These are good points. To expand a bit for even further nomenclature: > All algebraic numbers and some transcendental numbers are computable. Thus being irrati

These are good points. To expand a bit for even further nomenclature: > All algebraic numbers and some transcendental numbers are computable. Thus being irrational does not imply not being computable. Yes, there is a distinction to be made about set computability and element computability. The real numbers (rationals and irrationals) are not computable because they are uncountable. Hence floating point numbers only approximate the reals, but they are not real representations themselves. But individual real numbers are frequently computable in a finite number of steps, including irrationals. For example, any given digit of Pi or the sqrt(2) is computable in a finite number of steps. But Chaitan's constant is an uncomputable real number. Likewise, transcendental functions - such as logarithms - are not generally computable as a set even though individual instances can be computed. This is why an inequality including an expression with logarithmic exponents might be reducible in Mathematica, but when you set those exponents to be the ceiling or floor functions of those logarithms it suddenly becomes insoluble. > You can’t find a general, algebraic formula to solve polynomial equations of degree n where n > 4. Yes, and the terminology for something with no general algebraic formula is that there's no closed form solution for it. Mathematica's documentation specifically states it can solve any expression with exponents of 4 of less, but it cannot do so with exponents of greater than 4.