Result is instant because you know the result, this is called a verification. Otherwise on average you need precision to represent 2**n values for each range. In reality the precision needs to gradually increase according to log-scale rules. In even more reality floating points can't hold any rational numbers, just close-enough values (think about 1/3 which is impossible to represent as a non-fraction by a binary-based computer), so most likely you'll skip the value you're looking for unless you know its floating point formula (based on mantissa and exponent) can represent it exactly. High-level decimal libraries expansions can't save you from these problems.

Sure. Pure math is reasoning. Don't confuse that with arithmetics. Your examples are bad because 1 + 1 is the basis for subtraction, multiplication, and division. It's also the first rule in the definition of arithmetics (there exists a value 0, and every possible number x has a successor, hey, let's call that x + 1). So in a "pure math" way only an additive definition is enough, and you develop up from that. So a Riemann integral is also pure math. I really don't get where exactly you draw the line between "math" and "pure math". An EC equation is still pure math. Moving electrons in a circuit according to some principles is not math, it's physics or programming.