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.

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    $\begingroup$ Related: Is there a “positive” definition for irrational numbers? $\endgroup$ – projectilemotion Jan 5 '18 at 19:30
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    $\begingroup$ Honestly speaking, I don't care much. Already the name "irrational" is slander, they are more rational than... whatever. I care more for "computable". $\endgroup$ – Professor Vector Jan 5 '18 at 21:03
  • $\begingroup$ "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 $2$. $\endgroup$ – darij grinberg Jan 6 '18 at 0:04
  • $\begingroup$ @projectilemotion Thanks for the link. $\endgroup$ – Allawonder Jan 6 '18 at 1:33
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    $\begingroup$ @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. $\endgroup$ – Allawonder Jan 6 '18 at 1:35

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 $\mathbb{R}^2$.

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    $\begingroup$ 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. $\endgroup$ – timtfj Jan 31 at 18:20

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.

  • $\begingroup$ 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? $\endgroup$ – Allawonder Jan 6 '18 at 1:45
  • $\begingroup$ @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. $\endgroup$ – Mark Bennet Jan 6 '18 at 8:18

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