Non-integer base

A non-integer base is a base whose radix is not an integer. Such bases typically fail to represent all rationals without ending up with an irrational expansion, but can represent some rationals (mostly integers). The digit set of a non-integer base is the ceil function of the radix, or the radix rounded up. For example, the number 16 in base 3.5 is 110.0300221..., but the number 8 in base 3.5 is exactly 21. However, some

Terminating non-integer bases
Some non-integer bases can represent all rational numbers. These bases are the x values of equations of the form x + ax^-1 + bx^-2 + cx^-3... = y,  where a, b, c, d..., and y are integers. This is because, given the equation, the integer y represented in the base x will always be the terminating decimal 10.abcd...x. This also applies to y's powers, and sums of y's powers (the rational numbers), thus making y a catalyst for terminating decimal expansions. An example of this is the golden ratio base, base 1+sqrt(5)/2, the solution to x + x^-2 = 2, which can represent all integers with terminating decimlals containing 1s and 0s, and all fractions with repeating or terminating decimals.