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Geometry? Pi is an infinite series, as are the trigonometric functions. How can you explain the circumference of a circle without invoking pi?
by rlpb 9y ago
Geometry? Pi is an infinite series, as are the trigonometric functions. How can you explain the circumference of a circle without invoking pi?
- srean 9y agoAs long as \epsilon != 0 one can make rational_of(\Pi, \epsilon) an arbitrarily close approximation of \Pi, without the need for infinity. In fact that is precisely what we use in practice, by that I mean computation. Ultimately it comes down to a trade off. Which axiomatic system does one to work in ? Introducing infinity does have its fair share of warts (by warts I mean consequences that flies in the face of intuition).
- rlpb 9y agoHow do you approximate pi in this way without making use of an infinite series? Or are you saying that an approximation that doesn't use an infinity is fine even if an infinity was used to arrive at it?
- srean 9y agoIts the latter. You can truncate the series at an appropriate place of your choosing. But yes, \Pi as a number does not even exist unless you add the Reals, so that would need infinity. What I am saying is, one can work with the Rationals.
- widdma 9y agoPi isn't an infinite series, it is just a number. Sure, it can be represented as an infinite series, but so can any number! (Eg. 1 = 0.999999...)
- tinus_hn 9y agoThe difference obviously is that you can't represent pi as a finite series or a repeating infinite series while you can do that with the rational numbers.