Fractions using the same digits as their decimal representation
Christian Boyer,
V0.1, September 26th, 2006
V0.2 (more bases: 17 to 24), October 5th, 2006

"Is there a finite decimal that uses the same digits as its reduced fractional representation?"
David Wilson, September 19th, 2006

Here is the most compact possible example, the only solution using 2 digits:

5 / 2 = 2.5

With 3 digits, only two solutions:

( 5 / 20 = 0.25 )
59 / 2 = 29.5

As some other solutions in this page, 5/20 = 0.25 is written between parenthesis because it is not an irreducible fraction, 5/20 can be simplified by 5, reduced to 1/4: it is not a real solution to the initial problem. Because a solution has a finite decimal representation, a denominator of an irreducible solution can only be of the form: 2^k * 5^k'

My results found after the Wilson's question are summarized in this page:

A "compact" solution means a solution using a small number of digits. A "small" solution means a solution with a small value. The most beautiful solutions are in yellow. For any comment, send me a message!

• Using all digits, as a sudoku!

Using all digits from 1 to 9, as a sudoku, here is the smallest solution. Check each side exactly as a line in a sudoku puzzle: all digits -each used once- in the left side, and all digits -each used once- in the right side of the equality!

( 124983 / 576 = 216.984375 )

And using all digits from 0 to 9, here is the smallest solution:

( 2340819 / 576 = 4063.921875 )

Some other solutions exist using all digits, 1 to 9, or 0 to 9, as listed below in this page. But there is no solution using an irreducible fraction.

• Series -> ∞

Starting from 5/2=2.5, it is easy to construct this series of fractions:

5 / 2 = 2.5
59 / 2 = 29.5
599 / 2 = 299.5
5999 / 2 = 2999.5
...

Numerous other series -> ∞ exist. For example:

653 / 2 = 326.5
6053 / 2 = 3026.5
60053 / 2 = 30026.5
600053 / 2 = 300026.5
...

• Series -> 0

Starting again from 5/2=2.5, it is easy to construct this series of fraction:

5 / 2 = 2.5
( 5 / 20 = 0.25 )
( 5 / 200 = 0.025 )
( 5 / 2000 = 0.0025 )
...

Except its first term 5/2, this above series don't use irreducible fractions. Here is the most compact series using only irreducible fractions:

17537 / 12800 = 1.370078125
17537 / 128000 = 0.1370078125
17537 / 1280000 = 0.01370078125
...

17537 / 1280000.1370078125, and the next members of the series, have the same number of zeros on the left and on the right, counting the leading zero. Daniel Asimov asked if it is possible to have an irreducible fraction < 1 using the notation without the leading zero. Yes, here is the most compact solution, generating a new series:

322673 / 512000 = .630220703125
322673 / 5120000 = .0630220703125
322673 / 51200000 = .00630220703125
...

• Small solutions [1, 5/2]

We have seen series with infinite numbers of solutions < 1. And it easy quite easy to construct big solutions, with for example series -> ∞. But small solutions, say in [1, 5/2], are rare. And irreducible solutions are much more rare. Here are small known solutions:

( 12965 / 12800 = 1.012890625 )
( 6152 / 5120 = 1.2015625 )
17537 / 12800 = 1.370078125
( 2570 / 1280 = 2.0078125 )
( 57497 / 28160 = 2.041796875 )
5 / 2 = 2.5

Because their values are < 10, all of them generate series -> 0 when divided by 10, 100, 1000, and so on...

Who will give other solutions [1, 5/2]? And more particularly, solutions:
-using irreducible fractions smaller than 1.370078125?
-using reducible fraction smaller than 1.012890625?

• List of most compact solutions, from 2 to 8 digits

Download the full list of the 910 most compact solutions, from 2 to 8 digits (Excel file of 180Kb).

Some solutions from the Excel file:

Number of digits (number of solutions - number of solutions using only irreducible fractions)

2 digits (1 - 1)

5 / 2  =  2.5

3 digits (2 - 1)

( 5 / 20  =  0.25 )
59 / 2  =  29.5

4 digits (3 - 1)

( 5 / 200  =  0.025 )
599 / 2  =  299.5
653 / 2  =  326.5

5 digits (5 - 1)

( 5 / 2000  =  0.0025 )
5999 / 2  =  2999.5
6053 / 2  =  3026.5
6539 / 2  =  3269.5
6593 / 2  =  3296.5

6 digits (14 - 6)

( 5 / 20000  =  0.00025 )
( 9315 / 72  =  129.375 )
( 7052 / 32  =  220.375 )
( 22514 / 8  =  2814.25 )
( 35042 / 8  =  4380.25 )
( 65174 / 8  =  8146.75 )
59999 / 2  =  29999.5
60053 / 2  =  30026.5
60539 / 2  =  30269.5
60593 / 2  =  30296.5
65399 / 2  =  32699.5
65939 / 2  =  32969.5
65993 / 2  =  32996.5
66533 / 2  =  33266.5

7 digits (103 - 36)

( 5 / 200000 = 0.000025 )
( 9680 / 512 = 18.90625 )
15269 / 80 = 190.8625
35447 / 80 = 443.0875
( 35271 / 48 = 734.8125 )
...
665393 / 2 = 332696.5
665933 / 2 = 332966.5

8 digits (782 - 228)

( 5 / 2000000 = 0.0000025 )
( 6152 / 5120 = 1.2015625 )
( 2570 / 1280 = 2.0078125 )
( 37485 / 960 = 39.046875 )
( 38295 / 960 = 39.890625 )
...
8571425 / 2 = 4285712.5
8574251 / 2 = 4287125.5

• List of solutions using all digits

With all digits from 1 to 9, each used only once... exactly as a SUDOKU puzzle..., 2 solutions:

( 124983 / 576 = 216.984375 )
( 8954631 / 72 = 124369.875 )

With all digits from 0 to 9, each used only once, 5 solutions:

( 2340819 / 576 = 4063.921875 )
( 2409183 / 576 = 4182.609375 )
( 4023189 / 576 = 6984.703125 )
( 4138209 / 576 = 7184.390625 )
( 4309281 / 576 = 7481.390625 )

576 = 2^6 * 3^2 is the preferred denominator. The only other one is 72 = 2^3 * 3^2 = 576/8. With these denominators, all the numerators are of course multiples of 3^2, avoiding infinite decimal representation.

• Other bases

It is more natural for us to read and understand numbers when they are written in our classical base 10. But the same problem exists when numbers are written in other bases, and the solutions are of course completely different.

The results of the study on other bases are summarized in the table below. Two surprises:

1) I am unable to find at least one solution in base 4. Why? Who will provide at least one solution, or a proof of impossibility in base 4?

2) Our base 10 is the kindest base of all the analyzed bases, with its most compact solution 5/2 = 2.5 using only 2 digits. No other base 24 has a so compact solution with only 2 digits! The next kind bases allowing solutions with 2 digits are not in the table: big bases 30 (A/3=3.A), 36 (J/4=4.J), 60 (/6=6.), 68 (H/4=4.H),...

 Base b Nb digits Fraction written (*) inbase b Value written (*) inbase b Fraction written in base 10 Valuewritten in base 10 Comments 2 Impossible (**) 3 Impossible (**) 4 Unknown or impossible ??? 5 Impossible (**) 6 3 35 / 4 5.43 23 / 4 5.75 Only solution -in this base- using 3 digits 7 Impossible (**) 8 8 ( 6204501 / 6 ) 1026065.4 (1640769 / 6) 273461.5 Smallest of the 9 solutions (***) using 8 digits 9 6 ( 61014 / 6 ) 10146.6 (40108 / 6) 6684.666... Smallest of the 3 solutions (***) using 6 digits 10 2 5 / 2 2.5 5 / 2 2.5 Only solution -in this base- using 2 digits 11 Impossible (**) 12 3 14 / 9 1.94 16 / 9 1.777... Only solution -in this base- using 3 digits 13 Impossible (**) 14 4 ( A73 / C ) C3.A7 2061 / 12 171.75 Smallest of the 2 solutions (***) using 4 digits 15 6 ( 5C12 / 39 ) 192.C35 (19592 / 54) 362.814... Smallest of the 5 solutions (***) using 6 digits 16 5 ( D611 / C ) 11D6.C (54801 / 12) 4566.75 Only solution -in this base- using 5 digits 17 Impossible (**) 18 3 49 / 2 24.9 81 / 2 40.5 Only solution -in this base- using 3 digits 19 Impossible (**) 20 3 25 / G 2.G5 45 / 16 2.8125 Smallest of the 2 solutions (***) using 3 digits 21 5 (22EA / J) 2A2.JE (19708 / 18) 1094.888... Smallest of the 3 solutions (***) using 5 digits 22 4 (BGM / C) MC.GB (5697 / 12) 474.75 Only solution -in this base- using 2 digits 23 Impossible (**) 24 3 JF / 4 4F.J 447 / 4 111.75 Only solution -in this base- using 3 digits

(*) Fractions and values, when written in bases > 10, use this notation:

 A 10 H 17 B 11 J 18 C 12 K 19 D 13 L 20 E 14 M 21 F 15 N 22 G 16 P 23

(**) Prime bases can't have solutions, because the value of any fraction x/y,with x not a multiple of y, has infinite digits after the decimal point (hmm... in fact, we can't call it a "decimal" point...)

(***) Lists of multiple most compact solutions, bases 8, 9, 14, 15, 20, 21:

Base 8, using 8 digits, 9 solutions

(6204501 / 6 = 1026065.4)  (written in base 10 = 1640769 / 6 = 273461.5)
(6431065 / 6 = 1056663.4)  (written in base 10 = 1716789 / 6 = 286131.5)
(6504201 / 6 = 1066025.4)  (written in base 10 = 1738881 / 6 = 289813.5)
(6514027 / 6 = 1067256.4)  (written in base 10 = 1742871 / 6 = 290478.5)
(6541027 / 6 = 1072656.4)  (written in base 10 = 1753623 / 6 = 292270.5)
(7011423 / 6 = 1126730.4)  (written in base 10 = 1839891 / 6 = 306648.5)
(7041123 / 6 = 1132670.4)  (written in base 10 = 1851987 / 6 = 308664.5)
(7143145 / 6 = 1145673.4)  (written in base 10 = 1885797 / 6 = 314299.5)
(7314541 / 6 = 1167345.4)  (written in base 10 = 1939809 / 6 = 323301.5)

Base 9, using 6 digits, 3 solutions

(61014 / 6 = 10146.6)  (written in base 10 = 40108 / 6 = 6684.666...)
(64101 / 6 = 10614.6)  (written in base 10 = 42364 / 6 = 7060.666...)
(76214 / 6 = 12476.6)  (written in base 10 = 50476 / 6 = 8412.666...)

Base 14, using 4 digits, 2 solutions

(A73/C = C3.A7)  (written in base 10 = 2061 / 12 = 171.75)
A57/2 = 52A.7  (written in base 10 = 2037 / 2 = 1018.5)

Base 15, using 6 digits, 5 solutions

(5C12 / 39 = 192.C35)  (written in base 10 = 19592 / 54 = 362.814814...)
(9A12A / 6 = 192A6.A)  (written in base 10 = 489640 / 6 = 81606.666...)
(A29A1 / 6 = 1A692.A)  (written in base 10 = 515176 / 6 = 85862.666...)
(D252A / 6 = 22D56.A)  (written in base 10 = 666040 / 6 = 111006.666...)
(D25A2 / 6 = 22D6A.5)  (written in base 10 =  666152 / 6 = 111025.333...)

Base 20, using 3 digits, 2 solutions

25 / G = 2.G5  (written in base 10 = 45 / 16 = 2.8125)
A5 / 2 = 52.A  (written in base 10 = 205 / 2 = 102.5)

Base 21, using 5 digits, 3 solutions

(22EA / J = 2A2.JE)  (written in base 10 = 19708 / 18 = 1094.888...)
(G57L / J = JL5.G7) (written in base 10 = 150548 / 18 = 8363.777...)
(J17A / C = 1ACJ.7) (written in base 10 = 167296 / 12 = 13941.333...)