# Is there an accepted symbol for irrational numbers?

$\mathbb Q$ is used to represent rational numbers. $\mathbb R$ is used to represent reals.

Is there a symbol or convention that represents irrationals.

Possibly $\mathbb R - \mathbb Q$?

• Sometimes written as $\mathbb I$, but I doubt there is a generally used convention because - other than $\mathbb Q$ and $\mathbb R$ - the irrationals by themselves are not a nice structure (field, ring, group, ...) Jul 23, 2013 at 18:30
• I've seen "$\Bbb P$" before. Jul 23, 2013 at 18:30
• I've seen $\mathbb{T}$ before though it confused me and wasn't sure why they chose that. Jul 23, 2013 at 18:30
• I've seen $\mathbb{Q}^c$, for $\mathbb{Q}$'s complement (in $\mathbb{R}$). Jul 23, 2013 at 22:33
• I've also seen $\mathbb J$. Wikipedia: Blackboard bold Jul 24, 2013 at 2:08

Customarily, the set of irrational numbers is expressed as the set of all real numbers "minus" the set of rational numbers, which can be denoted by either of the following, which are equivalent:

• $\mathbb R \setminus \mathbb Q$, where the backward slash denotes "set minus".

• $\mathbb R - \mathbb Q,\;$ where we read the set of reals, "minus" the set of rationals.

Occasionally you'll see some authors use an alternative notation: e.g., $$\mathbb P = \{x\mid x \in \mathbb R \land x \notin \mathbb Q\}$$ or $$\mathbb I = \{x \mid x\in \mathbb R \land x \notin \mathbb Q\}$$ But if and when an alternative letter like $\mathbb P$ or $\mathbb I$ is used, it should be preceded by a clear statement as to the fact that it is being used to denote the set of irrational numbers.

The most common expression is just $\Bbb R\setminus\Bbb Q$. When a single letter is used, in my experience by far the most common is $\Bbb P$, though I have on very rare occasions seen $\Bbb I$. (Note, though, that $\Bbb I$ is also occasionally used to denote $[0,1]$.) If the context were clear enough, you could probably get away with using $\Bbb P$ without comment, but it would be much safer (and more courteous) to define the symbol explicitly.

In topological contexts (including descriptive set theory) the irrationals are often denoted by $\omega^\omega$ (or occasionally $\Bbb N^{\Bbb N}$), since in the topology that they inherit from $\Bbb R$ they are homeomorphic to the product space $\omega^\omega$; here no further comment is required.

When I first learned, the symbol $\mathbb{K}$ was used. Obviously, though, given the relative infrequency of the need of calling out the set by name, if you do use a symbol, introduce it with "$= \mathbb{R} - \mathbb{Q}$" or something similar.