Originally posted by polydigm
| Quote: | Originally posted by scallopino
| Quote: | Originally posted by polydigm
I'm not sure that song is entirely accurate in that I don't think the Ancient Greeks would have had the wherewithal to figure out that pi is
not rational. You can keep on constructing rational numbers that get more and more accurate and there's no limit to how accurate they can get.
For example 21991/7000 is accurate to 5 decimal places. Thus, they would have easily been able to find rational solutions more accurate than any
method available to them for measurement and so they would have needed some other method to show it was not rational. The proofs are actually quite
sophisticated. |
That's right...the metric space (Q,d) is not complete, and the sequence of rationals (3, 3.1, 3.14, 3.141, 3.1415...) is the usual example given
to introduce completeness. The sequence is a Cauchy sequence but doesn't converge in (Q,d) because the limit is not rational. |
Indeed, I actually understand this. The Ancient Greeks just didn't have mathematics that sophisticated. I dropped out of Pure Maths half way
through Honours level. I just couldn't see myself as a full time mathematician. The level of dedication required would have excluded too many of
my other interests for my liking. |