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    @Magus consider the irrational number e (2.71...). The natural log - ln(x) is log base e. Thus, irrational bases do exist and are useful. The particular usefulness of base pi, I'm not sure of - but that doesn't mean it isn't used somewhere. Commented Apr 25, 2014 at 15:53
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    @Max you are straying more and more into math questions. You may find If a number is irrational in base 10, is it irrational in other bases? to be a useful read and a starting point for more number theory questions. Commented Apr 25, 2014 at 15:59
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    1/3 isn't irrational. Commented Apr 25, 2014 at 16:00
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    The OP asked about base 10 (ten). Making a number system base of anything will allow you to express anything as 10. Based on the Wikipedia article, using an irrational number as a base does not make it rational. Rational numbers can be expressed as integers for both the numerator and denominator, repeating numbers in a decimal, or finite termination of numbers in a decimal. Commented Apr 25, 2014 at 16:39
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    @FrustratedWithFormsDesigner Irrationality has nothing whatsoever to do with bases. Well, that's an overstatement, but it's irrationality that has implications for the number's representation in various bases (e.g. whether it has infinite non-repeating digits), not the other way around. Read the math.se question linked to above: math.stackexchange.com/questions/625473/… Commented Apr 25, 2014 at 16:39