th digit after the decimal point is $w_n$. But some modification is made to ensure irrationality.

First we describe the $n$-th digit $e_n$ after the decimal point of $y$, where $w_n=5$. List these $n$ as $n_1,n_2,n_3,n_4,\dots$. Let $e_{n_1}=5$. Let $e_{n_2}=6$. Let $e_{n_3}=e_{n_4}=5$. Let $e_{n_6}=6$. Let $e_{n_7}=e_{n_8}=e^{n_9}=5$. Let $e_{n_{10}}=6$. Continue, leaving longer and longer strings of unchanged $5$'s.

The same procedure is used to produce the $n$-th digit $e_n$ after the decimal point of $y$, where $w_n=3$. Leave longer and longer strings of these unchanged, and switch the digit to $4$ occasionally.

The number $y$ we produce has decimal expansion which differs in the $n$-th place from the corresponding digit of $x_n$. The occasional switches from $5$ to $6$ or $3$ to $4$ ensure that the decimal expansion of $y$ is not ultimately periodic.

Answer 4

Let $b$ be an infinite binary sequence. Define $$ E(b) = \sum_{k=0}^\infty \left(\frac{1}{(2k)!} + \frac{b_k}{(2k+1)!}\right). $$ Wikipedia's proof of the irrationality of $e$ extends to show that $E(b)$ is irrational for every infinite binary sequence $b$. So if you believe that the set of all infinite binary sequences is uncountable, you must also believe that the set of irrational numbers is uncountable.

We could also define $$ E(b) = \sum_{k=0}^\infty \frac{b_k}{k!}, $$ but then $E(b)$ is irrational only if $b$ has infinitely many $1$s, or in other words, it doesn't end in an infinite sequence of zeroes.