Definition of irrational number

Question

What is a formal definition of a irrational number? Usually, we say that it is a number that it is not rational. Is it enough?

Answer 1

Uncle Google and auntie Wikipedia are your friends. Wikipedia correctly states:

In mathematics, an irrational number is any real number that cannot be expressed as a ratio of integers

In a way, it's not enough to say that any number that is not rational is irrational, because most complex numbers (like $i$) are neither rational nor irrational.

Answer 2

A real number is irrational if is not rational.

But the definition of real number is much less simple.

Answer 3

It is a number that cannot be written as $\frac{p}{q}$, for $(p,q)\in\mathbb{Z}\times\mathbb{N}^\ast$. Formally: $r$ is irrational iff (i) $r\in\mathbb{R}$; and (ii) $\forall(p,q)\in\mathbb{Z}\times\mathbb{N}^\ast, r\neq \frac{p}{q}$.

Answer 4

Yes, irrational = "not rational". It is well-defined in any context where "rational" is well-defined, e.g. in any ring containing the rational numbers $\,\Bbb Q.\,$ For example, as I mentoned in an answer, to the question Is $i$ irrational?, many algebraic number theorists use the terminology for complex numbers $\not\in \Bbb Q.\,$ In particular, be aware that irrational need not imply real (as in the classical case).