How do you think of irrational numbers? What are some ways by which you can characterise an irrational number? The basic way is as those real numbers inexpressible as integral fractions; another is as those reals with non-periodic decimal expansions; another would be as quantities (without loss of generality, focus only only positive numbers) which can never be perfect or precisely tuned, but always potentially approaching a value (think in terms of decimal expansions), etc. **_What are some other ways, images, mental pictures or aids in general for thinking about the irrational numbers?_** Thank you.

Here are some ways I think of irrational numbers:

(1) An irrational number is a number whose positive integer multiples never hit an integer. (But these multiples come arbitrarily close to integers.)

(2) Imagine a wheel that has a rotation rate of $\alpha$ revolutions per second. $\alpha$ is irrational if and only if there is never a nonzero whole number of revolutions after a nonzero whole number of seconds.

(3) A line through the origin has irrational slope if and only if it misses all other points on the integer grid in $\mathbb{R}^2$.