God

[Max Cohen]

884509627386359275033751967 943067599621731590401694134
434007629683591574337516791 197615733475195375920401694
343151239621353184932676605 800621596380716399501371459
954387507655892533875618750 354029981152863950711207613

[Patrick del Poker Grande]

1<font color="white"> Star </font>

[drudman]

n/0

[Quaalude]

0

[Zeno]

[ QUOTE ]
884509627386359275033751967 943067599621731590401694134
434007629683591574337516791 197615733475195375920401694
343151239621353184932676605 800621596380716399501371459
954387507655892533875618750 354029981152863950711207613

[/ QUOTE ]


Since God is a Trinity shouldn't that be cubed?


Perhaps God is just a series of ones and zeroes.

-Zeno

[kgrad5]

[ QUOTE ]


Since God is a Trinity shouldn't that be cubed?



[/ QUOTE ]

Is that a fact?

[drudman]

I meant to say:

i

[vulturesrow]

[ QUOTE ]
[ QUOTE ]


Since God is a Trinity shouldn't that be cubed?



[/ QUOTE ]

Is that a fact?

[/ QUOTE ]

In Christian theology, yes.

[CallMeIshmael]

[ QUOTE ]
I meant to say:

i

[/ QUOTE ]

/images/graemlins/grin.gif /images/graemlins/grin.gif /images/graemlins/grin.gif /images/graemlins/grin.gif

[jason_t]

[ QUOTE ]
884509627386359275033751967 943067599621731590401694134
434007629683591574337516791 197615733475195375920401694
343151239621353184932676605 800621596380716399501371459
954387507655892533875618750 354029981152863950711207613

[/ QUOTE ]

Max Cohen is the main character in Pi and these are the digits that he writes on the piece of paper with "Only God is Perfect" at the bottom.

[CallMeIshmael]

[ QUOTE ]
Max Cohen is the main character in Pi and these are the digits that he writes on the piece of paper with "Only God is Perfect" at the bottom.

[/ QUOTE ]

Nice catch fish.

[PairTheBoard]

[ QUOTE ]
884509627386359275033751967 943067599621731590401694134
434007629683591574337516791 197615733475195375920401694
343151239621353184932676605 800621596380716399501371459
954387507655892533875618750 354029981152863950711207613

[/ QUOTE ]

I suspect you may have a rounding error.

PairTheBoard

[brassnuts]

If God is a number, he is 1.61803398875...

[jason_t]

[ QUOTE ]
If God is a number, he is 1.61803398875...

[/ QUOTE ]

At least pick a transcendental number.

[gumpzilla]

[ QUOTE ]

At least pick a transcendental number.

[/ QUOTE ]

To his credit, he did pick the number least well approximable by rationals. I think that's a better choice.

[BruceZ]

[ QUOTE ]
Max Cohen is the main character in Pi and these are the digits that he writes on the piece of paper with "Only God is Perfect" at the bottom.

[/ QUOTE ]

I lost respect for that movie when at one point someone says something like "I'm sure you've written down every 216 digit number".

[mmbt0ne]

Mine was when he said that the numbers of the Fibonacci sequence approximated theta when they were divided instead of tau?

Or when Euclid spit out a 218 digit number that was nowhere similar to the 216 number he writes later.

Of course, starting out the move with a misrepresentation of Pi itself didn't help much.

[CallMeIshmael]

Im pretty sure the movie just sucked ass as a movie. With or without the math mistakes.


Though they didnt help /images/graemlins/grin.gif /images/graemlins/grin.gif

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]

At least pick a transcendental number.

[/ QUOTE ]

To his credit, he did pick the number least well approximable by rationals. I think that's a better choice.

[/ QUOTE ]

Least well approximable by rationals? How can that make any sense?

PairTheBoard

[gumpzilla]

[ QUOTE ]

Least well approximable by rationals? How can that make any sense?

[/ QUOTE ]

The golden ratio phi is an irrational number. It has an extremely simple continued fraction expansion:

1 + 1 / ( 1 + 1 / (1 + 1 / . . . . )

. . . ones all the way down. It turns out that there is a relationship between continued fraction expansions and how well you can approximate an irrational number; specifically, if you truncate the expansion at any point, the rational number that you get is the best approximation to the irrational number that is possible for all denominators equal to or smaller than the denominator of that partial summation. It further turns out that you can show that the approximations you get in this way are worse for phi than for any other rational number. Thus, phi is the irrational number that is worst approximable by rationals.

[jason_t]

[ QUOTE ]
[ QUOTE ]

Least well approximable by rationals? How can that make any sense?

[/ QUOTE ]

The golden ratio phi is an irrational number. It has an extremely simple continued fraction expansion:

1 + 1 / ( 1 + 1 / (1 + 1 / . . . . )

. . . ones all the way down. It turns out that there is a relationship between continued fraction expansions and how well you can approximate an irrational number; specifically, if you truncate the expansion at any point, the rational number that you get is the best approximation to the irrational number that is possible for all denominators equal to or smaller than the denominator of that partial summation. It further turns out that you can show that the approximations you get in this way are worse for phi than for any other rational number. Thus, phi is the irrational number that is worst approximable by rationals.

[/ QUOTE ]

Related to this is the famous Liouville Approximation Theorem.

If x is algebraic of degree n &gt;= 2 and p/q is rational then |x - p/q| &gt;= 1/q^n.

It's hard to approximate irrational algebraic numbers.

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]

Least well approximable by rationals? How can that make any sense?

[/ QUOTE ]

The golden ratio phi is an irrational number. It has an extremely simple continued fraction expansion:

1 + 1 / ( 1 + 1 / (1 + 1 / . . . . )

. . . ones all the way down. It turns out that there is a relationship between continued fraction expansions and how well you can approximate an irrational number; specifically, if you truncate the expansion at any point, the rational number that you get is the best approximation to the irrational number that is possible for all denominators equal to or smaller than the denominator of that partial summation. It further turns out that you can show that the approximations you get in this way are worse for phi than for any other rational number. Thus, phi is the irrational number that is worst approximable by rationals.

[/ QUOTE ]

Trying to understand this. Say Pn/Qn is the rational aproximation for phi at the nth truncation. Then ANY other irrational number has a better rational aproximation using denominators less than or equal to Qn?

You must mean something else because I just don't think that can be right. Let d1 be the distance between phi and Pn/Qn and let d2 be the distance to the next closest aproximation among all rationals with denominators less than or equal to Qn. Just choose an irrational close to phi and a small distance futher away from Pn/Qn, a smaller distance than say (d2-d1)/2.

PairTheBoard

[jason_t]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]

Least well approximable by rationals? How can that make any sense?

[/ QUOTE ]

The golden ratio phi is an irrational number. It has an extremely simple continued fraction expansion:

1 + 1 / ( 1 + 1 / (1 + 1 / . . . . )

. . . ones all the way down. It turns out that there is a relationship between continued fraction expansions and how well you can approximate an irrational number; specifically, if you truncate the expansion at any point, the rational number that you get is the best approximation to the irrational number that is possible for all denominators equal to or smaller than the denominator of that partial summation. It further turns out that you can show that the approximations you get in this way are worse for phi than for any other rational number. Thus, phi is the irrational number that is worst approximable by rationals.

[/ QUOTE ]

Trying to understand this. Say Pn/Qn is the rational aproximation for phi at the nth truncation. Then ANY other irrational number has a better rational aproximation using denominators less than or equal to Qn?

You must mean something else because I just don't think that can be right. Let d1 be the distance between phi and Pn/Qn and let d2 be the distance to the next closest aproximation among all rationals with denominators less than or equal to Qn. Just choose an irrational close to phi and a small distance futher away from Pn/Qn, a smaller distance than say (d2-d1)/2.

PairTheBoard

[/ QUOTE ]

Dirichlet proved that if x is irrational an in [0,1] then |x-p/q| &lt;= 1/q^2 for infinitely many rational p/q. So for each q lay out a bunch of intervals centered at 1/q, 2/q, ..., (q-1)/q with radius 1/q^2. Then each irrational lands in infinitely many intervals. Hurwitz proved that if each of the intervals is reduced in length by a factor of more than 1/sqrt(5) then suddenly some irrationals will only lie in finitely many intervals and the least irrational with this property is [sqrt(5) - 1]/2 = [1 + sqrt(5)]/2 - 1.

[jason_t]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]

Least well approximable by rationals? How can that make any sense?

[/ QUOTE ]

The golden ratio phi is an irrational number. It has an extremely simple continued fraction expansion:

1 + 1 / ( 1 + 1 / (1 + 1 / . . . . )

. . . ones all the way down. It turns out that there is a relationship between continued fraction expansions and how well you can approximate an irrational number; specifically, if you truncate the expansion at any point, the rational number that you get is the best approximation to the irrational number that is possible for all denominators equal to or smaller than the denominator of that partial summation. It further turns out that you can show that the approximations you get in this way are worse for phi than for any other rational number. Thus, phi is the irrational number that is worst approximable by rationals.

[/ QUOTE ]

Related to this is the famous Liouville Approximation Theorem.

If x is algebraic of degree n &gt;= 2 and p/q is rational then |x - p/q| &gt;= 1/q^n.

It's hard to approximate irrational algebraic numbers.

[/ QUOTE ]

For sufficiently large q.

[bohemian]

Sorry, first digit is a 7.

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]

Least well approximable by rationals? How can that make any sense?

[/ QUOTE ]

The golden ratio phi is an irrational number. It has an extremely simple continued fraction expansion:

1 + 1 / ( 1 + 1 / (1 + 1 / . . . . )

. . . ones all the way down. It turns out that there is a relationship between continued fraction expansions and how well you can approximate an irrational number; specifically, if you truncate the expansion at any point, the rational number that you get is the best approximation to the irrational number that is possible for all denominators equal to or smaller than the denominator of that partial summation. It further turns out that you can show that the approximations you get in this way are worse for phi than for any other rational number. Thus, phi is the irrational number that is worst approximable by rationals.

[/ QUOTE ]

Trying to understand this. Say Pn/Qn is the rational aproximation for phi at the nth truncation. Then ANY other irrational number has a better rational aproximation using denominators less than or equal to Qn?

You must mean something else because I just don't think that can be right. Let d1 be the distance between phi and Pn/Qn and let d2 be the distance to the next closest aproximation among all rationals with denominators less than or equal to Qn. Just choose an irrational close to phi and a small distance futher away from Pn/Qn, a smaller distance than say (d2-d1)/2.

PairTheBoard

[/ QUOTE ]

Dirichlet proved that if x is irrational an in [0,1] then |x-p/q| &lt;= 1/q^2 for infinitely many rational p/q. So for each q lay out a bunch of intervals centered at 1/q, 2/q, ..., (q-1)/q with radius 1/q^2. Then each irrational lands in infinitely many intervals. Hurwitz proved that if each of the intervals is reduced in length by a factor of more than 1/sqrt(5) then suddenly some irrationals will only lie in finitely many intervals and the least irrational with this property is [sqrt(5) - 1]/2 = [1 + sqrt(5)]/2 - 1.

[/ QUOTE ]

That's amazing. Maybe this is a trivial implication but it's one I never considered. Evidently then, it's possible to construct a collection of open balls of differing radii around all the rationals which will not cover the unit interval. This despite the rationals being dense in the interval.

My intuition tells me there's got to be something special about those finite number of rationals p/q whose epsilon balls approaching 1/(q^2)sqrt(5) include phi-1.

Also, the largest factor for which all such balls exclude phi-1 must be something special too.

Maybe God is the Cantor Set?

PairTheBoard

[jason_t]

[ QUOTE ]
[ QUOTE ]


Dirichlet proved that if x is irrational an in [0,1] then |x-p/q| &lt;= 1/q^2 for infinitely many rational p/q. So for each q lay out a bunch of intervals centered at 1/q, 2/q, ..., (q-1)/q with radius 1/q^2. Then each irrational lands in infinitely many intervals. Hurwitz proved that if each of the intervals is reduced in length by a factor of more than 1/sqrt(5) then suddenly some irrationals will only lie in finitely many intervals and the least irrational with this property is [sqrt(5) - 1]/2 = [1 + sqrt(5)]/2 - 1.

[/ QUOTE ]

That's amazing. Maybe this is a trivial implication but it's one I never considered. Evidently then, it's possible to construct a collection of open balls of differing radii around all the rationals which will not cover the unit interval. This despite the rationals being dense in the interval.


[/ QUOTE ]

Order the rationals in the unit interval r_1, r_2, r_3, ... and put an interval of length epsilon * 2^(-k) around r_k where 0 &lt; epsilon &lt; 1 is fixed. The total length of each of these intervals is

sum k=1 to infinity epsilon 2^(-k) = epsilon &lt; 1.

Hence we've put an interval of different length around each rational but didn't cover the unit interval.

In the language of topology and measure theory, we have constructed a nowhere dense subset of the unit interval with positive measure. The set is simultaneously small (nowhere dense) and big (not of zero measure).

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]


Dirichlet proved that if x is irrational an in [0,1] then |x-p/q| &lt;= 1/q^2 for infinitely many rational p/q. So for each q lay out a bunch of intervals centered at 1/q, 2/q, ..., (q-1)/q with radius 1/q^2. Then each irrational lands in infinitely many intervals. Hurwitz proved that if each of the intervals is reduced in length by a factor of more than 1/sqrt(5) then suddenly some irrationals will only lie in finitely many intervals and the least irrational with this property is [sqrt(5) - 1]/2 = [1 + sqrt(5)]/2 - 1.

[/ QUOTE ]

That's amazing. Maybe this is a trivial implication but it's one I never considered. Evidently then, it's possible to construct a collection of open balls of differing radii around all the rationals which will not cover the unit interval. This despite the rationals being dense in the interval.


[/ QUOTE ]

Order the rationals in the unit interval r_1, r_2, r_3, ... and put an interval of length epsilon * 2^(-k) around r_k where 0 &lt; epsilon &lt; 1 is fixed. The total length of each of these intervals is

sum k=1 to infinity epsilon 2^(-k) = epsilon &lt; 1.

Hence we've put an interval of different length around each rational but didn't cover the unit interval.

In the language of topology and measure theory, we have constructed a nowhere dense subset of the unit interval with positive measure. The set is simultaneously small (nowhere dense) and big (not of zero measure).

[/ QUOTE ]

yep

PTB