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Home | 18.013A | Chapter 1 | Section 1.2 |
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Prove that rational numbers repeat the same finite sequence of digits endlessly, and that irrational numbers do not.
Solution:
When you divide by there are only possible remainders.
This means that after at most steps in your long division the remainder must repeat itself.
Since and are integers, once your division gets past the decimal point, you always bring down a 0 in your division procedure.
If you always bring down a 0, the result of your division beyon any point depends only on the remainder at that point.
Thus once a remainder is repeated past the decimal point, the sequence of remainders between the repeat will continue repeating endlessly.
For example,
is .909090 ... or .(09)*.
is .142857142857142857 ... or .(142857)*.
If the decimal expansion of a number repeats endlessly, as for some sequence that is digits long, we can write
which is a rational number. If you have problems following this manipulation do not worry. We will talk about it later.
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