Reciprocal of 1089
Jump to navigation
Jump to search
Theorem
The reciprocal of $1089$ is:
- $\dfrac 1 {1089} = 0 \cdotp \dot 00091 \, 82736 \, 45546 \, 37281 \, 9 \dot 1$
This sequence is A113657 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Note that it consists mainly of the unit multiples of $9$ from $1$ to $9$ concatenated.
Proof
Performing the calculation using long division:
0.00091827364554637281910009... -------------------------------------------- 1089)1.00000000000000000000000000... 9801 5445 1089 ---- ---- ---- 1990 5950 9910 1089 5445 9801 ---- ---- ---- 9010 5050 1090 8712 4356 1089 ---- ---- ---- 2980 6940 10000 2178 6534 9801 ---- ---- ---- 8020 4060 .... 7623 3267 ---- ---- 3970 7930 3267 7623 ---- ---- 7030 3070 6534 2178 ---- ---- 4960 8920 4356 8712 ---- ---- 6040 2080 5445 1089
$\blacksquare$
Historical Note
According to David Wells in his $1997$ work Curious and Interesting Numbers, 2nd ed., this result can be attributed to N. Goddwin, but no corroboration or expansion of this has yet been found.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1089$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1089$