Proof that recurring number must be rational?

Let m,n be natural numbers with n >= 1 and 0 < m <= n. Im asked to give a clear and concise argument that shows that the decimal expansion of m/n found by long division is recurring.

Can anyone help me out with this one, came up on a recent past paper.

Cheers

Nappy

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Originally Posted by Nappy

Let m,n be natural numbers with n >= 1 and 0 < m <= n. Im asked to give a clear and concise argument that shows that the decimal expansion of m/n found by long division is recurring.

Can anyone help me out with this one, came up on a recent past paper.

Cheers

Nappy

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If you ignore the decimal point there are only n possible remainders at each stage of the log-division process. Once a remainder is repeated all the subsequent multipliers on to of the long division will repeat as will the remainders. hence eventually the decimal expansion will become recuring.

Note while that is what your question asked for it is not what the title of this thread is asking.

CB