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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 5-6, Math Kangaroo 7-8.

Rational numbers can be written in decimal form simply by performing long division, like in this example:

The long division above terminates, however, other long divisions do not.

Rational numbers generate decimal numbers of three kinds:

Name | Definition | Example | How to Generate |

terminating | they have a finite number of decimals | 0.12234995 | the denominator must have only factors of 2 and/or 5 |

non-terminating | they have a number of decimals that repeat forever | 0.591591591... | the denominator must not have any factors of 2 or 5 |

mixed | they have a non-repeating part and a repeating part | 0.4427591591... | the denominator must have factors of 2 and/or 5 as well as other factors |

Fact: Since 2 and 5 "divide" 1, fractions that have only factors of 2 and/or 5 at the denominator will always terminate:

Example:

Fact: Any non-terminating decimal resulting from dividing two integers must repeat.

Proof: The division by any non-zero natural number n generates only n possible remainders. As soon as one of these remainders comes up again, the whole sequence of remainders starts to repeat itself.

Fact: The non-repeating part of a decimal has as many digits as the largest of the powers of 2 and/or of 5 that occur in the prime factoring of the denominator.

Fact: No repeating decimal can have its repeating part equal to 9.

Proof:

Fact: Pretty obvious fact, considering the above: