Here are some cool arithmetic amusements and tricks that don't serve much of a practical purpose, but you can use them to amuse your friends. For each of these, you may find it interesting to continue the patterns shown, and also try to see why these patterns arise.

1⁄81 | = | 0.0123456789 |

1⁄9801 | = | 0.00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899 |

1⁄998001 | = | 0.000001002003004005006007008009010011012013014015016017018019020021022023...995996997998999 (i.e. the decimal expansion contains all 3-digit numbers from 000 to 999) |

Can you continue the pattern?

9 × 9 + 7 = 88

9 × 98 + 6 = 888

9 × 987 + 5 = 8,888

9 × 9,876 + 4 = 88,888

...

9 × 98 + 6 = 888

9 × 987 + 5 = 8,888

9 × 9,876 + 4 = 88,888

...

1 × 8 + 1 = 9

12 × 8 + 2 = 98

123 × 8 + 3 = 987

1,234 × 8 + 4 = 9,876

...

12 × 8 + 2 = 98

123 × 8 + 3 = 987

1,234 × 8 + 4 = 9,876

...

9 × 1 + 2 = 11

9 × 12 + 3 = 111

9 × 123 + 4 = 1,111

9 × 1,234 + 5 = 11,111

...

9 × 12 + 3 = 111

9 × 123 + 4 = 1,111

9 × 1,234 + 5 = 11,111

...