The most fundamental constants in mathematics are zero and one. The transcendentals are called such because they can't be expressed in terms of algebraic operat

The most fundamental constants in mathematics are zero and one. The transcendentals are called such because they can't be expressed in terms of algebraic operations using these two constants as starting point. Changing the bases of a number system doesn't change this fact. However, we can have alternative, more powerful formal systems that are able to capture well-known transcendental numbers in a finite amount of notation. The most powerful class of such systems are Turing-complete systems. However, even such systems won't capture all transcendentals, since there exists a subset of transcendentals called uncomputable numbers. Uncomputable numbers outnumber computable ones. More powerful formalisms also have the downsides that they lose some "nice" properties such as ability to always prove equality of two numbers, as they gain power. Algebraic numbers and especially rational numbers are super nice and well-behaved in comparison.