Are irrational numbers like π only approximations or “asymptotic” points on the real line?

Direct Answer

No. Numbers such as π are exact points on the real number line.
We can only write down ever‐better approximations (e.g. 3.14, 22/7, 355/113), but π itself is not “just” an approximation or some boundary you approach from afar. It is a well‐defined real number, typically characterized by a Dedekind cut or as the limit of a convergent sequence.

---

Why We Use Approximations

• Rational approximations let us compute and reason numerically: for any irrational α, there are infinitely many fractions p/q with |α – p/q| ≤ 1/q². Dirichlet’s theorem guarantees that as the denominator q grows, these fractions zero in on α ever more closely [A].
• In fact, the rational points are dense on the real line: in every interval however small, there’s a rational p/q arbitrarily close to any target real number α [B].

---

Exact Definitions of π and Other Irrationals

• Dedekind cut: π can be defined as the set of all rational numbers whose squares are less than π² (itself defined, say, as the area ratio of a unit circle).
• Cauchy sequence: π = lim_n→∞ a_n where (a_n) might be partial sums of the Leibniz series or perimeters of inscribed polygons.

In each case, π is the limit—unique and exact, not merely an asymptote.

---

No “Next” Real Number

Unlike the integers or rationals with fixed gaps, the reals form a continuous continuum. There is no immediate neighbor to the left or right of π—between π − ε and π + ε lie infinitely many other reals, regardless of how tiny ε is. In this sense π “fits” just like any other real: it has no predecessors or successors, but it is nonetheless a precise, singular point on the line [C].