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.

Related: Is there a “positive” definition for irrational numbers? — projectilemotion, Jan 05 '18 at 19:30
Honestly speaking, I don't care much. Already the name "irrational" is slander, they are more rational than... whatever. I care more for "computable". — , Jan 05 '18 at 21:03
"which can never be perfect or precisely tuned" This is not a meaningful understanding or even intuition. From the point of view of the definition of reals, $1 + \sqrt{5}$ isn't any more "vague" than $2$. — darij grinberg, Jan 06 '18 at 00:04
@ProfessorVector Note that the term irrational suggests that they are inexpressible as ratios of integers. It has nothing to do with their being unreasonable in the way you think. — Allawonder, Jan 06 '18 at 01:35
@darijgrinberg I meant that with respect to their decimal expansions, obviously. I have edited the question to reflect this point. — Allawonder, Jan 06 '18 at 01:36
For those who say the question is too broad, could you give suggestions as to improving the penultimate paragraph; I don't know how to make it more specific. Also, at least two people seemed to understand it well enough. — Allawonder, Jan 06 '18 at 01:43

score 4 · Accepted Answer · answered Jan 05 '18 at 23:44

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 $\mathbb{R}^2$.

Horrifyingly, $\sin(ax) \cdot \sin (bx) $ isn't even a periodic function if $\frac{a}{b}$ is irrational. Neither is $\sin(ax) + \sin (bx)$, for that matter. — timtfj, Jan 31 '19 at 18:20

score 1 · Answer 2 · answered Jan 05 '18 at 20:12

1

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.

answered Jan 05 '18 at 20:12

Mark Bennet

100,194

Thank you. I found this interesting: 'In the end, I think it is the concept of "integer" which turns out to be more subtle.' Could you say more about what you meant? Why are the integers subtler than the irrationals? – Allawonder Jan 06 '18 at 01:45
@Allawonder If you extend $\mathbb Q$ you get another field. If you look at the ring of integers within the field, you recover a notion of "prime" and ideas like factorisation and localisation. An integer in this context (an algebraic integer) satisfies a monic polynomial with integer coefficients. – Mark Bennet Jan 06 '18 at 08:18

How do you think of irrational numbers?

2 Answers2