[math-fun] Egyptian Fractions (was Re: Messages in pi (cont'd))
Bill Gosper <billgosper@gmail.com> wrote:
Off-list I grumbled that no self-respecting Deity would bother sending clues to worshipers dumb enough to use decimal instead of continued fractions.
Perhaps there's a hidden message in the continued fraction for e. :-) Why not Egyptian fractions? The Egyptians, ancient and modern, have always been very religious. Any positive real number can be approximated to any desired precision by an Egyptian fraction, i.e. the sum of reciprocals of distinct integers, a subset of the harmonic series. They aren't unique for a given real number, but the *greedy* Egyptian fraction for that real number is unique. Has anyone inspected the greedy Egyptian fraction for pi? The first few terms are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 744, 1173268, 2586625801171, 14348276635209672362238685, 1062286904072440687703470835520966381484062674280821 (A243020). See the pattern? Me neither. Except the first 12 terms. Maybe if it continued that way? After all, a friend of mine once confidently asserted that pi was infinite. (I think he may have been exaggerating.) Speaking of infinite, try to find an Egyptian fraction for a thousand. Good luck. Unlike the Egyptian fraction for pi, it's finite. But it contains more terms than there are atoms in the known universe, most of them consecutive integers starting with 1. What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original.
The Egyptians were ok with integers, so I think they would have called one thousand one thousand, as opposed to 1+1/2+1/3+…1/e^1000 :-) But it’s a lovely exercise that the greedy algorithm for Egyptian fractions — that is, subtract the largest reciprocal — terminates for any rational number between 0 and 1. There’s an inductive proof, but the nature of the induction is surprising. - Cris
On Dec 7, 2018, at 9:46 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
Bill Gosper <billgosper@gmail.com> wrote:
Off-list I grumbled that no self-respecting Deity would bother sending clues to worshipers dumb enough to use decimal instead of continued fractions.
Perhaps there's a hidden message in the continued fraction for e. :-)
Why not Egyptian fractions? The Egyptians, ancient and modern, have always been very religious.
Any positive real number can be approximated to any desired precision by an Egyptian fraction, i.e. the sum of reciprocals of distinct integers, a subset of the harmonic series. They aren't unique for a given real number, but the *greedy* Egyptian fraction for that real number is unique. Has anyone inspected the greedy Egyptian fraction for pi? The first few terms are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 744, 1173268, 2586625801171, 14348276635209672362238685, 1062286904072440687703470835520966381484062674280821 (A243020). See the pattern? Me neither. Except the first 12 terms. Maybe if it continued that way? After all, a friend of mine once confidently asserted that pi was infinite. (I think he may have been exaggerating.)
Speaking of infinite, try to find an Egyptian fraction for a thousand. Good luck. Unlike the Egyptian fraction for pi, it's finite. But it contains more terms than there are atoms in the known universe, most of them consecutive integers starting with 1.
What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original.
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Hello, the Egyptian fraction of Pi - 3 is 8, 61, 5020, 128541455, 162924332716605980, 28783052231699298507846309644849796, 871295615653899563300996782209332544845605756266650946342214549769447, ... that is Pi = 3 + 1/8 + 1/61 + 1/5020 + ... see A001466 for details. Simon Plouffe Le 2018-12-08 à 06:01, Cris Moore a écrit :
The Egyptians were ok with integers, so I think they would have called one thousand one thousand, as opposed to 1+1/2+1/3+…1/e^1000 :-)
But it’s a lovely exercise that the greedy algorithm for Egyptian fractions — that is, subtract the largest reciprocal — terminates for any rational number between 0 and 1. There’s an inductive proof, but the nature of the induction is surprising.
- Cris
On Dec 7, 2018, at 9:46 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
Bill Gosper <billgosper@gmail.com> wrote:
Off-list I grumbled that no self-respecting Deity would bother sending clues to worshipers dumb enough to use decimal instead of continued fractions. Perhaps there's a hidden message in the continued fraction for e. :-)
Why not Egyptian fractions? The Egyptians, ancient and modern, have always been very religious.
Any positive real number can be approximated to any desired precision by an Egyptian fraction, i.e. the sum of reciprocals of distinct integers, a subset of the harmonic series. They aren't unique for a given real number, but the *greedy* Egyptian fraction for that real number is unique. Has anyone inspected the greedy Egyptian fraction for pi? The first few terms are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 744, 1173268, 2586625801171, 14348276635209672362238685, 1062286904072440687703470835520966381484062674280821 (A243020). See the pattern? Me neither. Except the first 12 terms. Maybe if it continued that way? After all, a friend of mine once confidently asserted that pi was infinite. (I think he may have been exaggerating.)
Speaking of infinite, try to find an Egyptian fraction for a thousand. Good luck. Unlike the Egyptian fraction for pi, it's finite. But it contains more terms than there are atoms in the known universe, most of them consecutive integers starting with 1.
What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original.
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I take it you mean this is the greedy Egyptian fraction of pi-3, since the series isn’t unique? - Cris
On Dec 7, 2018, at 11:57 PM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello, the Egyptian fraction of Pi - 3 is
8, 61, 5020, 128541455, 162924332716605980, 28783052231699298507846309644849796, 871295615653899563300996782209332544845605756266650946342214549769447, ...
that is Pi = 3 + 1/8 + 1/61 + 1/5020 + ...
see A001466 for details.
Simon Plouffe
Le 2018-12-08 à 06:01, Cris Moore a écrit :
The Egyptians were ok with integers, so I think they would have called one thousand one thousand, as opposed to 1+1/2+1/3+…1/e^1000 :-)
But it’s a lovely exercise that the greedy algorithm for Egyptian fractions — that is, subtract the largest reciprocal — terminates for any rational number between 0 and 1. There’s an inductive proof, but the nature of the induction is surprising.
- Cris
On Dec 7, 2018, at 9:46 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
Bill Gosper <billgosper@gmail.com> wrote:
Off-list I grumbled that no self-respecting Deity would bother sending clues to worshipers dumb enough to use decimal instead of continued fractions. Perhaps there's a hidden message in the continued fraction for e. :-)
Why not Egyptian fractions? The Egyptians, ancient and modern, have always been very religious.
Any positive real number can be approximated to any desired precision by an Egyptian fraction, i.e. the sum of reciprocals of distinct integers, a subset of the harmonic series. They aren't unique for a given real number, but the *greedy* Egyptian fraction for that real number is unique. Has anyone inspected the greedy Egyptian fraction for pi? The first few terms are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 744, 1173268, 2586625801171, 14348276635209672362238685, 1062286904072440687703470835520966381484062674280821 (A243020). See the pattern? Me neither. Except the first 12 terms. Maybe if it continued that way? After all, a friend of mine once confidently asserted that pi was infinite. (I think he may have been exaggerating.)
Speaking of infinite, try to find an Egyptian fraction for a thousand. Good luck. Unlike the Egyptian fraction for pi, it's finite. But it contains more terms than there are atoms in the known universe, most of them consecutive integers starting with 1.
What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original.
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What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original.
While there has been plenty commentary on Egyptian fractions, I’d like to suggest my favorite alternative, the Pierce and Engel fractions. Engel are defined by x= 1/a_1 + 1/(a_1a_2) + 1/(a_1a_2a_3) + … where u_1=x, a_k = ceil(1/u_k), u_{k+1}=u_ka_k-1. And a lovely obscure number representation was introduced by Marshall Hall Jr in the 1940s. Define a bounded continued fraction as one where the partial quotients can only be so big. For example, F(4) is the set of all continued fractions with partial quotients from 1 to 4. Hall proved that every real can be represented as the sum of two bounded continued fractions with bound 4, which he wrote as R=F(4)+F(4).
participants (4)
-
Cris Moore -
Keith F. Lynch -
Lucas, Stephen K - lucassk -
Simon Plouffe