Expansions of unit fractions

Introduction

Here are expansions of unit fractions in different bases. Notes:

In every table, on each row, all the cells have the same values (represented in a different way).
In bases greater than 10, letters a–j denote digits 10–19.
In the “expansion” columns, the non-repeating and repeating parts have been separated with a space because text-based browsers may not display the overlines over the repeating parts. For example, 0.0 45 means 0.0454545…
In each table, all numbers are in the specified base. For example, in the octal table, 10¹² means 8¹⁰.
The tables were created with this Python program.
See also “Repeating decimal” on Wikipedia.

In decimal

Unit fraction	Intermediate form	Expansion
1/2	5 / 10	0.5
1/3	3 / 9	0. 3
1/4	25 / 100	0.25
1/5	2 / 10	0.2
1/6	(1 + 6 / 9) / 10	0.1 6
1/7	142857 / (10⁶−1)	0. 142857
1/8	125 / 1000	0.125
1/9	1 / 9	0. 1
1/10	1 / 10	0.1
1/11	9 / 99	0. 09
1/12	(8 + 3 / 9) / 100	0.08 3
1/13	76923 / (10⁶−1)	0. 076923
1/14	714285 / (10⁶−1) / 10	0.0 714285
1/15	6 / 9 / 10	0.0 6
1/16	625 / 10⁴	0.0625
1/17	588235294117647 / (10¹⁶−1)	0. 0588235294117647
1/18	5 / 9 / 10	0.0 5
1/19	52631578947368421 / (10¹⁸−1)	0. 052631578947368421
1/20	5 / 100	0.05
1/21	47619 / (10⁶−1)	0. 047619
1/22	45 / 99 / 10	0.0 45
1/23	434782608695652173913 / (10²²−1)	0. 0434782608695652173913
1/24	(41 + 6 / 9) / 1000	0.041 6
1/25	4 / 100	0.04
1/26	384615 / (10⁶−1) / 10	0.0 384615
1/27	37 / 999	0. 037
1/28	(3 + 571428 / (10⁶−1)) / 100	0.03 571428
1/29	344827586206896551724137931 / (10²⁸−1)	0. 0344827586206896551724137931
1/30	3 / 9 / 10	0.0 3
1/31	32258064516129 / (10¹⁵−1)	0. 032258064516129
1/32	3125 / 10⁵	0.03125

In binary

Unit fraction	Intermediate form	Expansion
1/10	1 / 10	0.1
1/11	1 / 11	0. 01
1/100	1 / 100	0.01
1/101	11 / (10¹⁰⁰−1)	0. 0011
1/110	1 / 11 / 10	0.0 01
1/111	1 / 111	0. 001
1/1000	1 / 1000	0.001
1/1001	111 / (10¹¹⁰−1)	0. 000111
1/1010	11 / (10¹⁰⁰−1) / 10	0.0 0011
1/1011	1011101 / (10¹⁰¹⁰−1)	0. 0001011101
1/1100	1 / 11 / 100	0.00 01
1/1101	100111011 / (10¹¹⁰⁰−1)	0. 000100111011
1/1110	1 / 111 / 10	0.0 001
1/1111	1 / (10¹⁰⁰−1)	0. 0001
1/10000	1 / 10¹⁰⁰	0.0001
1/10001	1111 / (10¹⁰⁰⁰−1)	0. 00001111
1/10010	111 / (10¹¹⁰−1) / 10	0.0 000111
1/10011	11010111100101 / (10¹⁰⁰¹⁰−1)	0. 000011010111100101
1/10100	11 / (10¹⁰⁰−1) / 100	0.00 0011
1/10101	11 / (10¹¹⁰−1)	0. 000011
1/10110	1011101 / (10¹⁰¹⁰−1) / 10	0.0 0001011101
1/10111	1011001 / (10¹⁰¹¹−1)	0. 00001011001
1/11000	1 / 11 / 1000	0.000 01
1/11001	1010001111010111 / (10¹⁰¹⁰⁰−1)	0. 00001010001111010111
1/11010	100111011 / (10¹¹⁰⁰−1) / 10	0.0 000100111011
1/11011	10010111101101 / (10¹⁰⁰¹⁰−1)	0. 000010010111101101
1/11100	1 / 111 / 100	0.00 001
1/11101	100011010011110111001011 / (10¹¹¹⁰⁰−1)	0. 0000100011010011110111001011
1/11110	1 / (10¹⁰⁰−1) / 10	0.0 0001
1/11111	1 / (10¹⁰¹−1)	0. 00001
1/100000	1 / 10¹⁰¹	0.00001

In octal

Unit fraction	Intermediate form	Expansion
1/2	4 / 10	0.4
1/3	25 / 77	0. 25
1/4	2 / 10	0.2
1/5	1463 / (10⁴−1)	0. 1463
1/6	(1 + 25 / 77) / 10	0.1 25
1/7	1 / 7	0. 1
1/10	1 / 10	0.1
1/11	7 / 77	0. 07
1/12	6314 / (10⁴−1) / 10	0.0 6314
1/13	564272135 / (10¹²−1)	0. 0564272135
1/14	52 / 77 / 10	0.0 52
1/15	473 / (10⁴−1)	0. 0473
1/16	4 / 7 / 10	0.0 4
1/17	421 / (10⁴−1)	0. 0421
1/20	4 / 100	0.04
1/21	3607417 / (10¹⁰−1)	0. 03607417
1/22	34 / 77 / 10	0.0 34
1/23	32745 / (10⁶−1)	0. 032745
1/24	3146 / (10⁴−1) / 10	0.0 3146
1/25	3 / 77	0. 03
1/26	2721350564 / (10¹²−1) / 10	0.0 2721350564
1/27	2620544131 / (10¹³−1)	0. 02620544131
1/30	25 / 77 / 10	0.0 25
1/31	2436560507534121727 / (10²⁴−1)	0. 02436560507534121727
1/32	2354 / (10⁴−1) / 10	0.0 2354
1/33	22755 / (10⁶−1)	0. 022755
1/34	2 / 7 / 10	0.0 2
1/35	215173454106475626043236713 / (10³⁴−1)	0. 0215173454106475626043236713
1/36	2104 / (10⁴−1) / 10	0.0 2104
1/37	2041 / (10⁵−1)	0. 02041
1/40	2 / 100	0.02

In hexadecimal

Unit fraction	Intermediate form	Expansion
1/2	8 / 10	0.8
1/3	5 / f	0. 5
1/4	4 / 10	0.4
1/5	3 / f	0. 3
1/6	(2 + a / f) / 10	0.2 a
1/7	249 / fff	0. 249
1/8	2 / 10	0.2
1/9	1c7 / fff	0. 1c7
1/a	(1 + 9 / f) / 10	0.1 9
1/b	1745d / (10⁵−1)	0. 1745d
1/c	(1 + 5 / f) / 10	0.1 5
1/d	13b / fff	0. 13b
1/e	(1 + 249 / fff) / 10	0.1 249
1/f	1 / f	0. 1
1/10	1 / 10	0.1
1/11	f / ff	0. 0f
1/12	e38 / fff / 10	0.0 e38
1/13	d79435e5 / (10⁹−1)	0. 0d79435e5
1/14	c / f / 10	0.0 c
1/15	c3 / fff	0. 0c3
1/16	ba2e8 / (10⁵−1) / 10	0.0 ba2e8
1/17	b21642c859 / (10^b−1)	0. 0b21642c859
1/18	a / f / 10	0.0 a
1/19	a3d7 / (10⁵−1)	0. 0a3d7
1/1a	9d8 / fff / 10	0.0 9d8
1/1b	97b425ed / (10⁹−1)	0. 097b425ed
1/1c	924 / fff / 10	0.0 924
1/1d	8d3dcb / (10⁷−1)	0. 08d3dcb
1/1e	8 / f / 10	0.0 8
1/1f	8421 / (10⁵−1)	0. 08421
1/20	8 / 100	0.08

In base 4

Unit fraction	Intermediate form	Expansion
1/2	2 / 10	0.2
1/3	1 / 3	0. 1
1/10	1 / 10	0.1
1/11	3 / 33	0. 03
1/12	2 / 3 / 10	0.0 2
1/13	21 / 333	0. 021
1/20	2 / 100	0.02
1/21	13 / 333	0. 013
1/22	12 / 33 / 10	0.0 12
1/23	1131 / (10¹¹−1)	0. 01131
1/30	1 / 3 / 10	0.0 1
1/31	10323 / (10¹²−1)	0. 010323
1/32	102 / 333 / 10	0.0 102
1/33	1 / 33	0. 01
1/100	1 / 100	0.01
1/101	33 / (10¹⁰−1)	0. 0033
1/102	32 / 333 / 10	0.0 032
1/103	3113211 / (10²¹−1)	0. 003113211
1/110	3 / 33 / 10	0.0 03
1/111	3 / 333	0. 003
1/112	2322 / (10¹¹−1) / 10	0.0 02322
1/113	230201121 / (10²³−1)	0. 00230201121
1/120	2 / 3 / 100	0.00 2
1/121	22033113 / (10²²−1)	0. 0022033113
1/122	21312 / (10¹²−1) / 10	0.0 021312
1/123	2113231 / (10²¹−1)	0. 002113231
1/130	21 / 333 / 10	0.0 021
1/131	203103313023 / (10³²−1)	0. 00203103313023
1/132	2 / 33 / 10	0.0 02
1/133	201 / (10¹¹−1)	0. 00201
1/200	2 / 1000	0.002

In base 6

Unit fraction	Intermediate form	Expansion
1/2	3 / 10	0.3
1/3	2 / 10	0.2
1/4	13 / 100	0.13
1/5	1 / 5	0. 1
1/10	1 / 10	0.1
1/11	5 / 55	0. 05
1/12	43 / 1000	0.043
1/13	4 / 100	0.04
1/14	3 / 5 / 10	0.0 3
1/15	313452421 / (10¹⁴−1)	0. 0313452421
1/20	3 / 100	0.03
1/21	24340531215 / (10²⁰−1)	0. 024340531215
1/22	23 / 55 / 10	0.0 23
1/23	2 / 5 / 10	0.0 2
1/24	213 / 10⁴	0.0213
1/25	204122453514331 / (10²⁴−1)	0. 0204122453514331
1/30	2 / 100	0.02
1/31	15211325 / (10¹³−1)	0. 015211325
1/32	(1 + 4 / 5) / 100	0.01 4
1/33	14 / 55 / 10	0.0 14
1/34	1345242103 / (10¹⁴−1) / 10	0.0 1345242103
1/35	1322030441 / (10¹⁵−1)	0. 01322030441
1/40	13 / 1000	0.013
1/41	1235 / (10⁵−1)	0. 01235
1/42	121502434053 / (10²⁰−1) / 10	0.0 121502434053
1/43	12 / 1000	0.012
1/44	(1 + 14 / 55) / 100	0.01 14
1/45	1124045443151 / (10²²−1)	0. 01124045443151
1/50	1 / 5 / 10	0.0 1
1/51	10545 / (10¹⁰−1)	0. 010545
1/52	1043 / 10⁵	0.01043
1/53	1031345242 / (10¹⁴−1) / 10	0.0 1031345242
1/54	1020412245351433 / (10²⁴−1) / 10	0.0 1020412245351433
1/55	1 / 55	0. 01
1/100	1 / 100	0.01

In base 12

Unit fraction	Intermediate form	Expansion
1/2	6 / 10	0.6
1/3	4 / 10	0.4
1/4	3 / 10	0.3
1/5	2497 / (10⁴−1)	0. 2497
1/6	2 / 10	0.2
1/7	186a35 / (10⁶−1)	0. 186a35
1/8	16 / 100	0.16
1/9	14 / 100	0.14
1/a	(1 + 2497 / (10⁴−1)) / 10	0.1 2497
1/b	1 / b	0. 1
1/10	1 / 10	0.1
1/11	b / bb	0. 0b
1/12	a35186 / (10⁶−1) / 10	0.0 a35186
1/13	9724 / (10⁴−1) / 10	0.0 9724
1/14	9 / 100	0.09
1/15	8579214b36429a7 / (10¹⁴−1)	0. 08579214b36429a7
1/16	8 / 100	0.08
1/17	76b45 / (10⁶−1)	0. 076b45
1/18	7249 / (10⁴−1) / 10	0.0 7249
1/19	6a3518 / (10⁶−1) / 10	0.0 6a3518
1/1a	6 / b / 10	0.0 6
1/1b	6316948421 / (10^b−1)	0. 06316948421
1/20	6 / 100	0.06
1/21	5915343a0b62a68781b / (10¹⁸−1)	0. 05915343a0b62a68781b
1/22	56 / bb / 10	0.0 56
1/23	54 / 1000	0.054
1/24	5186a3 / (10⁶−1) / 10	0.0 5186a3
1/25	4b7 / (10⁴−1)	0. 04b7
1/26	4972 / (10⁴−1) / 10	0.0 4972
1/27	478aa093598166b74311b28623a55 / (10²⁶−1)	0. 0478aa093598166b74311b28623a55
1/28	46 / 1000	0.046
1/29	4 / b / 10	0.0 4
1/2a	429a708579214b36 / (10¹⁴−1) / 10	0.0 429a708579214b36
1/2b	414559b3931 / (10¹⁰−1)	0. 0414559b3931
1/30	4 / 100	0.04

In base 20

Unit fraction	Intermediate form	Expansion
1/2	a / 10	0.a
1/3	6d / jj	0. 6d
1/4	5 / 10	0.5
1/5	4 / 10	0.4
1/6	(3 + 6d / jj) / 10	0.3 6d
1/7	2h / jj	0. 2h
1/8	2a / 100	0.2a
1/9	248hfb / (10⁶−1)	0. 248hfb
1/a	2 / 10	0.2
1/b	1g759 / (10⁵−1)	0. 1g759
1/c	(1 + d6 / jj) / 10	0.1 d6
1/d	1af7dgi94c63 / (10^c−1)	0. 1af7dgi94c63
1/e	(1 + 8b / jj) / 10	0.1 8b
1/f	(1 + 6d / jj) / 10	0.1 6d
1/g	15 / 100	0.15
1/h	13abf5hcig984e27 / (10^g−1)	0. 13abf5hcig984e27
1/i	(1 + 248hfb / (10⁶−1)) / 10	0.1 248hfb
1/j	1 / j	0. 1
1/10	1 / 10	0.1