How do you think of irrational numbers?

Question

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.

Answer 1

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 $\mathbb{R}^2$.

Answer 2

I think that for me I don't often think of individual irrationals. There are perhaps two basic contexts in which I mainly go beyond rationals:

(i) Algebraic numbers for solving polynomial equations

(ii) Real and Complex numbers for taking limits (eg infinite sums) (and Reals for the Intermediate Value property for continuous functions)

In each case I am looking for a rich enough context in which I can be confident that what I get at the end of my work will exist and be well-defined.

In the end, I think it is the concept of "integer" which turns out to be more subtle. The rationals are "just" the prime subfield of any field of characteristic zero.

So I think of these numbers as an abundance which gives me a rich enough context.