Despite L proving that transcendental numbers exist, π does not satisfy Liouville’s criterion (it can’t be well approximated by rational numbers), so its classification remained elusive.
Pi can be approximated VERY well with rational numbers, so much so that Mathologer has an entire video about it! Definitely take a look for his "most irrational number" video, in which pi is shown to be not very irrational at all.
It is not quite to the same extent as Liouville's manufactured number, but the fractions 22/7 and 355/113 both get far closer to pi than their digit count suggests, which is exactly what said criterion was.
Edit: Unlike the golden ratio, which cannot be approximated efficiently at all with a fraction, but being a polynomial root (unlike pi) it follows the pattern perfectly.
(it can’t be well approximated by rational numbers)
garbles Liouville's criterion pretty badly.
Notice first that 355/113 is wrong by 0.000000266. The error is about one part in four million. Yet the denominator is only 113. The accuracy is roughly the cube of the denominator. With a typical number, you can go looking for good rational approximations, a/b, and with the right choice of b the accuracy is around the square of the denominator. So 355/113 does get closer to pi than the digit count suggests.
Does this suggest a route to proving that pi is transcendental? Not really. Picture a polynomial with integer coefficients with degree 4. Say 7x^4 + 11x^3 - 6x + 12. Evaluate it at a rational number a/b. You get
Either the whole number is zero (congratulations you have found a rational root) or the error is some multiple of 1/b^4. There is a granularity here. In this case the grain size is 1/b^4. For an nth degree polynomial the granularity is 1/b^n.
Since 355/113 is close to pi, around the cube of the denominator, that hints that pi is a little special because rational approximations are usually accurate to around the square of the denominator. On the other hand, the root of a cube is sometimes approximated to roughly the cube of the denominator. And the root of a quartic equation is sometimes approximated to roughly the fourth power of the denominator. We are not close to ruling out pi being the root of polynomial of third or fourth degree :-(
A further complication is that Liouville looked at sequences of approximations. a1/b1, a2/b2, a3/b3,... and concerned himself with the general trend. How accurate are the approximations? What power of b_i is involved. The thing he noticed was that with a root of an nth degree polynomial, you cannot have the accuracy trend being better than 1/b^n.
So Liouville rolled up his sleeves and set to work contriving a really fucking weird number. Think about factorials, 1, 2, 6, 24, 120, 720, 5040,... Now create a sequence of numbers. Start with one in the first decimal place 0.1. Then add a 1 to the second decimal place = 0.11. Next add a one to the sixth decimal place 0.110001. After that add a one to the twenty fourth decimal place = 0.110001000000000000000001. That is starting to look weird. The next one has a one in the 120th decimal place = 0.110001000000000000000001000 ... lots and lots of zeros omitted to avoid doing a page widening attack on vote ... 0001
But we are only just getting start. After 0.110001000000000000000001000... one hundred zeros ...00100 ... six hundred zeros ... 00100 ... nearly five thousand zeros ... 00100 ... oh god, way too many zeros ...00100 ... even more zeros, seriously WTF! ... 00100 ...
He has his limit number L when he takes this to infinity, and he has his sequence, a sequence of rational approximations. The accuracy of rational approximations is approximately 1/denominator^n where n keeps getting bigger. If L is the root of a polynomial, then n is limited by the degree of the polynomial. But Liouville has done the work of a mad scientist very well, the n for his number just keeps growing. So it is not the root of a polynomial.
That is brilliant work for 1844. But the underlying idea is quite limited. Yes, he can prove that L is transcendental, but only because of the extremely weird way that L was constructed. What about pi? That is going to need a new and different idea
[ + ] SithEmpire
[ - ] SithEmpire 2 points 9 monthsJul 31, 2023 03:14:55 ago (+2/-0)*
Pi can be approximated VERY well with rational numbers, so much so that Mathologer has an entire video about it! Definitely take a look for his "most irrational number" video, in which pi is shown to be not very irrational at all.
It is not quite to the same extent as Liouville's manufactured number, but the fractions 22/7 and 355/113 both get far closer to pi than their digit count suggests, which is exactly what said criterion was.
Edit: Unlike the golden ratio, which cannot be approximated efficiently at all with a fraction, but being a polynomial root (unlike pi) it follows the pattern perfectly.
[ + ] happytoes
[ - ] happytoes 0 points 9 monthsAug 2, 2023 10:49:42 ago (+0/-0)
garbles Liouville's criterion pretty badly.
Notice first that 355/113 is wrong by 0.000000266. The error is about one part in four million. Yet the denominator is only 113. The accuracy is roughly the cube of the denominator. With a typical number, you can go looking for good rational approximations, a/b, and with the right choice of b the accuracy is around the square of the denominator. So 355/113 does get closer to pi than the digit count suggests.
Does this suggest a route to proving that pi is transcendental? Not really. Picture a polynomial with integer coefficients with degree 4. Say 7x^4 + 11x^3 - 6x + 12. Evaluate it at a rational number a/b. You get
( 7 a^4 + 11 a^3 b - 6 a b^3 + 12 b^4 )/ b^4 = (some whole number) / b^4
Either the whole number is zero (congratulations you have found a rational root) or the error is some multiple of 1/b^4. There is a granularity here. In this case the grain size is 1/b^4. For an nth degree polynomial the granularity is 1/b^n.
Since 355/113 is close to pi, around the cube of the denominator, that hints that pi is a little special because rational approximations are usually accurate to around the square of the denominator. On the other hand, the root of a cube is sometimes approximated to roughly the cube of the denominator. And the root of a quartic equation is sometimes approximated to roughly the fourth power of the denominator. We are not close to ruling out pi being the root of polynomial of third or fourth degree :-(
A further complication is that Liouville looked at sequences of approximations. a1/b1, a2/b2, a3/b3,... and concerned himself with the general trend. How accurate are the approximations? What power of b_i is involved. The thing he noticed was that with a root of an nth degree polynomial, you cannot have the accuracy trend being better than 1/b^n.
So Liouville rolled up his sleeves and set to work contriving a really fucking weird number. Think about factorials, 1, 2, 6, 24, 120, 720, 5040,... Now create a sequence of numbers. Start with one in the first decimal place 0.1. Then add a 1 to the second decimal place = 0.11. Next add a one to the sixth decimal place 0.110001. After that add a one to the twenty fourth decimal place = 0.110001000000000000000001. That is starting to look weird. The next one has a one in the 120th decimal place = 0.110001000000000000000001000 ... lots and lots of zeros omitted to avoid doing a page widening attack on vote ... 0001
But we are only just getting start. After 0.110001000000000000000001000... one hundred zeros ...00100 ...
six hundred zeros ... 00100 ... nearly five thousand zeros ... 00100 ... oh god, way too many zeros ...00100 ... even more zeros, seriously WTF! ... 00100 ...
He has his limit number L when he takes this to infinity, and he has his sequence, a sequence of rational approximations. The accuracy of rational approximations is approximately 1/denominator^n where n keeps getting bigger. If L is the root of a polynomial, then n is limited by the degree of the polynomial. But Liouville has done the work of a mad scientist very well, the n for his number just keeps growing. So it is not the root of a polynomial.
That is brilliant work for 1844. But the underlying idea is quite limited. Yes, he can prove that L is transcendental, but only because of the extremely weird way that L was constructed. What about pi? That is going to need a new and different idea
[ + ] shitface9000
[ - ] shitface9000 0 points 9 monthsJul 30, 2023 22:04:49 ago (+0/-0)