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0.13131313 is a rational number, as it's a recurring number, and all recurring numbers are rational. To prove this, you first multiply by 10 until the decimal part of the number is the same. In this case, 0.13131313...x100=13.13131313....
Then you take off the original number 0.131313...x100 - 0.131313... = 13
Now you can simplify the left hand side 0.131313...x99=13
And divide both sides by 99, to get 0.131313....=13/99
So the original number can be expressed as 13/99, proving it's rational.
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∙ 12y ago
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