Egyptians had a number system that had no fractions the way we know them. Instead, they used integers, and the inverse numbers of integers. To express other fractions, they used sums of inverse numbers of integers, and to make it even more challenging, required that the sums did not involve duplicates. Eventually the number system evolved the include a symbol for 2/3. I've heard it said that this peculiar kind of math kept Egyptian priests in power for thousands of years as only they knew how to do fractions like that, and math was not allowed to get easier so as to keep others from taking the jobs. I've also heard that this was one of the more important reasons about why the Egyptians managed to build the pyramids, and very little after that for the next couple of millenniums until the Romans came.
Thus, for example Egyptians wrote 3/4 as 1/2 + 1/4; and 7/12 as 1/3 + 1/4.
Questions are..
* how do you write 2/3 as a sum of different inverse numbers?
* is there a method to write any fraction m/n as a sum of inverse numbers?
Originally posted by talzamir2/3 = 1/2 + 1/6 which is kind of trivial (use 4/6 which is 1/6 + 3/6)
Egyptians had a number system that had no fractions the way we know them. Instead, they used integers, and the inverse numbers of integers. To express other fractions, they used sums of inverse numbers of integers, and to make it even more challenging, required that the sums did not involve duplicates. Eventually the number system evolved the include a symb ...[text shortened]... t inverse numbers?
* is there a method to write any fraction m/n as a sum of inverse numbers?
Probably you can, given enough numbers, but it will be quite difficult. To prove that it is possible is probably very difficult. But I would try to expand any fraction you care to give me (not more than 3 examples in all)
Originally posted by PonderableI think I figured it out.
2/3 = 1/2 + 1/6 which is kind of trivial (use 4/6 which is 1/6 + 3/6)
Probably you can, given enough numbers, but it will be quite difficult. To prove that it is possible is probably very difficult. But I would try to expand any fraction you care to give me (not more than 3 examples in all)
I'll try it on the fraction 10/17.
10/17 is greater than 1/2, so I'll use 1/2 as my first term.
1/2 + 3/34 = 10/17
Now to work on the 3/34. I need to multiply the numerator by something and make it bigger than the denominator to see which term I can extract next. 3*12 = 36 does the trick. 1/12 is term #2.
1/2 + 1/12 + 2/408
But the last term reduces to 1/204, so I'm done.
1/2 + 1/12 + 1/204 = 10/17
Originally posted by wolfgang59Oh, certainly the Roman number system had its flaws too. Not fun at all to try LXVI times XCIX. But the 12-based system by which Romans handled fractions is actually pretty neat.
And the Romans were better mathematicians!?!?!
I don't think so!
SwissGambit's method.. is this how it works?
Starting with 11/17.
17/11 = 1 6/11 which rounds up to two so that's the first denominator.
11/17 - 1/2 = 22/34 - 17/34 = 5/34
34/5 = 6 4/5 which rounds up to seven, so
5/34 - 1/7 = 35/238 - 34/238 = 1/238
so 11/17 = 1/2 + 1/7 + 1/238.
Or,
10/17; 17/10 = 1.7 -> first denominator is 2.
10/17 - 1/2 = 20/34 - 17/34 = 3 /34
34 / 3 = 11 1/3 -> next denominator is 12.
3/34 - 1/12 = 18/204 -17/204 = 1/204
so 10/17 = 1/2 + 1/12 + 1/204
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Nice method! it's most efficient if you round to nearest, and allow terms to be subtracted as well as added
i.e.
3 + 1/7 - 1/791 = pi to 6 decimal places
which is entry A001567 in the encyclopedia of integer sequences
http://oeis.org/A001467
if you round up and therefore only allow terms to be added you get
1 + 1/8 + 1/61
which is right to 5 decimal places and is entry A001466 in the encyclopedia
http://oeis.org/A001466
there is a common bad example of 5/121
Originally posted by iamatigerFor 3 + 1/8 + 1/61, I get 3.1413934426, which is only right to 3 decimal places?!
Nice method! it's most efficient if you round to nearest, and allow terms to be subtracted as well as added
i.e.
3 + 1/7 - 1/791 = pi to 6 decimal places
which is entry A001567 in the encyclopedia of integer sequences
http://oeis.org/A001467
if you round up and therefore only allow terms to be added you get
1 + 1/8 + 1/61
which is right to ...[text shortened]... A001466 in the encyclopedia
http://oeis.org/A001466
there is a common bad example of 5/121
Originally posted by iamatigerI'm not sure allowing negative terms makes it more efficient.
Nice method! it's most efficient if you round to nearest, and allow terms to be subtracted as well as added
i.e.
3 + 1/7 - 1/791 = pi to 6 decimal places
which is entry A001567 in the encyclopedia of integer sequences
http://oeis.org/A001467
if you round up and therefore only allow terms to be added you get
1 + 1/8 + 1/61
which is right to ...[text shortened]... A001466 in the encyclopedia
http://oeis.org/A001466
there is a common bad example of 5/121
Let's take 1/8 + 1/61 = 69/488 as an example.
With negative terms allowed, and always rounding to the nearest fraction, I get 1/7 - 1/683 +1/2333128 = 69/488, which is a very ugly result!
Originally posted by talzamirWell they could go as small as 1/1728 (1/12 * 1/12 * 1/12) and that is all well
Oh, certainly the Roman number system had its flaws too. ... But the 12-based system by which Romans handled fractions is actually pretty neat.
and good for practical work but not useful for math.
Incidently this site
http://www.curiousnotions.com/fractions/index.asp#fractions
has a converter to change any fraction into Roman fractions.
0.1 equates to 1/12 + 1/72 + 5/1728 (=173/1728)
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Originally posted by SwissGambitNice counter example, neither method is guaranteed to find the best although I suspect rounding to nearest is better on average.
I'm not sure allowing negative terms makes it more efficient.
Let's take 1/8 + 1/61 = 69/488 as an example.
With negative terms allowed, and always rounding to the nearest fraction, I get 1/7 - 1/683 +1/2333128 = 69/488, which is a very ugly result!
I can easily make the rounding up method look worse by picking e.g
1/7 - 1/13 = 6/91
rounding to nearest finds
1/15 - 1/1365
rounding up finds
1/6 + 1/292 + 1/106288
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After a small simulation of all fractions with denominator less than 123 and enumerator less than denominator, I found only a small difference between the two methods, with the rounding-up method giving on average 3.2 fractions and the rounding-nearest method giving on average 3.0 fractions.
However looking at the largest denominator there was a huge difference:
Rounding up does worst for 8/97 which it turns to:
1/13 + 1/181 + 1/38041 + 1/1736503177 + 1/3769304102927363485
+ 1/18943537893793408504192074528154430149
+ 1/538286441900380211365817285104907086347439746130226973253778132494225813153
+ 1/[WORD TOO LONG]
whereas the worst case for rounding to nearest is much better at 47/73 with
1/1 - 1/4 - 1/13 - 1/543 - 1/412246 - 1/424866499045 -1/180511542010330119412980
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I've just noticed wolfram alpha does it, which proves my perl simulation is right at least!
http://www.wolframalpha.com/input/?i=8%2F97+egyptian+fraction
and the super cool doofer here: (I wonder how it works?)
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fractions/egyptian.html#shorttable
finds 8/97 = 1/13 + 1/182 + 1/17654 to be the best possible with all positive terms
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Rounding to nearest still wins when the denominator of the fraction to be decomposed ranges from 2..1000
on average to nearest needs 3.956 fractions, but rounding up needs 4.213 fractions
The smallest term in rounding to nearest is at 317/739 and is:
1/915546455684058748900662753016244979642885301214026384244483872977
53186075546741559438139320114993643646791204462214307919040483553384
9454145910939822549839466435632130
The smallest term in rounding up is at 36/457 and is:
1/839018826833450186636781520007011999269820404906753180244759299287
83737889539760561326146999562649871928983511239253043084051410214699
86256665947569952734180156000234940492081088941857817740026830632042
52356172520941088783702738286944210460710059319691268110283467445381
02665362859976568473910538864231004478584490215707691900373523154378
17850733931761441676882524465414164664186084654585029979714254283427
69433127784560570193376772878336217849260872114137931351960543608384
24400950566425317387570523488957085392410564019361930133277698968824
85550270543952379075819512618682808991505743601648001879641672743230
78311078867593844043149124596271281252530924719121766925749760855109
10006673184147826281268664269339589622998374522627779305582060905834
82691521900836957046857696220116551591742723266473426955898181271263
03038171968768650476413027459205291075571637957597356820188031655122
74974365230126839454212397089242294433585791764163604189219254713517
81536020388776776143582815811036855260413298414968634103058882552344
95015115912388514981113593387572720476744188169200130515719608747338
81013672826778401335239691097990454591345853624332731197780512641006
55769612376408248521143288840865815420914926003128384256669276276742
27053793897767395465326589843035773944346372949759909905561209334216
84715815664488428130051269991053009287091906187661577070851924381867
63662454774620422942676746779547837269903493861174680719328740210237
14524610740225814235147693954027910741673103980749749728106483987721
60273867317300936280233709290884779749947589534711288933950292840780
80586702977221756866386787887386898039455740028056772504632864793636
70076942509109589495377221095405979217163821481666646160815221224686
56253053611661364530533592281952403782987896151817017796876836485339
90573577721416556223812801969086370315564364614042859304264369836581
06288733881761514992109680298995922754466040011586713812553117621857
10951725894384600417943252113184415624242835127018880391955439862008
46685140545044140622760122924973752382108865950062494534604147901476
11422121782194848803348777061816460876697945418158442269512987729152
44194032646663161042490615823728821870644796311301923955788548664731
40853576518952261173647603153943546245479192091385391808078296725459
24239541758108877100331729470119526373928796447673951888289511964811
63302536982115669593455710342992106338796504671507010291681197655258
44641539812142776225973081134493204623416830552005765719102416866159
24531368198770946893858410058348221985603151428153382461711196734214
08585252377842263090764623590075231757102213156942123119632908002395
23647885443014954220610660369117723857396599976655038324445297135442
86955548310166168837889046149061296461059432238621602179724809510024
77212749708025840169492997310518483221462278567965155036846552482106
2859837409907538269572622296774545103747438431266995525592705