21–30 of 38 posts
Rather, rational numbers awfully close (in ordinary human terms) to specific, well known irrational numbers. There are, I think, just as many irrational numbers comparably close to any rational number.
Earlier quoted context omitted.
Well some numbers expose patterns when written as a continued fraction. In particular e becomes pretty regular. You can modify the continued fraction slightly to make pi regular as well, but the normal continued fraction sequence doesn't give much of an insight. Other than the fact that 3 + 1/(7 + 1/16)) is a damn good approximation (7 digits, pretty good for something that can be written using only 4 digits total: […
Phi/golden ratio also has a cool continued fraction sequence...it's only 1's all the way down
But phi is indeed especially interesting because of what its sequence implies for rational approximations of phi.
I thought that colouring the pattern by the instantaneous velocity of the ball would be an obvious improvement and might uncover further structure.
I always wondered if some hidden pattern would be exposed when visualising numbers in unconventional ways in numbers with no known pattern such as Pi or prime numbers. A sort of multi-dimensional rendering that suddenly reveals a hidden pattern.
> Here are some more irrational numbers expressed in this way Rather, rational numbers awfully close (in ordinary human terms) to specific, well known irrational numbers. There are, I think, just as many irrational numbers comparably close to any rational number.
> Here are some more irrational numbers expressed in this way Rather, rational numbers awfully close (in ordinary human terms) to specific, well known irrational numbers. There are, I think, just as many irrational numbers comparably close to any rational number.
If we want to open the floodgates on being too pedantic, I think there are uncountably more irrational numbers close to any rational number than there are rational numbers close to an irrational number. But in both cases, it's definitely a bunch.
It's math. There's no such thing as "too pedantic", as long as you're being interesting and not mean about it.
> I think there are uncountably more irrational numbers close to any rational number than there are rational numbers close to an irrational number.
I think that's right.
Irrationals near a rational are almost certainly uncountable, as otherwise I think we can force all the irrationals to be countable by bucketing them. I think that concern is countered if any bucket has to be uncountable, but if it's not all that makes some rationals special in a way they probably aren't.
Rationals near an irrational is definitely countable, as all the rationals is countable.
> The colors are arbitrary, and have no deeper meaning I thought that colouring the pattern by the instantaneous velocity of the ball would be an obvious improvement and might uncover further structure.