# Proof that recurring number must be rational?

• Jun 10th 2010, 04:00 AM
Nappy
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
• Jun 10th 2010, 04:20 AM
CaptainBlack
Quote:

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

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.