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π is defined, not measured, so it has no uncertainty; the value stored here is the nearest IEEE-754 double to the true constant, correct to about 16 significant figures, which is more than enough to compute the circumference of the observable universe to within the width of a hydrogen atom. Archimedes squeezed it between 223/71 and 22/7 around 250 BC using inscribed and circumscribed 96-gons, and the method he invented — trap the answer between two computable bounds and tighten — is the ancestor of the whole idea of a limit.
Johann Lambert proved π irrational in 1761, but the decisive result came in 1882, when Ferdinand von Lindemann proved it transcendental: it satisfies no polynomial equation with rational coefficients. That single theorem closed a problem that had been open for 2000 years. Squaring the circle — constructing, with compass and straightedge alone, a square equal in area to a given circle — requires constructing √π, and constructible numbers are all algebraic. Lindemann's proof did not merely show that nobody had yet succeeded; it showed that nobody ever could, and the Paris Academy had already stopped accepting submissions on the problem by then out of sheer exhaustion.