Keep in mind that fractions of rational numbers and decimal repeats are all considered rational numbers. So, ⅓ = 0.33333.... is a rational number.

If you chose

base 12, then ⅓ no longer gives a repeat decimal, but rather is 0.4 (still rational).

The problem with an irrational base is that many irrational numbers are not whole factors of other irrational numbers.

So, if one had base pi, then [tex]\sqrt{2}[/tex] is still irrational.

If one used base [tex]\sqrt{2}[/tex], then [tex]\sqrt{3}[/tex], as well as [tex]\sqrt[3]{2}[/tex] would still be irrational.