# Multiplication by 9, 99, 999, etc.

One way to multiply a number by 9 is to multiply by 10 and then subtract the number from the product. There is another way to multiply fast by 9 and as the first one it has an analogue for multiplication by 99, 999 and all such numbers. Let's start with the multiplication by 9.

To multiply a one digit number *a* by 9, first subtract 1 and form *b* = *a* - 1.*b* from 9: *c* = 9 - *b*.*b* and *c* next to each other:

9*a* = *b**c*.

For example, find 6×9 (so that *a* = 6.) First subtract:

Next, find 37×99. First, subtract 1:

Why does this work? For the multiplication by 9, *b**c* = 10*b* + *c*:

bc | = 10b + c |

= 10(a - 1) + (9 - (a - 1)) | |

= 10a - 10 + 10 - a | |

= 9a, |

as required. Similarly, for a 2-digit *a*:

bc | = 100b + c |

= 100(a - 1) + (99 - (a - 1)) | |

= 100a - 100 + 100 - a | |

= 99a. |

Do try the same derivation for a three digit number. As an example,

543×999 | = 1000×542 + (999 - 542) |

= 999×542 + 999 | |

= 999×543 |

just by using the distributive law twice.

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Copyright © 1996-2018 Alexander Bogomolny

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