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Wersja z 2015-09-12

Grzegorz Jagodziński

Ułamki egipskie

1 2 3 4 5 6

Tabela rozkładu niektórych ułamków na dwa składniki

ułamek warunek `a` różnica rozkład  
mianownik 3
`2/3` `1.5 < a < 3` `2` `2/3 - 1/2 = 1/6` `2/3 = 1/2 + 1/6`  
mianownik 4
`3/4` `1.33 < a < 2.67` `2` `3/4 - 1/2 = 1/4` `3/4 = 1/2 + 1/4`  
mianownik 5
`2/5` `2.5 < a < 5` `4` `2/5 - 1/4 = 3/20`  
`3` `2/5 - 1/3 = 1/15` `2/5 = 1/3 + 1/15`
`3/5` `1.67 < a < 3.33` `3` `3/5 - 1/3 = 4/15`  
`2` `3/5 - 1/2 = 1/10` `3/5 = 1/2 + 1/10`
`4/5` `1.25 < a < 2.5` `2` `4/5 - 1/2 = 3/10` *
mianownik 6
`5/6` `1.2 < a < 2.4` `2` `5/6 - 1/2 = 1/3` `5/6 = 1/2 + 1/3`  
mianownik 7
`2/7` `3.5 < a < 7` `6` `2/7 - 1/6 = 5/42`  
`5` `2/7 - 1/5 = 3/35`
`4` `2/7 - 1/4 = 1/28` `2/7 = 1/4 + 1/28`
`3/7` `2.33 < a < 4.67` `4` `3/7 - 1/4 = 5/28` *
`3` `3/7 - 1/3 = 2/21`
`4/7` `1.75 < a < 3.5` `3` `4/7 - 1/3 = 5/21`  
`2` `4/7 - 1/2 = 1/14` `4/7 = 1/2 + 1/14`
`5/7` `1.4 < a < 2.8` `2` `5/7 - 1/2 = 3/14` *
`6/7` `1.17 < a < 2.33` `2` `6/7 - 1/2 = 5/14` *
mianownik 8
`3/8` `2.67 < a < 5.33` `5` `3/8 - 1/5 = 7/40`  
`4` `3/8 - 1/4 = 1/8` `3/8 = 1/4 + 1/8`
`3` `3/8 - 1/3 = 1/24` `3/8 = 1/3 + 1/24`
`5/8` `1.6 < a < 3.2` `3` `5/8 - 1/3 = 7/24`  
`2` `5/8 - 1/2 = 1/8` `5/8 = 1/2 + 1/8`
`7/8` `1.14 < a < 2.29` `2` `7/8 - 1/2 = 3/8` *
mianownik 9
`2/9` `4.5 < a < 9` `8` `2/9 - 1/8 = 7/72`  
`7` `2/9 - 1/7 = 5/63`
`6` `2/9 - 1/6 = 1/18` `2/9 = 1/6 + 1/18`
`5` `2/9 - 1/5 = 1/45` `2/9 = 1/5 + 1/45`
`4/9` `2.25 < a < 4.5` `4` `4/9 - 1/4 = 7/36`  
`3` `4/9 - 1/3 = 1/9` `4/9 = 1/3 + 1/9`
`5/9` `1.8 < a < 3.6` `3` `5/9 - 1/3 = 2/9`  
`2` `5/9 - 1/2 = 1/18` `5/9 = 1/2 + 1/18`
`7/9` `1.29 < a < 2.57` `2` `7/9 - 1/2 = 5/18` *
`8/9` `1.13 < a < 2.25` `2` `8/9 - 1/2 = 7/18` *
mianownik 10
`3/10` `3.33 < a < 6.67` `6` `3/10 - 1/6 = 2/15`  
`5` `3/10 - 1/5 = 1/10` `3/10 = 1/5 + 1/10`
`4` `3/10 - 1/4 = 1/20` `3/10 = 1/4 + 1/20`
`7/10` `1.43 < a < 2.86` `2` `7/10 - 1/2 = 1/5` `7/10 = 1/2 + 1/5`  
`9/10` `1.11 < a < 2.22` `2` `9/10 - 1/2 = 2/5` *
mianownik 11
`2/11` `5.5 < a < 11` `10` `2/11 - 1/10 = 9/110`  
`9` `2/11 - 1/9 = 7/99`
`8` `2/11 - 1/8 = 5/88`
`7` `2/11 - 1/7 = 3/77`
`6` `2/11 - 1/6 = 1/66` `2/11 = 1/6 + 1/66`
`3/11` `3.67 < a < 7.33` `7` `3/11 - 1/7 = 10/77`  
`6` `3/11 - 1/6 = 7/66`
`5` `3/11 - 1/5 = 4/55`
`4` `3/11 - 1/4 = 1/44` `3/11 = 1/4 + 1/44`
`4/11` `2.75 < a < 5.5` `5` `4/11 - 1/5 = 9/55`  
`4` `4/11 - 1/4 = 5/44`
`3` `4/11 - 1/3 = 1/33` `4/11 = 1/3 + 1/33`
`5/11` `2.2 < a < 4.4` `4` `5/11 - 1/4 = 9/44` *
`3` `5/11 - 1/3 = 4/33`
`6/11` `1.83 < a < 3.67` `3` `6/11 - 1/3 = 7/33`  
`2` `6/11 - 1/2 = 1/22` `6/11 = 1/2 + 1/22`
`7/11` `1.57 < a < 3.14` `3` `7/11 - 1/3 = 10/33` *
`2` `7/11 - 1/2 = 3/22`
`8/11` `1.38 < a < 2.75` `2` `8/11 - 1/2 = 5/22` *
`9/11` `1.22 < a < 2.44` `2` `9/11 - 1/2 = 7/22` *
`10/11` `1.1 < a < 2.2` `2` `10/11 - 1/2 = 9/22` *
mianownik 12
`5/12` `2.4 < a < 4.8` `4` `5/12 - 1/4 = 1/6` `5/12 = 1/4 + 1/6`  
`3` `5/12 - 1/3 = 1/12` `5/12 = 1/3 + 1/12`
`7/12` `1.71 < a < 3.43` `3` `7/12 - 1/3 = 1/4` `7/12 = 1/3 + 1/4`  
`2` `7/12 - 1/2 = 1/12` `7/12 = 1/2 + 1/12`
`11/12` `1.09 < a < 2.18` `2` `11/12 - 1/2 = 5/12` *

Gwiazdka w ostatniej kolumnie oznacza, że rozkład na dwa składniki nie istnieje i należy odszukać ułamek w następnej tabeli.