8,092,015 is a 19,933-gonal and 809,203-gonal number.
Farey
Fractions and Funy Farey Fraction Addition:
The 775 fractions in the box below are
“Farey Fractions”:
They are all fractions between zero and
one, written in numeric order (smallest to largest), and written in simplest
form.
The numerators and denominators are all
integers between zero and 50.
They have an interesting property: if you
choose any three consecutive fractions, and you “Farey add” the first and last
fraction of the three, you will get the middle fraction (though you may have to
simplify the fraction).
Instead of adding fractions the normal way
we usually do, we add them the way we would add them if we did not know how to
add fractions. (It's funny but it works.) In this case we add the
denominators together to become the denominator of the answer, and we add the
numerators together to get the numerator of the answer. (I will note Farey addition by “+F”.)
Examples:
You see the three fractions highlighted in
yellow – 1/18, 2/35, and 1/17? Take the
first fraction and the third fraction and “Farey Add” them: 1/18 +F
1/17 = (1 + 1)/(18 + 17) = 2/35.
The numbers highlighted in blue are 5/41, 6/49,
1/8: 5/49 +F 1/8 = (5 + 1)/(41 + 8) = 6/49.
The numbers highlighted in green are 9/38,
5/21, and 11/46. 9/38 +F 11/46
= (9 + 11)/(38 + 46) = 20/84, which simplifies to 5/21.
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0/1, 1/50,
1/49, 1/48, 1/47,
1/46, 1/45, 1/44,
1/43, 1/42, 1/41,
1/40, 1/39, 1/38,
1/37, 1/36, 1/35,
1/34, 1/33, 1/32,
1/31, 1/30, 1/29,
1/28, 1/27, 1/26,
1/25, 2/49, 1/24,
2/47, 1/23, 2/45,
1/22, 2/43, 1/21,
2/41, 1/20, 2/39,
1/19, 2/37, 1/18, 2/35, 1/17, 3/50,
2/33, 3/49, 1/16,
3/47, 2/31, 3/46,
1/15, 3/44, 2/29,
3/43, 1/14, 3/41,
2/27, 3/40, 1/13,
3/38, 2/25, 3/37,
4/49, 1/12, 4/47,
3/35, 2/23, 3/34,
4/45, 1/11, 4/43,
3/32, 2/21, 3/31,
4/41, 1/10, 5/49,
4/39, 3/29, 5/48,
2/19, 5/47, 3/28,
4/37, 5/46, 1/9,
5/44, 4/35, 3/26,
5/43, 2/17, 5/42,
3/25, 4/33, 5/41, 6/49, 1/8,
6/47, 5/39, 4/31,
3/23, 5/38, 2/15,
5/37, 3/22, 4/29,
5/36, 6/43, 7/50,
1/7, 7/48, 6/41,
5/34, 4/27, 7/47,
3/20, 5/33, 7/46,
2/13, 7/45, 5/32,
3/19, 7/44, 4/25,
5/31, 6/37, 7/43,
8/49, 1/6, 8/47,
7/41, 6/35, 5/29,
4/23, 7/40, 3/17,
8/45, 5/28, 7/39,
9/50, 2/11, 9/49,
7/38, 5/27, 8/43,
3/16, 7/37, 4/21,
9/47, 5/26, 6/31,
7/36, 8/41, 9/46,
1/5, 10/49, 9/44,
8/39, 7/34, 6/29,
5/24, 9/43, 4/19,
7/33, 10/47, 3/14,
8/37, 5/23, 7/32,
9/41, 11/50, 2/9,
11/49, 9/40, 7/31,
5/22, 8/35, 11/48,
3/13, 10/43, 7/30,
11/47, 4/17, 9/38, 5/21, 11/46, 6/25,
7/29, 8/33, 9/37,
10/41, 11/45, 12/49,
1/4, 12/47, 11/43,
10/39, 9/35, 8/31,
7/27, 13/50, 6/23,
11/42, 5/19, 9/34,
13/49, 4/15, 11/41,
7/26, 10/37, 13/48,
3/11, 11/40, 8/29,
13/47, 5/18, 12/43,
7/25, 9/32, 11/39,
13/46, 2/7, 13/45,
11/38, 9/31, 7/24,
12/41, 5/17, 13/44,
8/27, 11/37, 14/47,
3/10, 13/43, 10/33,
7/23, 11/36, 15/49,
4/13, 13/42, 9/29,
14/45, 5/16, 11/35,
6/19, 13/41, 7/22,
15/47, 8/25, 9/28,
10/31, 11/34, 12/37,
13/40, 14/43, 15/46,
16/49, 1/3, 17/50,
16/47, 15/44, 14/41,
13/38, 12/35, 11/32,
10/29, 9/26, 17/49,
8/23, 15/43, 7/20,
13/37, 6/17, 17/48,
11/31, 16/45, 5/14,
14/39, 9/25, 13/36,
17/47, 4/11, 15/41,
11/30, 18/49, 7/19,
17/46, 10/27, 13/35,
16/43, 3/8, 17/45,
14/37, 11/29, 19/50,
8/21, 13/34, 18/47,
5/13, 17/44, 12/31,
19/49, 7/18, 16/41,
9/23, 11/28, 13/33,
15/38, 17/43, 19/48,
2/5, 19/47, 17/42,
15/37, 13/32, 11/27,
20/49, 9/22, 16/39,
7/17, 19/46, 12/29,
17/41, 5/12, 18/43,
13/31, 21/50, 8/19,
19/45, 11/26, 14/33,
17/40, 20/47, 3/7,
19/44, 16/37, 13/30,
10/23, 17/39, 7/16,
18/41, 11/25, 15/34,
19/43, 4/9, 21/47,
17/38, 13/29, 22/49,
9/20, 14/31, 19/42,
5/11, 21/46, 16/35,
11/24, 17/37, 23/50,
6/13, 19/41, 13/28,
20/43, 7/15, 22/47,
15/32, 23/49, 8/17,
17/36, 9/19, 19/40,
10/21, 21/44, 11/23,
23/48, 12/25, 13/27,
14/29, 15/31, 16/33,
17/35, 18/37, 19/39,
20/41, 21/43, 22/45,
23/47, 24/49, 1/2,
25/49, 24/47, 23/45,
22/43, 21/41, 20/39,
19/37, 18/35, 17/33,
16/31, 15/29, 14/27,
13/25, 25/48, 12/23,
23/44, 11/21, 21/40,
10/19, 19/36, 9/17,
26/49, 17/32, 25/47,
8/15, 23/43, 15/28,
22/41, 7/13, 27/50,
20/37, 13/24, 19/35,
25/46, 6/11, 23/42,
17/31, 11/20, 27/49,
16/29, 21/38, 26/47,
5/9, 24/43, 19/34,
14/25, 23/41, 9/16,
22/39, 13/23, 17/30,
21/37, 25/44, 4/7,
27/47, 23/40, 19/33,
15/26, 26/45, 11/19,
29/50, 18/31, 25/43,
7/12, 24/41, 17/29,
27/46, 10/17, 23/39,
13/22, 29/49, 16/27,
19/32, 22/37, 25/42,
28/47, 3/5, 29/48,
26/43, 23/38, 20/33,
17/28, 14/23, 25/41,
11/18, 30/49, 19/31,
27/44, 8/13, 29/47,
21/34, 13/21, 31/50,
18/29, 23/37, 28/45,
5/8, 27/43, 22/35,
17/27, 29/46, 12/19,
31/49, 19/30, 26/41,
7/11, 30/47, 23/36,
16/25, 25/39, 9/14,
29/45, 20/31, 31/48,
11/17, 24/37, 13/20,
28/43, 15/23, 32/49,
17/26, 19/29, 21/32,
23/35, 25/38, 27/41, 29/44,
31/47, 33/50, 2/3,
33/49, 31/46, 29/43,
27/40, 25/37, 23/34,
21/31, 19/28, 17/25,
32/47, 15/22, 28/41,
13/19, 24/35, 11/16,
31/45, 20/29, 29/42,
9/13, 34/49, 25/36,
16/23, 23/33, 30/43,
7/10, 33/47, 26/37,
19/27, 31/44, 12/17,
29/41, 17/24, 22/31,
27/38, 32/45, 5/7,
33/46, 28/39, 23/32,
18/25, 31/43, 13/18,
34/47, 21/29, 29/40,
8/11, 35/48, 27/37,
19/26, 30/41, 11/15,
36/49, 25/34, 14/19,
31/42, 17/23, 37/50,
20/27, 23/31, 26/35,
29/39, 32/43, 35/47,
3/4, 37/49, 34/45,
31/41, 28/37, 25/33,
22/29, 19/25, 35/46,
16/21, 29/38, 13/17,
36/47, 23/30, 33/43,
10/13, 37/48, 27/35,
17/22, 24/31, 31/40,
38/49, 7/9, 39/50,
32/41, 25/32, 18/23,
29/37, 11/14, 37/47,
26/33, 15/19, 34/43,
19/24, 23/29, 27/34,
31/39, 35/44, 39/49,
4/5, 37/46, 33/41,
29/36, 25/31, 21/26,
38/47, 17/21, 30/37,
13/16, 35/43, 22/27,
31/38, 40/49, 9/11,
41/50, 32/39, 23/28,
37/45, 14/17, 33/40,
19/23, 24/29, 29/35,
34/41, 39/47, 5/6,
41/49, 36/43, 31/37,
26/31, 21/25, 37/44,
16/19, 27/32, 38/45,
11/13, 39/46, 28/33,
17/20, 40/47, 23/27,
29/34, 35/41, 41/48,
6/7, 43/50, 37/43,
31/36, 25/29, 19/22,
32/37, 13/15, 33/38,
20/23, 27/31, 34/39,
41/47, 7/8, 43/49,
36/41, 29/33, 22/25,
37/42, 15/17, 38/43,
23/26, 31/35, 39/44,
8/9, 41/46, 33/37,
25/28, 42/47, 17/19,
43/48, 26/29, 35/39,
44/49, 9/10, 37/41,
28/31, 19/21, 29/32,
39/43, 10/11, 41/45,
31/34, 21/23, 32/35,
43/47, 11/12, 45/49,
34/37, 23/25, 35/38,
12/13, 37/40, 25/27,
38/41, 13/14, 40/43,
27/29, 41/44, 14/15,
43/46, 29/31, 44/47,
15/16, 46/49, 31/33,
47/50, 16/17, 33/35,
17/18, 35/37, 18/19,
37/39, 19/20, 39/41,
20/21, 41/43, 21/22,
43/45, 22/23, 45/47,
23/24, 47/49, 24/25,
25/26, 26/27, 27/28,
28/29, 29/30, 30/31,
31/32, 32/33, 33/34,
34/35, 35/36, 36/37,
37/38, 38/39, 39/40,
40/41, 41/42, 42/43,
43/44, 44/45, 45/46,
46/47, 47/48, 48/49,
49/50, 1/1
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Sources:
David
For two consecutive fractions in the sequence a/b and c/d, cross multiplying them gives consecutive numbers, i.e. ad=bc-1.
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