Yet our discussion here has given me an idea (which is so simple, I’m sure it’s been done).
A right angle triange can be drawn on a sphere (like a globe), and it can have 3 90° angles, when the interior angles of a triangle add up to 180° when depicted on a flat surface.
So, if we have a very large right angle triange, will we see angles that add up to more than 180°? If space is curved?
So, does measuring the angles of a very large triangle tell us if space is flat or curved, and–if curved–how large the Universe actually is?
Like determining the size of a globe by measuring the length of the sides and interior angles of the triangle that’s drawn on it?
Assuming that the Universe is a hypersphere, and not elipsoid or saddle-shaped?
That is exactly how it is done. The last time I looked it up the measurements were still showing no curvature; indicated that if the universe was indeed curved, it must be at least 20x the observable volume to hide this fact from the best experiments at the time.
What are you saying here? If you’re saying that mathematicians haven’t proven that Pi is both irrational and transcendental, then you are wrong, because there are formal proofs for both.
Yes, I made an assumption without actually understanding. As Pi was ever expanding, I assumed it was not known… my bad. This has nothing to do with rational or irrational. Contrary to popular opinion, the definition of an [Irrational Number has nothing to do with decimal expansions, non-repeating or otherwise. Instead, an Irrational Number is a Real Number that is NOT a Rational Number.
All rational numbers are real, but all real numbers are not rational . Real numbers contain irrational numbers also. Irrational numbers are non-terminating and non-repeating while rational numbers are terminating or non-terminating but repeating.
I stand corrected and learned something about numbers.
For sure. These decimals that go on forever are just an artifact of the base of the number system. There really isn’t anything special about them. It’s why I always facepalm when I meet someone who tries to memorize dozens of digits of one (like pi).
Actually, remembering decimals of pi and e to the precision of float64 can be useful, for numerical programming purposes. Yes, those constants are accessible through any modern programming language, but I type pi and e from memory quite often out of pure laziness, when I can’t be assed to look up the name of the constant in the required language. Mostly in code that is only used once, for quick calculations.
What do you mean “real world”? In my professional life, I’ve encountered quite real navigation-related situations where circles with diameters on the scales of several km are involved, where curvature of the earth must be considered, and where precision on the order of 0.1 m is required. Clearly, an accuracy of pi better than 3.1 is required in these cases.
