Reciprocal of 1,233 in decimal

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Monday, August 11, 2003 

On Monday, mathematicians are marking a special numerical date: 8/11/03, or 00081103, is the repeating sequence of 1/1233 when that fraction is written as a decimal.

1 / 1233 = 0.00081103, where those eight digits repeat infinitely.

If you multiply 1,233 by 81,103, then add one, you get exactly 100 million; no more, no less.

In base ten (decimal), to turn a repeating decimal of n digits into a fraction, make a fraction whose numerator is the repeating portion and whose denominator is the number 10ⁿ − 1, which is written with n nines. For example, use 999 for a 3-digit string, 9,999 for a 4-digit string, and so on.

• Example 1: Turn 0.[405] = 0.405405… into a fraction. 0.405405… = ${\frac {405}{999}}$ = ${\frac {135}{333}}$ = ${\frac {45}{111}}$ = ${\frac {15}{37}}$ With the help of a calculator, you can enter “15 / 37” and see that it indeed returns 0.405405…
• Example 2: Turn 0.[0003] into a fraction. ${\frac {3}{9,999}}$ = ${\frac {1}{3,333}}$

For a terminating decimal, the procedure is very similar, except that you’ll use 10ⁿ instead.

• Example 3: Turn 0.425 into a fraction. ${\frac {425}{1,000}}$ = ${\frac {85}{200}}$ = ${\frac {17}{40}}$

• Example 4: Turn 0.715 into a fraction. ${\frac {715}{1,000}}$ = ${\frac {143}{200}}$

Sources

• "[ ]" —  e.g. December 31, 1999
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